algorithms for uniform distribution of solutions over pareto set, and

VYTAUTAS MAGNUS UNIVERSITY INSTITUTE OF MATHEMATICS AND INFORMATICS

Ingrida Steponavičė

ALGORITHMS FOR UNIFORM DISTRIBUTION OF SOLUTIONS OVER PARETO SET, AND THEIR APPLICATIONS IN RISK MANAGEMENT

Doctoral dissertation Physical Science, Informatics (09P)

Kaunas, 2010

Dissertation prepared in 2005 - 2010 at Vytautas Magnus University Advisor: prof. habil. dr. Antanas Žilinskas (Mathematics and Informatics Institute, physical science, informatics - 09P) Co-advisor: prof. dr. Panos M. Pardalos (University of Florida, physical science, informatics - 09P)

VYTAUTO DIDŽIOJO UNIVERSITETAS MATEMATIKOS IR INFORMATIKOS INSTITUTAS

Ingrida Steponavičė

TOLYGAUS SPRENDINIŲ IŠDĖSTYMO PARETO AIBĖJE ALGORITMAI IR JŲ TAIKYMAI RIZIKOS VALDYME

Daktaro disertacija Fiziniai mokslai, Informatika (09P)

Kaunas, 2010

Disertacija rengta 2005 - 2010 metais Vytauto Didžiojo universitete Mokslinis vadovas: prof. habil. dr. Antanas Žilinskas (Matematikos ir informatikos institutas, fiziniai mokslai, informatika - 09P) Konsultantas: prof. dr. Panos Pardalos (Floridos universitetas, fiziniai mokslai, informatika - 09P)

Acknowledgments I am grateful to many people for supporting me throughout my graduate study. First of all, I would like to express my earnest gratitude to my advisor, Prof. Dr. Antanas ˘ Zilinskas, for directing this study and reading previous drafts of this work. Without his invaluable guidance, inspiration, and support throughout my research, this work would not be complete. This dissertation is more accurate and more clearly expressed because of his input. I also would like to express my gratitude to Prof. Dr. Panos M. Pardalos, for his inspiring and encouraging comments. His technical and editorial advice was essential to the completion of this thesis. Many thanks to my husband Arturas who has always been there for me, listening to me complain and encouraging me to continue. Last but not least, I am forever indebted to my sun Matas, parents, sisters, and brother for their endless love and support.

Notations Abbreviations AbYSS

Archive-based hybrid Scatter Search algorithm

AW

Adjustable weights method

CellDE

Cellular Differential Evolution algorithm

CP

Compromise programming

CVaR

Conditional Value at Risk

DM

A decision-maker

DSD

Directed search domain

EA

Evolutionary algorithm

FastPGA Fast Pareto genetic algorithm GA

Genetic algorithm

GD

Generational distance

GP

Goal programming

HV

Hypervolume

IGD

Inverted generational distance

LGP

Lexicographic goal programming

MADM

A decision-maker

MCDA

Multi-Criteria Decision Aid

MMP

Multi-objective mathematical programming

MO

Multi-objective optimization

MOCell

Multi-Objective Cellular genetic algorithm

MCDM

Multicriteria decision making

MOP

A decision-maker

MR

Mean-reverting

MRJD

Mean-reverting jump-diffusion

NBI

Normal boundary intersection

NC

Normal constraint

NSGA-II Nondominated Sorting Genetic Algorithm II PP

Physical programming

S

Spread

SLGP

Successive lexicographic goal programming

SOP

A decision-maker

SPEA2

Strenght Pareto Evolutionary Algorithm 2

WS

Weighted sum method

VaR

Value at Risk

Content 1. INTRODUCTION 2. MULTI-OBJECTIVE OPTIMIZATION 2.1. BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. PERFORMANCE METRICS . . . . . . . . . . . . . . . . . . 2.2.1. GENERATIONAL DISTANCE . . . . . . . . . . . . . 2.2.2. INVERTED GENERATIONAL DISTANCE . . . . . . 2.2.3. HYPERVOLUME . . . . . . . . . . . . . . . . . . . . . 2.2.4. SPREAD AND GENERALIZED SPREAD . . . . . . . 2.3. SCALARIZATION METHODS . . . . . . . . . . . . . . . . . 2.3.1. WEIGHTED SUM METHOD . . . . . . . . . . . . . . 2.3.2. GOAL PROGRAMMING . . . . . . . . . . . . . . . . 2.4. INTERACTIVE METHODS . . . . . . . . . . . . . . . . . . . 2.5. FUZZY METHODS . . . . . . . . . . . . . . . . . . . . . . . 2.6. METAHEURISTIC ALGORITHMS . . . . . . . . . . . . . . 2.6.1. FastPGA . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2. MOCeLL . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3. AbYSS . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4. NSGA-II . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5. SPEA2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.6. CellDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. DECISION AID METHODS . . . . . . . . . . . . . . . . . . . 2.8. REVIEW OF RELATED WORKS . . . . . . . . . . . . . . . 2.8.1. NORMAL-BOUNDARY INTERSECTION METHOD 2.8.2. NORMAL CONSTRAINT METHOD . . . . . . . . . 2.8.3. PHYSICAL PROGRAMMING METHOD . . . . . . . 2.8.4. OTHER METHODS . . . . . . . . . . . . . . . . . . . 2.9. APPLICATIONS IN RISK MANAGEMENT . . . . . . . . . 2.9.1. OPTIMAL PORTFOLIO SELECTION . . . . . . . . . 2.9.2. PRICING TOLLING AGREEMENTS . . . . . . . . .

13

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

3. ALGORITHMS FOR UNIFORM PARETO SET APPROXIMATION 3.1. CONCEPT OF UNIFORM DISTRIBUTION . . . . . . . . . . . . . . . . . 3.2. ALGORITHM OF ADJUSTABLE WEIGHTS . . . . . . . . . . . . . . . . . 3.3. ALGORITHM OF SUCCESSIVE LEXICOGRAPHIC GOAL PROGRAMMING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. EXPERIMENTS DESCRIPTION . . . . . . . . . . . . . . . . . . . . . . . .

17 17 20 21 21 22 23 24 24 26 27 28 29 30 31 32 32 33 33 34 35 35 36 36 37 39 39 41 44 44 46 48 52

3.4.1. TEST PROBLEMS . . . . . . 3.4.2. ALGORITHMS SETUP . . . 3.4.3. TESTS RESULTS . . . . . . 3.4.4. QUALITY OF PARETO SET 3.5. CHAPTER CONCLUSIONS . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPROXIMATIONS . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

4. OPTIMAL PORTFOLIO SELECTION 4.1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. RISK MEASURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. TRANSACTION COSTS . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. PROBLEM FORMULATION . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. SINGLE PERIOD PORTFOLIO OPTIMIZATION . . . . . . . 4.4.2. MULTI-STAGE STOCHASTIC PORTFOLIO OPTIMIZATION 4.5. DISCUSSION ON EXPERIMENTAL RESULTS . . . . . . . . . . . . 4.5.1. CASE 1: DETERMINISTIC PORTFOLIO OPTIMIZATION . 4.5.1.1. TWO CRITERIA PROBLEM . . . . . . . . . . . . . . 4.5.1.2. THREE CRITERIA PROBLEM . . . . . . . . . . . . 4.5.2. CASE 2: STOCHASTIC PORTFOLIO OPTIMIZATION . . . . 4.5.2.1. DATA AND SAMPLE PATHS GENERATION . . . . 4.5.2.2. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. CHAPTER CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . 5. PRICING TOLLING AGREEMENTS 5.1. PROBLEM FORMULATION . . . . . . . 5.1.1. SINGLE OBJECTIVE . . . . . . . 5.1.2. BI-OBJECTIVE FORMULATION 5.2. EXPERIMENTAL RESULTS . . . . . . . 5.2.1. SCENARIOS GENERATION . . . 5.2.2. CASE STUDY . . . . . . . . . . . 5.3. CHAPTER CONCLUSIONS . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . .

. . . . . . . . . . . . . .

. . . . . . .

. . . . .

. . . . . . . . . . . . . .

. . . . . . .

. . . . .

52 52 54 72 76

. . . . . . . . . . . . . .

77 77 78 82 84 84 85 87 87 87 90 92 92 95 97

. . . . . . .

99 99 99 101 103 103 104 108

6. MAIN RESULTS AND CONCLUSIONS

109

REFERENCES

111

List of tables 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.1 4.2 4.3 4.4 4.5 5.1 5.2 5.3 5.4 5.5 5.6

MOPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimates of inverted generational distance (IGD) metric obtained by WS, AW and SLGP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimates of hypervolume (HV) metric obtained by WS, AW and SLGP . . Estimates of spread (∆) metric obtained by WS, AW and SLGP . . . . . . . Average and standard deviation of generational distance (GD) metric of metaheuristic algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average and standard deviation of inverted generational distance (IGD) metric of metaheuristic algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . Average and standard deviation of hypervolume (HV) metric of metaheuristic algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average and standard deviation of spread (∆) metric of metaheuristic algorithms Estimates of performance metrics obtained with different step size/number of solutions of problem ZDT1 and Viennet2 . . . . . . . . . . . . . . . . . . . .

53

Performance metrics for bi-objective portfolio problem . . . . . . . . . . . . Performance metrics for three criteria portfolio problem; F denotes the number of function evaluations by the meta-heuristic methods . . . . . . . . . . Average daily returns×(10e-4) and sigma×(10e-2) of assets . . . . . . . . . . Proportional trading costs to trading volume . . . . . . . . . . . . . . . . . . Performance metrics for two and three criteria stochastic portfolio problems with different risk level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Parameters used for short-term study . . . . . . . . . . . . . . . . . . . . . . Setup parameters used in short-term study . . . . . . . . . . . . . . . . . . . Prices (in $) and CVaR of 1-year tolling agreement contracts calculated on 80 × 80 grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prices, CVaR0.95 and VaR0.95 of 1-year tolling agreement contracts calculated on 80 × 80 grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The average values of performance metrics of tolling agreement problem, CVaR0.95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prices, CVaR0.99 and VaR0.99 of 1-year tolling agreement contracts . . . . . .

104 104

55 56 56 58 59 60 61 73

92 94 95 96

105 105 106 107

List of figures 2.1 2.2 2.3 2.4

Illustration of Euclidean distance for calculation of GD and Graphical illustration of HV . . . . . . . . . . . . . . . . . Illustration of spread calculation . . . . . . . . . . . . . . . Solution using the weighted sum method . . . . . . . . . .

IGD . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23

Graphical explanation of distribution . . . . . . . . . . . . . . . . . . . . . . Graphical interpretation of simplex partitioning . . . . . . . . . . . . . . . . Pareto set and solutions of problem ZDT1 obtained by WS, AW and SLGP . Solutions of problem ZDT1 obtained by metaheuristic algorithms . . . . . . Pareto set and solutions of problem ZDT2 obtained by SLGP . . . . . . . . Solutions of problem ZDT2 obtained by metaheuristic algorithms . . . . . . Pareto set and solutions of problem ZDT3 obtained by SLGP . . . . . . . . Solutions of problem ZDT3 obtained by metaheuristic algorithms . . . . . . Pareto set and solutions of problem ZDT4 obtained by WS, AW, and SLGP Solutions of problem ZDT4 obtained by metaheuristic algorithms . . . . . . Pareto set and solutions of problem Problem1 obtained by WS, AW, and SLGP Solutions of problem Problem1 obtained by metaheuristic algorithms . . . . Pareto set and solutions of problem Kursawe obtained by SLGP . . . . . . . Solutions of problem Kursawe obtained by metaheuristic algorithms . . . . . Pareto set and solutions of problem Fonseca obtained by SLGP . . . . . . . Solutions of problem Fonseca obtained by metaheuristic algorithms . . . . . Pareto set and solutions of problem Tanaka obtained by SLGP . . . . . . . . Solutions of problem Tanaka obtained by metaheuristic algorithms . . . . . . Solutions of problem Viennet2 obtained by WS, AW, and SLGP . . . . . . . Solutions of problem Viennet2 obtained by metaheuristic algorithms . . . . . Pareto set and solutions of problem Viennet4 obtained by WS, AW, and SLGP Solutions of problem Viennet4 obtained by metaheuristic algorithms . . . . . Distribution of Pareto subsets composed by different number of ZDT1 solutions obtained by WS, AW, and SLGP . . . . . . . . . . . . . . . . . . . . . 3.24 Distribution of Pareto subsets composed by different number of Viennet2 solutions obtained by WS, AW, and SLGP . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5

Graphical interpretation of CVaR for given x ∈ X . . . . . . . . . . . . . . . Investment horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solutions in Pareto set generated by the WS method and AW algorithm . . . Solutions found by all algorithms . . . . . . . . . . . . . . . . . . . . . . . . Solutions of bi-objective portfolio problem generated by WS, AW, and SLGP

21 23 23 25 45 47 62 62 63 63 64 64 65 65 66 66 67 67 68 68 69 69 70 70 71 71 74 75 81 86 88 89 90

4.6

4.9 4.10 4.11 4.12

Solutions of bi-objective portfolio problem generated by meta-heuristic algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pareto set and solutions of three-objective portfolio problem generated by WS, AW, and SLGP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solutions of three-objective portfolio problem generated by meta-heuristic algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scenarios of HYB prices for two years . . . . . . . . . . . . . . . . . . . . . . Case 1. Pareto solutions obtained with different risk level α = 0.9, 0.95, 0.99 Case 3: Solutions obtained by SLGP with different risk level α = 0.9, 0.95, 0.99 Pareto frontiers with risk level α = 0.95 . . . . . . . . . . . . . . . . . . . .

5.1 5.2 5.3 5.4 5.5 5.6 5.7

Interpolation on a grid on a logarithmic plane . . . . . . . . . . . . . Optimal power plant operation on 80 × 80 grid in (lnG, lnP ) plane . CVaR of revenue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electricity and gas prices, $ for 1-year . . . . . . . . . . . . . . . . . . Solutions generated by WS, AW and SLGP algorithms . . . . . . . . Average of maximum CVaR values ($) with different confidence levels ¯ = 8.5 . . . . . . . . Sensitivity to different confidence levels, when H

4.7 4.8

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

91 93 94 95 96 97 97 101 102 102 104 106 107 107

chapter 1 INTRODUCTION Multi-objective optimization (MO) is very important area that have made significant contributions to engineering design theory and practice, because many engineering design problems are multi-objective and often require trade-offs among conflicting objectives. Generating Pareto optimal solutions is main issue in multi-objective optimization. In continuous problems, the number of Pareto optimal solutions is usually infinite and only in relatively simple cases the entire Pareto optimal set can be determined analytically. Thought financial risk has increased significantly in recent years, measuring and controlling risk is one of the major concern across all modern human activities. The financial and electricity markets that are very sensitive to economical and political changes are no exception. It is crucial to identify and manage the financial risk, because in global world information is available instantaneously, which means that markets can be affected very quickly by changes in exchange rates, interest rates and commodity prices. Therefore, risk management is a very important topic for researchers and practitioners. The modelling of financial risks and investor preferences as well as the solution of large-scale portfolio optimization problems are challenging tasks [Kuhn et al., 2008].

Relevance of the Problem Engineering design optimization and many other practical optimization applications are generally multi-objective. In real industrial design the decision-maker (DM) is able to consider only a few possible solutions. Also, sometimes the cost of generating Pareto points may become so high that the designer can afford only a few Pareto optimal solutions. Therefore, DM must be satisfied by obtaining enough Pareto optima to cover the minimal set in the criteria space properly. In such a context, it is crucial to have an uniform distribution of representative Pareto set to obtain maximum information on the whole Pareto surface at minimum computational cost. Here, the term ‘uniform distribution’ corresponds to the lo-

14

CHAPTER 1. INTRODUCTION

cation of centers of spheres of optimal covering of the Pareto optimal solution set by spheres of equal radius. There are developed several methods to yield a well-distributed subset of Pareto solutions. Nevertheless, there is some space for new approaches, which can better deal with some of the difficulties encountered by the existing approaches.

The Object of Research There is a need for methods generating uniformly distributed Pareto optimal solutions of multi-objective optimization problems that are computationally efficient and simple to implement. Uniform distribution is advantageous since it minimizes the size of not researched holes in the Pareto set, in other words it minimizes guaranteed distance between a potentially favorable solution and one of generated points. It is well known, however, that the uniform distribution of points is a difficult problem even for explicitly defined simple sets. Moreover, such distributions are not composite, i.e. the increase of the number of points by one implies complete relocation of points of the previous distribution.

The Aim and Tasks of the Thesis The purpose of this research is to develop algorithms obtaining uniformly distributed solutions approximation in Pareto set and applicable to multi-objective optimization problems arising in risk management area. Specific tasks of this research are: 1. Propose algorithms to obtain well-distributed solutions in Pareto set. 2. Develop algorithmic implementations of the new solution methodology. Test the implementations on a range of appropriate test problems and evaluate efficiency: i) Test the algorithm on a set of test problems, ii) Compare the solution quality and the efficiency to the obtained results of other solution methods according to several performance measures, and iii) Examine the impact of increasing number of generated solutions on uniform Pareto subset approximation of the new algorithms. 3. Formulate multi-criteria problems in risk management, namely optimal portfolio selection and pricing tolling agreements problems, and apply developed algorithms.

Scientific Novelty and Practical Significance Two methods for generating solutions uniformly distributed in the Pareto set are prosed. By the first method Pareto solutions are found minimizing weighted sums of objectives where weights are properly selected using branch and bound approach. The second method

15 solves a sequence of lexicographic goal programming problems with different reference points producing different solutions. Performed test of multi-criteria problems proved that proposed algorithms possess desired characteristics and outperform other algorithms according to selected performance metrics. It is explored the convergence of Pareto approximation distribution obtained by developed algorithms regarding the increasing number of solutions. The problem of pricing tolling agreement was formulated as bi-criteria problem including into objective function CVaR as a risk measure.

The Defended Statements 1. Two promising approaches, namely, adjustable weights method and sequential LGP method, were proposed for uniformly distributed optimal solutions in Pareto set generation. 2. These algorithms were applied to solve bi-criteria and three-criteria single and multiperiod optimal portfolio selection and pricing tolling agreement problems and proved their suitability. 3. The comparison of developed methods with traditional WS method and heuristic algorithms demonstrated their advantages and superiority regarding to performance measures. 4. The bi-criteria problem formulation of pricing tolling agreements was proposed and the sensitivity analysis of profit to confidence level was performed.

Approbation and Publications of the Research The main results of the research were presented and discussed at 4 conferences and published in 6 scientific publications. In addition, other two publications related with the thesis were published as well. Presentations at conferences: 1. “Electricity Financial Instruments”. INFORMS Annual Meeting, San Diego, California, US, October 2009. 2. “Evolutionary methods for multi-objective portfolio optimization”, World Congress on Engineering WCE 2008, Imperial College London, London, UK, July 2008. 3. “Multi-objective portfolio optimization”, Informatics summer school Modern Data Mining, Druskininkai, Lithuania, September 9-15 2007.

16

CHAPTER 1. INTRODUCTION

4. “Optimization methods application in portfolio selection”, Lithuanian young scientists conference Operational Research and Application 2007, Vilnius, Lithuania, May 18 2007. Publications in international periodicals: 1. P.M. Pardalos, I. Radziukyniene and A. Žilinskas. Adjustable Weights and Successive Lexicographic Goal Programming Methods for Pareto Set Generation. Optimization Letters, ISSN: 18624472, 18624480. (accepted) 2. I. Radziukyniene, A. Žilinskas. Comparison of Several Methods for Pareto Set Generation in Multi-Criteria Portfolio Optimization. International Journal of Computing, vol 7, Nr. 3, 2008, p. 22-29. ISSN 1727-6209. 3. I. Radziukyniene and A. Žilinskas. Approximation of Pareto Set in Multi Objective Portfolio Optimization. In Advances in Electrical Engineering and Computational Science. Series: Lecture Notes in Electrical Engineering, Vol. 39. Sio-Iong and Gelman (Eds.) 2009, p. 551-562. ISSN: 1876-1100. 4. V. Radziukynas and I. Radziukyniene. Optimization Methods Application to Optimal Power Flow in Electric Power Systems. In J. Kallrath et all (Eds.) Optimization in the Energy Industry. Energy Systems 2009, III, 409-436, ISSN: 1867-8998. Publications in reviewed proceedings of international conferences: 1. I. Radziukyniene, A. Žilinskas. Evolutionary Methods for Multi-Objective Portfolio Optimization. Proceedings of the 2008 International Conference of Financial Engineering (ICFE-08), London, UK, 2-4 July, 2008, p. 1155-1159, ISBN (Vol I): 978-98898671-9-5. 2. I. Radziukyniene, A. Žilinskas. On Pareto Set Generation in Multi-criteria Portfolio Optimization. Proceedings of the Fifth International Conference on Neural Networks and Artificial Intelligence May 27-30, 2008, Minsk, Belarus. Other reviewed publications on the topic of the thesis: 1. I. Radziukyniene, P. Xanthopoulos and P.M. Pardalos. Combining forecasts. Wiley Encyclopedia of Operations Research and Management Science, 2010. ISBN: 9780470400531. 2. I. Radziukyniene, N. Boyko, and P.M. Pardalos. Model-Based Forecasting. Wiley Encyclopedia of Operations Research and Management Science, 2010. ISBN: 9780470400531.

chapter 2 MULTI-OBJECTIVE OPTIMIZATION 2.1.

BACKGROUND

Multiple-objective optimization problems have received increased interest from researchers with various backgrounds since early 1960. Multi-objective optimization (MO) is very important area that have made significant contributions to engineering design theory and practice, because many engineering design problems are multiobjective and often require trade-offs among conflicting objectives. In the case of multiple-objectives, there does not necessarily exist a solution that is best with respect to all objectives because of incomparability and conflict among objectives [Lin and Gen, 2007]. Multi-objective optimization is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints. The general multi-objective optimization problem is posed as follows: Minimize F(x) = [F1 (x), F2 (x), . . . , Fk (x)]T x

Subject to gj (x) ≤ 0, hl (x) = 0,

j = 1, 2, . . . , m,

(2.1)

l = 1, 2, . . . , e,

where k is the number of objective functions, m is the number of inequality constraints, and e is the number of equality constraints. x ∈ E n is a vector of decision variables, where n is the number of independent variables xi . F(x) ∈ E k is a vector of objective functions Fi (x) : E n − → E 1. In a multi-objective context, the idea of this optimization process is that the result is shown as a set of optimal solutions (optimal Pareto frontier) for the different objective functions jointly [Donoso and Fabregat, 2007]. The goal of MOP is to identify a particular

18

CHAPTER 2. MULTI-OBJECTIVE OPTIMIZATION

class of solutions, i.e. the set of which comprises the Pareto frontier. A solution point is Pareto optimal if it is not possible to move from that point and improve at least one objective function without worsening any other objective function. We will provide some definitions of Pareto optimality in multiobjective framework using Pareto dominance relation as given in [Ryu et al., 2009]: Definition 1 If xu , xv ∈ X is two decision vectors, then F (xu ) is said to dominate F (xv ) (denoted F (xu ) ≺ F (xv )) if and only if fi (xu ) ≤ fi (xv ), ∀i ∈ {1, 2, ...k} and fj (xu ) < fj (xv ), ∃j ∈ {1, 2, ...k}. Definition 2 A point x∗ ∈ X is said to be globally Pareto optimal if and only if no x ∈ X such that F (x) ≺ F (x∗ ). Then F (x∗ ) is called globally efficient. The image of the set of globally efficient points is called the Pareto front. Definition 3 A point x∗ ∈ X is said to be locally Pareto optimal if and only if there exists an open neighborhood of x∗ , B(x∗ ), such that there is no x ∈ B(x∗ )∩X satisfying F (x) ≺ F (x∗ ). Then F (x∗ ) is called locally efficient. The image of the set of locally efficient points is called the local Pareto front. Generating Pareto optimal solutions is main issue in multiobjective optimization. In continuous problems, the number of Pareto optimal solutions is usually infinite and only in relatively simple cases the entire Pareto optimal set can be determined analytically [Grosan and Abraham, 2008]. The overall goal is to find well-converged and well-diversified set of Pareto optimal solutions. Therefore, for Pareto optimal set generation, it is desirable for method to posses following characteristic [Messac et al., 2003]: 1. It should generate a uniformly distributed set of Pareto points in criteria space and do not skip any region. This feature ensures that all regions of feasible solution space are adequately represented in the generated sampling. 2. It should be able generate all available Pareto solution. This trait is of critical importance, because the method incapable to generate desired Pareto optimal solution results in a fundamental failure. 3. It should generate only Pareto optimal solutions. A non-Pareto solution means that a better solution exists and trade-off is not reached. There are reported many methods for solving Multiobjective Optimization Problem (MOP) that convert it into a Single Objective Optimization Problem (SOP), a sequence of SOPs, or into another MOP. The aggregation of objectives into a single, parameterized objective function is a simple and widely used approach [Grosan and Abraham, 2010], where

2.1. BACKGROUND

19

an approximation of the Pareto-optimal set is obtained by means of the optimization of the aggregate objective function with different settings of parameters. However, in many cases decision can not be made without some information about the whole Pareto set. The theoretical problem to find the whole Pareto set normally (e.g. in case of continuum cardinality of Pareto set) can not be solved algorithmically, thus the theoretical problem of finding of whole Pareto set should be reformulated to algorithmic construction of an appropriate its approximation. Weight-based approaches are very difficult to implement in practice for realistic problems that comprise many weights. These methods typically require many iterations on the choice of weights; and often provide no clear guidance how to converge to the right set of weights [Messac, 1996]. Preemptive or priori approaches prioritize the various competing objectives and suffer from two major problems. First, it is often very unrealistic to declare one objective infinitely more important than another. In the real world, preferences take the form of degrees. Second, there are n! ways of ordering n objectives/criteria/constraints. Only a small number of possibilities can therefore be explored. Moreover, it may also happen that all the solutions produced by a preemptive method (an ordering of the criteria) are unacceptable to the designer. It can easily be shown that a solution obtained by an Archimedian method is unobtainable by preemptive methods. For those reasons, weight-based methods have enjoyed significantly more popularity than the preemptive methods in the engineering design community. In a general sense, if one is not using a preemptive method, then one is using a weight-based method that requires some choice of weight from the part of the designer, implicitly or explicitly. Over the past decades evolutionary algorithms have received much attention owing to its intrinsic ability to handle optimization problems with both single and multiple objectives including problems of financial optimization [Li and Taiwo, 2006, Chang et al., 2000, Wang et al., 2006, Ehrgott et al., 2004, Xia et al., 2001, Mukerjee et al., 2002, Lin and Gen, 2008]. Let us note that financial optimization problems were attacked also by means of other metaheuristics, e.g. by simulated annealing in [Chang et al., 2000], and by Tabu search in [Stummer and Sun, 2005]. Comparisons of the performance of different heuristic techniques applied to solve one criterion portfolio choice problems are given by Chang et al. [Chang et al., 2000]. However, to the best knowledge of the authors similar comparative analysis of performance of recently proposed evolutionary multicriteria algorithms has not been yet reported. MO methods can be classified into three families: a priori methods, progressive methods, and a posteriori methods as given in [Miettinen, 1999]. In a priori methods, the DM has to define the trade-off to be applied (preferences) before running the optimization method. In progressive methods, the decision maker provides more information about his preferences as the method progresses. And finally, in a posteriori methods, the DM chooses a solution

20


after examining all those obtained by the optimization method [Miettinen, 1999]. However, the same method can depend not only to one family. The other way to categorize and describe the methods is to divide them into five sets: scalarization methods, interactive methods, fuzzy methods, meta-heuristic methods, and decision aid methods. Again, not all methods can be fitted neatly into a single set. Below, we will provide short description of each set. For additional information on multi-objective solution methods we refer to [Collette and Siarry, 2004, Marler and Arora, 2004, Andersson, 2000]. This chapter is organized as follows. Some of important metrics of performance are reviewed in Section 2.2.. Section 2.3. gives a short overview of scalarization methods applied in multi-objective optimization and their characteristics. The idea of interactive methods is given in Section 2.4., while the fuzzy methods are discussed in Section 2.5.. Description of meta-heuristic methods and their merits and demerits are outlined in Section 2.6.. Finally, this chapter is completed with an introduction of decision aid methods and conclusions.

2.2.

PERFORMANCE METRICS

In the last years many multiobjective optimizations algorithms have been developed. Comparing different optimization techniques experimentally always involves the notion of performance. The concept of a performance metric is usually used to form an understanding of which algorithm is the better and in what aspect [Bui et al., 2004]. In the case of multiobjective optimization, the definition of quality is substantially more complex than for single-objective optimization problems, because the optimization goal itself consists of multiple objectives [Zitzler et al., 2000]: • The distance of the resulting nondominated set to the Pareto-optimal front should be minimized. • A uniform distribution of the solutions found is desirable. The assessment of this criterion might be based on a certain distance metric. • The extent of the obtained nondominated front should be maximized, i.e., for each objective, a wide range of values should be covered by the nondominated solutions. However, it is hard to define a concise definition of algorithmic performance in multiobjective optimization and it is impossible to find a single metric for all criteria. It has been shown [Zitzler, 1999] that for an M -objective optimization problem, at least M performance metrics must be used. There are some publications on experimental comparison of different performance aspects of evolutionary methods. For example, in the study presented in ˘ ˘ [Zilinskas and Zilinskas, 2002] some general conclusions are drawn, however they could not

∑ Δ

=

+

=

∑ −

= =

∈

∑

∈

Figure 4. Illustration of Euclidean distance for calculati

2.1 Fig.: Illustration of Euclidean distance for calculation of GD and IGD be directly applied to a specific problem. Zitzler et al. [Zitzler et al., 2000] use criteria as an average distance to the Pareto-optimal set, the spread of obtained solutions and the number of solutions in the set. In this section we will list metrics applied here.

GENERATIONAL DISTANCE

Generational distance (GD) was introduced by Van Veldhuizen and Lamont [Veldhuizen and Lamont, 1998] to measure how far the elements are in the set of non-dominated vectors found from those in the Pareto optimal set and it is defined as: ∑n GD =

i=1

n

d2i

,

(2.2)

where n is the number of vectors in the set of nondominated solutions found so far and di is the Euclidean distance between each of these solutions and the nearest member of the Pareto optimal set as it is shown in Figure 2.1. In order to obtain reliable results, non-dominated sets are normalized before calculating this distance measure.

2.2.2.

+

21

2.2. PERFORMANCE METRICS

2.2.1.

∑

∈

INVERTED GENERATIONAL DISTANCE

Another metric giving information about closeness and diversity at the same time is inverted generational distance (IGD) [Villalobos-Arias et al., 2005, Nebro et al., 2006b]. This quality indicator is used to measure how far the elements are in the Pareto optimal set from those in the set of non-dominated vectors found. For IGD calculation, for every solution in real Pareto optimal solutions, the closest point in set of found solutions is located. It is defined as follows v u n u∑ IGD = (t d2i )/n, (2.3) i=1

≠

−

22


where n is equal to number of solutions consisting true Pareto front and di is the Euclidean distance between each vector of true Pareto frontier and the nearest member of a generated Pareto frontier. If a part of the Pareto front is missing or very poorly approximated by found solutions, the value of IGD grows. Let us assume two obtained set of solutions of a given problem: one only has two points, and the other has the two dominating points in the first set and many other non-dominated points. It is clear that from the multi-objective perspective that the second set is preferable to the first one. However, GD cannot differentiate the two sets because it only measure the distances from the obtained solutions to the true Pareto set. Meanwhile IGD can tell that the second one is better because the distances are measured from true Pareto set to the obtained solutions. Obviously, to reliably compute either GD or IGD, a sufficiently dense set of real Pareto optimal points is needed [Aittokoski and Miettinen, 2008]. Thus, these metrics are usable only with problems where the exact Pareto front is known.

2.2.3.

HYPERVOLUME

Recently, a hypervolume (HV) indicator also known as an S-metric [Zitzler and Thiele, 1998] has gained popularity. This quality indicator calculates the volume (in the objective space) covered by members of a non-dominated set of solutions [Zitzler et al., 2000], and as such it can give information about both closeness and diversity at the same time. In addition, it possesses a desirable property, i.e., whenever one approximation completely dominates another approximation, the hypervolume of the former will be greater than the hypervolume of the latter [Zitzler et al., 2007]. The goal of this measure is to identify the proportion of the volume enclosed by reference point and Pareto optimal front covered by the nondominated solutions obtained at the end of the search. The dominated hypervolume corresponds to the size of the region of the objective space (bounded by a reference point) which contains solutions being weakly dominated by at least one of the members of the set [Beume and Rudolph, 2006]. The Figure 2.2 presents a non-dominated set of nine 3-dimensional points. The dominated volume is bounded by the reference point r. Along each d-th coordinate, the d-dimensional space is cut into (d1) -dimensional slices that are stored in the orthogonal partition tree. The ddimensional volume is calculated by computing the (d1)-dimensional volume with the help of the orthogonal partition tree and sweeping along the slices in dimension d. Each member y of a set M weakly dominates a region in the objective space shaped like an infinite hypercuboid h(y) = [y1 , inf] × · · · × [yd , inf] (in case the domain is infinite). These hypercuboids become finite by bounding them with a reference point r, which has to be dominated by each member of the set M : h(y) = [y1 , r1 ] × · · · × [yd , rd ]. The S-metric is the hypervolume of the union

23

2.2. PERFORMANCE METRICS

f2

r

a g b c

e

f3

i f

h

d

f1

2.2 Fig.: Graphical illustration of HV

⎛ ⎜U ⎜= ⎝

=

Figure 3. Illustration of variables used in calculating sprea

2.3 Fig.: Illustration of spread calculation of the weakly dominated hypercuboids, whereas severally covered regions are counted once. Methods with larger values of HV are desirable. The formal definition of S-metric is based = on the Lebesgue measure Λ: = HV (M, r) = λ(∪h(y)|y ∈ M )

2.2.4.

(2.4)

SPREAD AND GENERALIZED SPREAD

The spread or ∆ metric calculation method is as suggested by Deb [Deb, 2001]. To quantify the spread, Euclidean distance between any two neighbor solutions in non-dominated solution set is calculated and then the average of these distances is obtained. Then the extreme points of the true Pareto frontier are found and Euclidean distance between these∑ Δ = = points and the boundary solutions of the obtained Pareto solution is calculated. Figure 2.3 demonstrates how this metric is calculated. The spread metric calculates the distance between two consecutive solutions, and only works for 2-objective problems. = =

+

∑ −

∈

∑

≠

∑

∈

24


A generalized metric was proposed in [Zhou et al., 2006] to calculate the distance from a point to its nearest neighbor in more than two objective case. The nonuniformity in the distribution, ∆, is calculated as follows: ∑m ∗

∆(S, S ) =

i=1

∑ ¯ d(ei , S) + X∈S ∗ |d(X, S) − d| ∑m , ∗ ¯ i=1 d(ei , S) + |S |d

(2.5)

where {e1 , ..., em } are m extreme solutions in S ∗ and d(X, S) =

min

Y ∈S,Y ̸=X

||F (X) − F (Y )||2 ,

1 ∑ d¯ = ∗ d(X, S). |S | X∈S ∗ Here, S is a set of solutions and S ∗ is a set of known Pareto-optimal solutions. Uniform spread of the solutions means that we have made a good exploration of the search space and no regions are left unexplored. The preferable value of this metrics is equal to 0, i.e. it means that the achieved solutions are well distributed and include those extreme solutions.

2.3.

SCALARIZATION METHODS

Scalarization methods attempt to transform the multiobjective problem into a single, parameterized objective function, so that standard optimization techniques can be applied. In order to generate the Pareto optimal set, the parameters of this function are systematically varied by the optimizer. There exists methods that can be used to solve multi-objective problems using single-objective approximations. They include weighted sum, goal programming, ϵ-constraint, weighted metrics, Benson, lexicographic, and min-max, among others. All of these methods try to find the optimal Pareto front using different approximation techniques.

2.3.1.

WEIGHTED SUM METHOD

The most widely used method for MOP is the weighted sum (WS) method. It transforms multiple objectives into an aggregated scalar objective function by multiplying each of m objective functions by a weighting factor and summing up all contributors [Kim and De Weck, 2006]. The mathematical model of the weighted sum method takes the form of: minf (x), f (x) =

m ∑

ωi fi (x),

(2.6)

i=1

where ωi is the weight of i-th criterion, 0 ≤ ωi ≤ 1, i = 1, . . . , m, and

∑m i=1

ωi = 1.

2.3. SCALARIZATION METHODS

25

2.4 Fig.: Solution using the weighted sum method

This method finds the points of the optimal Pareto front (if set of solutions is convex), which consists of all the optimal solutions in the multi-objective optimization, through the combinations given by the weighted vector ω = {ω1 , ω2 , . . . , ωm }. It is very simple to implement and continues to be used extensively not only to provide multiple solution points by varying the weights consistently, but also to provide a single solution point that reflects preferences presumably incorporated in the selection of a single set of weights [Marler and Arora, 2010]. The weakness of this approach is varying the weights consistently and continuously may not necessarily result in an even distribution of Pareto optimal points and an accurate, complete representation of the Pareto optimal set. This was proved by Das and Dennis [Das and Dennis, 1997]. Also, the spread of the points strongly depends on the relative scaling of the objectives. When the ranges of the functions have different magnitudes, and especially when such differences are very large, one having the highest range values will predominate in the result. Moreover, the WS method is that not all of the Pareto optimal solution can be found unless the problem is convex. Otherwise, only solutions located on the convex part of the Pareto frontier can be identified. In Figure 2.4, there are points of sets that cannot be represented by a linear combination, and therefore, such points cannot be found by means of this method [Miettinen, 1999, Donoso and Fabregat, 2007]. This feature can be relaxed to some extent by convexifying the non-convex Pareto optimal set by raising the objective functions to a high enough power under certain assumptions [Miettinen, 1999]. [Messac et al., 2000, Messac and Ismail-Yahaya, 2001] provide the necessary and sufficient conditions for an AOF for capturing any Pareto point. Finally, varying the weights consistently and continuously may not necessarily result in an even distribution of Pareto optimal points and an accurate, complete representation of the Pareto optimal set. It was reported that compromise programming (CP) method overcomes some of the

26


drawbacks of the WS method, namely it is able to produce solutions on the non-convex regions of the Pareto frontier [Chen et al., 1999]. In [Messac and Ismail-Yahaya, 2001], it was proven the existence of a relationship between the order of the AOF and that of the Pareto frontier for the CP method to successfully generate Pareto solutions. However, both WS and CP as well as most methods still shares some deficiencies, i.e. they fails to generate a set of well-distributed solutions for a corresponding even distribution of weights. In order to overcome the last two flaws, we proposed a method of adjustable weights that is discussed in Section 3.2..

2.3.2.

GOAL PROGRAMMING

Goal Programming (GP) is an important technique for decision making problems where the decision maker aims to minimize the deviation between the achievement of goals and their aspiration levels. GP was first used by Charnes, Cooper and Ferguson in 1955 [Charnes et al., 1955], although the actual name first appear in 1961 [Charnes and Cooper, 1961]. It can be said that GP has been, and still is, the most widely used multi-objective technique in management science because of its inherent flexibility in handling decision-making problems with several conflicting objectives and incomplete or imprecise information [Romero, 2004, Chang, 2007]. The examples of GP and its variants applications for optimal portfolio selection can be found in [Arenas-Parra et al., 2001, Kaminski et al., 2009, Azmi and Tamiz, 2010]. In this method, goals bj specified for each objective function fi act as constraints. Then, ∑ the total deviation from the goals m j=1 |dj | is to minimized, where dj is the deviation from the goal bj for the jth objective. In order to model absolute value, dj is split into positive − + − + − and negative parts such that dj = d+ j − dj with dj , dj ≥ 0 and dj dj = 0. This methods allows multiple objectives and slack in the constraint (not hard). Generally, the problem is formulated as given in (2.7). Minimize − + x∈X,d ,d

m ∑

− (d+ i + di )

i=1

− Subject to fj (x) − d+ j + dj = bj , − d+ j , dj

≥ 0,

− d+ j dj = 0,

j = 1, 2, . . . , m,

(2.7)

j = 1, 2, . . . , m, j = 1, 2, . . . , m,

There are used various extended or modified forms of GP achievement functions [Romero, 2004, Jones and Tamiz, 2010]: • weighted GP, also known as Archimedean GP, • lexicographic GP, also known as non-Archimedean or pre-emptive GP [Ignizio, 1983],

2.4. INTERACTIVE METHODS

27

• MINMAX GP, also known as Chebyshev or Fuzzy Programming, • lexicographic MINMAX GP formulation, • extended GP model, • extended lexicographic GP [Romero, 2001], • fuzzy extended lexicographic GP [Arenas-Parra et al., 2009] Despite the popularity of the GP model, it has some weakness such as the fact it can not guarantee that the obtained solution is Pareto optimal, especially when the decision maker desires this property [Larbani and Aouni, 2007]. Till more general test for all GP model having the compact set of decisions and continuous objective functions was presented in [Larbani and Aouni, 2007], only some test for specific type of GP were developed. Moreover, this method has some limitations like complexity of the “overall objective” [Ignizio, 1978], the decision maker must elicit goal values, often must elicit weights as well. It has received many criticisms, especially, in the case of the aggregation’s procedure of deviations related to objective having incommensurable units of measurement [Aouni and Kettani, 2001].

2.4.

INTERACTIVE METHODS

Interactive methods belong to the progressive methods family and thus allow the decision maker to tune preferences with regard to trade-offs as the methods progress. They allow to find one and only one solution. In these methods, the DM works together with an analyst or other interactive computer program. Only part of the Pareto optimal points has to be generated and evaluated and the DM can specify or correct her/his preferences as the solution process continues and she/he gets to know the problem and its potentialities better. This implies that the DM may not know any global preference structure. A solution pattern is formed and repeated several times. After every iteration some information is given to the DM and she/he has to provide some information. The basic steps in interactive algorithm can be expressed as 1. Find an initial feasible solution, 2. Interact with the DM 3. Obtain a new solution (or a set of new solutions). If the new solution (or one of them) or one of the previous solutions is acceptable to the DM, stop. Otherwise, go to step 2.

28


Interactive methods differ from each other by the form in which information is given to the DM, by the form in which information is provided by the DM, and how the problem is transformed into a single objective optimization problem. Interactive methods have been classified in many ways, mainly according to their solution approaches. Miettinen [Miettinen, 1999] distinguishes two groups: searching-oriented and learning-oriented methods. In the first group, a converging sequences of solution proposals is presented to the DM and it is assumed that she/he provides consistent preference information. In the second group, a free exploration of alternatives is possible allowing trial and error. It doesn’t guide the DM and convergence is not guaranteed. Therefore, the best option would be the combination of these two approaches supporting the learning of preferences and including guiding properties. However, there are some critical components in application of such methods. Consistency of the responses of the DM is one of the crucial factors guaranteeing the success of many interactive solutions methods. In solving practical problems, knowledge about decision processes and analysis is needed to guarantee fruitful cooperation between the DM and the analyst. Other one aspect is that it is unrealistic to assume that DMs can provide precise information and inputs. Moreover, it is not clear such issues like convergence of an interactive method, how to select the starting point and etc.

2.5.

FUZZY METHODS

Fuzzy logic is a branch of mathematics that allows a computer to model the real world the same way that people do. It provides a simple way to deal with vague, ambiguous, and imprecise input or knowledge. Let’s consider Boolean logic, where every statement is true or false; i.e., it has a truth value 1 or 0. Thus, Boolean sets impose rigid membership requirements. In contrast, fuzzy sets have more flexible membership requirements that allow for partial membership in a set. Everything is a matter of degree, and exact reasoning is viewed as a limiting case of approximate reasoning. Hence, Boolean logic is a subset of Fuzzy logic. Fuzzy sets have powerful features to be incorporated into many optimization techniques. Therefore, it is not surprising that the fuzzy MCDM modeling will be seen more often in the literature over the next few years. The most favorable situation for solving a MADM problem is when all estimates of the criteria and their degree of importance are known precisely, which makes it possible to solve then with classic methods. They generally assume that all criteria and their respective weights are expressed in crisp values and, thus, that the rating and the ranking of the alternatives can be carried out without any problem. However, in a real-world decision problems, the application of the classic MADM method may face serious practical constraints because most problems arise in a complex environment where conflicting systems of logic, uncertain and imprecise knowledge, and possibly vague preferences have to be considered. It

2.6. METAHEURISTIC ALGORITHMS

29

is possible that some of parameters of a decision-making problem like objectives/constraints or attainments/weights because of their complexity, are not known precisely and can only be expressed qualitatively which demands the use of specific tools, techniques, and concepts which allow the available information to be represented with the appropriate granularity. To deal with the kind of qualitative, imprecise information or even ill-structured decision problems, Zadeh [Zadeh, 1965] suggested employing the fuzzy set theory as a modeling tool for complex systems that can be controlled by humans but are hard to define exactly. The use of fuzzy set theory enable to incorporate unquantifiable, incomplete and non obtainable information into the decision model. In this context, fuzzy set theory has attracted a lot of attention in MCDA for a long time. Fuzzy MCDM methods are based on the perception that the attributes and/or their attainments cannot be defined precisely, but only as fuzzy values. There are two way to cope with such vagueness: fuzzy to crisp and fuzzy to fuzzy approaches.The first approach proposes methodologies to convert fuzzy data to crisp values to use them in a classic MADM model. It is less complex and needs less computation. However, the conversion of fuzzy data to a crisp can result in significant degrees of error as an uncertain interval of values is replaced by a representative crisp value. The second one proposes algorithms to develop MADM methodologies capable of using fuzzy data. In this approach, a more complicated model capable of aggregating fuzzy values has to be created, because fuzzy or qualitative data are operationally more difficult to manipulate than crisp data, and they certainly increase the computational requirements in particular during the process of ranking when searching for the preferred alternatives [Chen and Hwang, 1992].

2.6.

METAHEURISTIC ALGORITHMS

In contrast to the classical, preference-based, approaches, another very large class of metaheuristic methods generate a set of Pareto solutions simultaneously, i.e. solve MOP directly [Abraham and Jain, 2005, Carlos, 2005, Coello, 2004]. The term metaheuristics in general means “to find beyond in an upper level.” Metaheuristic algorithms comprise Tabu Search (TS), Simulated Annealing (SA), Ant Colony Optimization (ACO), Evolutionary Algorithms (EAs), Memetic Algorithms (MAs) and others.This class of methods seems to be very promising. Although, metaheuristic algorithms do not guarantee that near-optimal solutions will be found quickly for all problem instances, they do find near optimal solutions for many problem instances that arise in practice. Other motivation to study and apply them is their wide range of applicability. This is the most appealing aspect of metaheuristics. However, they are usually computationally expensive because a massive number of nonPareto set points have to be evaluated [Shan and Wang, 2005]. Moreover, these do not guarantee either the generation of a well-distributed Pareto set or the representation of the

30


entire Pareto frontier. In this work we will focus on evolutionary algorithms (EAs) that are based on Darwinian principles of natural selection are called. They are inspired by nature’s capability to evolve living beings well adapted to their environment. There has been a variety of slightly different EAs proposed over the years. Three different strands of EAs developed independently of each other over time are evolutionary programming (EP) introduced by Fogel [Fogel, 1962] and Fogel et al. [Fogel et al., 1966], evolutionary strategies (ES) proposed by Rechenberg [Rechenberg, 1965], and genetic algorithms (GAs) initiated by Holland [Holland, 1975]. GAs are mainly applied to solve discrete problems. Genetic programming (GP) and Scatter search (SS) are more recent members of the EA family. EAs can be distinguished according to their main components and the way they explore the search space. There are different multi-objective EAs, which can be divided into elitist (SPEA, PAES, NSGA II, DPGA, MOMGA, etc.) and nonelitist (VEGA, VOES, MOGA, NSGA, NPGA, etc.) [Donoso and Fabregat, 2007]. They can also be classified based on the generation in which they were developed, i.e. the first generation includes those that do not work with Pareto dominance and the second generation consists of those that do work with the Pareto dominance concept [Gonzalez, 2007]. The methods: Fast Pareto genetic algorithm (FastPGA) [Eskandari and Geiger, 2008], Multi-Objective Cellular genetic algorithm (MOCeLL) [Nebro et al., 2006a], Archive-based hybrid Scatter Search algorithm (AbYSS) [Nebro et al., 2006b], Strength Pareto Evolutionary Algorithm (SPEA2) [Zitzler et al., 2002], and Cellular Differential Evolution algorithm (CellDE) [Durillo et al., 2008] were proposed recently. Their efficiency for various problems has been shown in original papers. Non-dominating Sorting Genetic Algorithm II (NSGAII) [Deb et al., 2005], the state-of-the-art evolutionary method, was chosen following many authors who use it as a standard for comparisons. Next, we will outline each of selected method for our study. Comprehensive surveys on metaheuristic algorithms can be found in [Boyd and Vandenberghe, 2004, Miettinen, 1999, Ehrgott and Wiecek, 2005, Ruzika and Wiecek, 2005, Deb, 2001, Tripathi et al., 2007, Ruzika and Wiecek, 2005, Siirola et al., 2004].

2.6.1.

FastPGA

Eskandari and Geiger [Eskandari and Geiger, 2008] have proposed framework named fast Pareto genetic algorithm that incorporates a new fitness assignment and solution ranking strategy for multi-objective optimization problems where each solution evaluation is relatively computationally expensive. The new ranking strategy is based on the classification of solution into two different categories according to dominance. The fitness of non-dominated solutions in the first rank is calculated by comparing each non-dominated solution with one another and assigning a fitness value computed using crowding distance. Each dominated


31

solution in the second rank is assigned a fitness value taking into account the number of both dominating and dominated solutions. New search operators are introduced to improve the proposed method’s convergence behavior and to reduce the required computational effort. A population regulation operator is introduced to dynamically adapt the population size as needed up to a user-specified maximum population size, which is the size of the set of nondominated solutions. FastPGA is capable of saving a significant number of solution evaluations early in the search and utilizes exploitation in a more efficient manner at later generations. Characteristics of FastPGA: the regulation operator employed in FastPGA improves its performance for fast convergence; proximity to the Pareto optimal set, and solution diversity maintenance

2.6.2.

MOCeLL

Nebro et al [Nebro et al., 2006a] presented MOCeLL, a multi-objective method based on cellular model of GAs, where the concept of small neighborhood is intensively used, i.e., population member may only interact with its nearby neighbors in the breeding loop. MOCell uses an external archive to store the non-dominated solutions found during the execution of the method, however, the main feature characterizing MOCell is that a number of solutions are moved back into the population from the archive, replacing randomly selected existing population members. This is carried out with the hope of taking advantage of the search experience in order to find a Pareto set with good convergence and spread. MOCell starts by creating an empty Pareto set. The Pareto set is just an additional population (the external archive) composed of a number of the non-dominated solutions found. Population members are arranged in a 2-dimensional toroidal grid, and the genetic operators are successively applied to them until the termination condition is met. Hence, for each population member, the method consists of selecting two parents from its neighborhood for producing an offspring. An offspring is obtained applying operators of recombination and mutation. After evaluating of the offspring as the new population member it is inserted in both the auxiliary population (if it is not dominated by the current population member) and the Pareto set. Finally, after each generation, the old population is replaced by the auxiliary one, and a feedback procedure is invoked to replace a fixed number of randomly chosen population members of the population by solutions from the archive. In order to manage the insertion of solutions in the Pareto set with the goal to obtain a diverse set, a density estimator based on the crowding distance has been used. This measure is also used to remove solutions from the archive when this becomes full. Characteristics of MOCeLL: the method uses an external archive to store the non-

32


dominated population members found during the search; the most salient feature of MOCeLL with respect to the other cellular approaches for multi-objective optimization is the feedback of members from archive to population.

2.6.3.

AbYSS

This method was introduced by Nebro et al [Nebro et al., 2006b]. It is based on the scatter search using a small population, known as the reference set, whose population members are combined to construct new solutions. Furthermore, these new population members can be improved by applying a local search method. For local search the authors proposed to use a simple (1+1) Evolution Strategy which is based on a mutation operator and a Pareto dominance test. The reference set is initialized from an initial population composed of disperse solutions, and it is updated by taking into account the solutions resulting from the local search improvement. AbYSS combines ideas of three state-of-the-art evolutionary methods for multi criteria optimization. On the one hand, an external archive is used to store the nondominated solutions found during the search, following the scheme applied by PAES [Knowles and Corne, 2000], but using the crowding distance of NSGA-II [Deb et al., 2005] as a niching measure instead of the adaptive grid used by PAES; on the other hand, the selection of solutions from the initial set to build the reference set applies the density estimation used by SPEA2 [Nebro et al., 2006b]. Characteristics of AbYSS: it uses an external archive to store the non-dominated population members found during the search; salient features of AbYSS are the feedback of population members from the archive to the initial set in the restart phase of the scatter search, as well as the combination of two different density estimators in different parts of the search.

2.6.4.

NSGA-II

The evolutionary method for multi-criteria optimization NGSA-II contains three main operators: a non-dominated sorting, density estimation, and a crowded comparison [Deb et al., 2005]. Starting from a random population the mentioned operators govern evolution whose aim is uniform covering of Pareto set. Non-dominated sorting maintains a population of non dominated members: if a descendant is dominated, it immediately dies, otherwise it becomes a member of population; all members of parent generation who are dominated by descendants die. The density at the particular point is measured as the average distance between the considered point and two points representing the neighbor (left and right) population members.


33

The crowded comparison operator (≺n ) defines selection for crossover oriented to increase the spread of current approximation of Pareto front. Population members are ranked taking into account ”seniority” (generation number) and local crowding distance. The worst-case complexity of NSGA-II algorithm is O(mN 2 ), where N is the population size and m is the number of objectives [Deb et al., 2005]. Characteristics of NSGA-II: this method is of the lower computational complexity than that of its predecessor NSGA; elitism is maintained, no sharing parameter needs to be chosen because sharing is replaced by crowded-comparison to reduce computations.

2.6.5.

SPEA2

As the multi-objective meta-heuristics for the Pareto solutions, SPEA2 is well-known for good performance. It was proposed by Zitzler et al [Zitzler et al., 2002] as an improved elitist multi-objective evolutionary algorithm that employs an enhanced fitness assignment strategy compared to its predecessor SPEA as well as new techniques for archive truncation and density-based selection. The main difference between SPEA and SPEA2 is that SPEA2 has advantage to link archive with the population efficiently. In this algorithm individuals i and j are selected in the archive. A comparison is made between the k-th neighbor individuals to individuals i and j. The individual with smaller distance is excluded. The process is repeated until the number of individuals in the archive reaches a certain number. Furthermore, the archive size is fixed, i.e., whenever the number of nondominated individuals is less than the predefined archive size, the archive is filled up by dominated individuals; with SPEA, the archive size may vary over time. In addition, the clustering technique, which is invoked when the nondominated front exceeds the archive limit, has been replaced by an alternative truncation method which has similar features but does not loose boundary. That constructs the solution set with solutions uniformly scattered. Thus, SPEA2 has better performance to keep the diversity of solution candidates and keep the Pareto solutions.

2.6.6.

CellDE

A new hybrid metaheuristic, called CellDE, combining the advantages of both MOCell and Generalized Differential Evolution 3 (GDE3) was presented in [Kukkonen and Lampinen, 2005]. The idea is to use MOCell as search engine and hybridizing it with DE, by replacing the typical genetic operators of crossover and mutation of GAs by the reproductive mechanism used in DE [Durillo et al., 2008]. The main difference between CellDE and MOCell arises in the creation of new individuals. Instead of using the classical GA operators to generate new individuals, CellDE takes the operator used in DE: three different individuals are chosen and the new offspring solution is

34


obtained based on the differences between them. The newly generated offspring is evaluated and then it replaces the original solution if dominates it, or, if both are non-dominated, it replaces the worst individual in the neighborhood. After that, the new individual is sent to the archive, where it is checked for its insertion. Finally, after each generation, a feedback procedure is performed to replace a number of randomly chosen individuals by a number of solutions taken from the archive.

2.7.

DECISION AID METHODS

Multicriteria decision aid (MCDA) as one of major disciplines in operations research has been developed over the last three decades. The MCDA is based on the idea that a single objective, goal, or criterion is rarely used to make real world decisions and is used to support the DM in situations where multiple conflicting decision factors have to be considered simultaneously [Zopounidis and Doumpos, 2002]. In this context, DM considers a set of alternatives and attempts to make an ‘optimal’ decision considering all the factors that are relevant to the analysis. However, the ‘optimal’ decision is rather a satisfactory nondominated decision, i.e., it is not dominated by other possible decisions in accordance with the DM’s system of values. The MCDA enables the DM to participate actively in the decision-making process and provide support him in understanding the peculiarities and the special features of the realworld problem. Thus, the DM is not restricted to a passive role implementing automatically the “optimal” solutions obtained from a mathematical model. Instead, he participates in the model formulation process as well as in the analysis and implementation of the results, according to his judgment policy. Several appropriate solution procedures interactive and iterative have been proposed to solve multiobjective mathematical programming (MMP) problems. The general framework within which these procedures operate can be considered as a two stage process.In the first stage an initial efficient solution is obtained and it is presented to the DM. If this solution is considered acceptable by the DM (i.e. if it satisfies his expectations on the given objectives), then the solution procedure stops. If this is not the case, then the DM is asked to provide information regarding his preferences on the prespecified objectives. This information involves the objectives that need to be improved as well as the trade-offs that he is willing to undertake to achieve these improvements. The objective of defining such information is to specify a new search direction for the development of new improved solutions. This process is repeated until a solution is obtained that is in accordance with the DMs preferences, or until no further improvement of the current, solution is possible. According to the part that can be problematic to the DM in MCDA, Roy [Roy, 1985] excluded the following problematic decision making: i) Problematic choosing one alterna-

2.8. REVIEW OF RELATED WORKS

35

tive, ii) problematic sorting the alternatives in homogeneous groups defined in a preference order, iii) problematic ranking the alternatives from the best one to the worst one, and iv) problematic describing the alternatives in terms of their performance on the criteria. Decision aid methods are different from the others in following aspects. First, they do not filter the elements of the solution set and keep the elements that could be compared to themselves, but set up an order relation among elements of the solution set and thus obtain a set of solutions with a partial order relation, or a single solution with a total order relation. Second, these methods only work with discrete sets of points. The decision aid methods choose or sort actions with respect to a set of criteria. Each method produces its own definition of the value of a criterion. Decision aid methods include ELECTRE methods (I, IS, II, III, IV, and TRI) [Roy, 1968, Miettinen, 1999] and PROMETHEE (I and II) [Zopounidis and Doumpos, 2002, Miettinen, 1999].

2.8.

REVIEW OF RELATED WORKS

In real industrial design the decision-maker is able to consider only a few possible solutions. Also, sometimes the cost of generating Pareto points may become so high that the designer can afford only a few Pareto optimal solutions. Therefore, we must be satisfied by obtaining enough Pareto optima to cover the minimal set in the criteria space properly [Grosan and Abraham, 2008]. In such a context, it is crucial to have an even distribution of representative Pareto set to obtain maximum information on the whole Pareto surface at minimum computational cost [Utyuzhnikov et al., 2009]. Morever, a well-distributed Pareto set can also be a good foundation for visualizing the Pareto frontier. Since WS methods have difficulty in finding Pareto frontiers in non-convex part and generating evenly distributed Pareto sets, many researchers have developed other methods. These include physical programming (PP) [Messac, 1996, 1900, Messac et al., 2001, Messac and Ismail-Yahaya, 2002, Messac and Mattson, 2002], normal boundary intersection (NBI) [Das and Dennis, 1998, Das, 1999, Shukla, 2007], normal constraint (NC) [Messac et al., 2003, Messac and Mattson, 2004], adaptive weighted sum (PAWS ) [Ryu et al., 2009] methods, directed search domain (DSD) method [Erfani and Utyuzhnikov, 2010], Successive Pareto Optimization (SPO) method [Hintermüller and Kopacka, 2009] and their modifications [Messac and Mattson, 2004, Shukla, 2007, Utyuzhnikov et al., 2009]. Let’s shortly discuss the underlying idea of each method and its limitations.

2.8.1.

NORMAL-BOUNDARY INTERSECTION METHOD

This method was introduced by Das and Dennis [Das and Dennis, 1998]. The NBI generates evenly spaced Pareto points for an even spread of weights, and the spacing of the

36


points is independent of the relative scaling of the objectives [Das and Dennis, 1998, Eddy and Lewis, 2001, Messac and Mattson, 2002]. This method is based on the idea the Pareto surface is then obtained by the intersection of lines normal to the utopia plane and the boundary of the feasible space. NBI can be applied for MOP with more than two criteria, however the visualization process may be complicated for more than three objectives, and how helpful it will be in guiding the user towards a better choice may depend on factors like the psychological aspects of the visualization [Das and Dennis, 1998]. The main problem with NBI method is that the solutions of the sub-problems need not be Pareto-optimal (not even locally). This method aims at getting boundary points rather than Pareto-optimal points, that are a subset of boundary points, that requires a filtering procedure [Messac et al., 2003]. Therefore, the method might be non-robust.

2.8.2.

NORMAL CONSTRAINT METHOD

It was introduced by Messac et al [Messac et al., 2003] to overcome the drawbacks of the NBI approach. As the NBI method, it has a clear geometrical interpretation and is based on the well-known fact that a Pareto surface belongs to the boundary of the feasible space towards minimization of the objective functions [Miettinen, 1999]. First, the so-called anchor points are obtained in the feasible objective space, first. An anchor point corresponds to the optimal value of one and only one objective function in the feasible space. Second, the utopia plane passing through the anchor points is considered. The main difference of NC method is that in the inequality constraints are used instead of equality. Therefore, this makes the method more flexible and stable in comparison with NBI. The NC may fail to generate Pareto solutions over the entire Pareto frontier in multidimensional case when the number of objectives exceeds two. However, the recent modification of the NC eliminates this drawback and guarantees the complete representation of a Pareto frontier [Messac and Mattson, 2004]. The NC approach looks at all possible regions in which Pareto-optimal performance vectors might be located to ensure that the periphery of the Pareto front is covered. However, the NC approach is not efficient for the engineering applications and it requires costly scalar optimization runs to obtain additional non-optimal solutions [Hintermüller and Kopacka, 2009]. The modified NC method generates additional redundant Pareto points near the extreme boundaries of the Pareto frontier [Messac and Mattson, 2004].

2.8.3.

PHYSICAL PROGRAMMING METHOD

Physical Programming method was proposed by Messac in [Messac, 1996]. It generates Pareto points on both convex and non-convex Pareto frontiers as shown in [Messac and Mattson, 2002, Messac and Ismail-Yahaya, 2002]. The method does not use any weight

2.8. REVIEW OF RELATED WORKS

37

coefficients but uses designer preferences in the form of metric classes in the optimization process [Eddy and Lewis, 2001]. In this sense, it appears to be the most interesting method for practical applications under the above stated conditions. In the PP, the designer assigns each objective to one of four categories (class functions). The optimization is based on minimization of an aggregate preference function determined by the preference functions (class functions) with the preferences set a priori. The notion of the generalized Pareto optimal solution is introduced in the PP-based method [Messac et al., 2001] on the basis of the PP class functions. To provide a well-distributed Pareto set, the off-set strategy is introduced in the PP-based algorithm in [Messac and Mattson, 2002]. The algorithm includes a few free parameters. Some evaluations of these parameters are given which nevertheless do not fully remove the uncertainties in their determination.

2.8.4.

OTHER METHODS

In previous sections we gave a short description of methods that are considered as state of arts among methods for generation of uniformly distributed Pareto set approximation of MOPs. Here we present their modifications and other not well-known methods. Modified NBI (mNBI) method was developed by Shukla [Shukla, 2007] for getting an even spread of efficient points. This method unlike NBI method does not produce dominated points and is theoretically equivalent to weighted-sum or ϵ-constraint method. It turned out that mNBI method does not require any unusual assumption compared to relationship of NBI method with WS method and GP method. Since some other class of methods like the NC or PAWS use similar line or inclined search based constraint in their sub-problems, the solutions of the sub-problems of these method are also in general not Pareto-optimal and hence the mNBI method is superior to them. Utyuzhnikov et al [Utyuzhnikov et al., 2009] modified the PP method to generate an even distribution of the entire Pareto set. The new approach also shares conceptual similarity with the NBI and the NC methods. The proposed method is based on the generalization of the class functions which leads to the manageable orientation of the search domain in the objective space, i.e. shrinks the search domain to a “hypercone” and make its location in the objective space better. The generation is performed for both convex and non-convex Pareto frontiers. The method does not generate non-Pareto solutions because local Pareto solutions are removed. Adaptive Weigted Sum method (AWS), presented by Kim and Weck [Kim and De Weck, 2006] differs from other multiobjective optimization methods in that it does not explore the Pareto front in a predetermined fashion, but determines the Pareto front adaptively. In the first phase, the usual WS method is performed to approximate the Pareto surface quickly, and a mesh of Pareto front patches is identified. Each Pareto front patch is then refined by

38


imposing additional equality constraints that connect the pseudonadir point and the expected Pareto optimal solutions on a piecewise planar hypersurface in the m-dimensional objective space. The main difference between AWS and NBI, in addition to adaptive refinement, is the fact that equality constraints in AWS are imposed radially from the pseudonadir point rather than normally to the utopia plane defined by the anchor points as is done in NBI [Kim and De Weck, 2006]. However, in case of very irregularly shaped Pareto front, it would be difficult to represent the Pareto front using only quadrilateral patches. Probably, it would be more effective to use a combination of triangular and quadrilateral meshes, or triangular meshes alone. In this context, more study should be necessary. Other WS based approach for approximating the Pareto front of a multi-objective simulation optimization problem was given in [Ryu et al., 2009]. PAWS combines WS and trust region (TR) methods. The method iteratively approximates each objective function using a metamodeling scheme and employs a WS method to convert the MOP into a set of single objective optimization problems. The weight on each single objective function is adaptively determined by accessing newly introduced points at the current iteration and the non-dominated points so far. The TRM methods is applied to the single objective function to search for the points on the Pareto frontier. It provides an ability to PAWS to find some points in the non-convex part of Pareto frontier because the trust region around a center point can be small enough. To generate the Pareto frontier, Erfani and Utyuzhnikov [Erfani and Utyuzhnikov, 2010] suggested a strategy based on a Directed search domain algorithm, which was first suggested and applied for the modification of the PP method [Utyuzhnikov et al., 2009]. The main idea of DSD is to shrink a search domain to obtain a Pareto solution in a selected area of objective space, i.e. a search cone is constructed and is rotated to search the solution on the peripheral region. A well-spread distribution of the selected search domains should provide a quasi-even Pareto set. It was showed that the DSD method works reasonably more efficient than its other counterparts, NC and NBI methods on some test cases in literature [Utyuzhnikov et al., 2009, Erfani and Utyuzhnikov, 2010]. Successive Pareto optimization method has been proposed by Mueller-Gritschneder et all [Hintermüller and Kopacka, 2009]. It is reported that it is able to generate a discretized Pareto front that represents all regions of the Pareto front for any number of performances. SPO is based on the idea that the boundary of a Pareto front consists of so-called trade-off limits. A trade-off limit is found by marking one performance irrelevant and optimizing the remaining performances. This is equal to generating the Pareto front of a subset of performances. The presented method uses the goal-attainment method. The approach based on line search to generate a uniform distribution of the optimal solutions along the Pareto frontier is presented in [Grosan and Abraham, 2008, 2010]. The approach is composed of two stages: convergence and spreading. In first phase, the problem

2.9. APPLICATIONS IN RISK MANAGEMENT

39

is transformed into a SOP and a solution is found using a line search based approach. In the second one, a set of Pareto solutions are generated starting with the solution obtained at the end of convergence phase. However, this method employs some parameters, whose values has to be adjusted experimentally. A straightforward approach applying a grid search algorithm is employed to develop Pareto frontiers in [Kasprzak and Lewis, 2000]. In this approach, first the sampling of the Pareto set without using a weighted objective function is being generated, i.e. a grid of the design space is taken and the values of the individual objective functions are evaluated at each point. Then fitting of a polynomial equation through the Pareto points, approximating the entire Pareto set, is performed. It is used to predict the optimal combination of objective weights and corresponding design variables of a preferred solution. Despite the developed methods there is some gap left which will be filled in Chapter 3.

2.9.

APPLICATIONS IN RISK MANAGEMENT

Many computational finance problems ranging from asset allocation to risk management, from option pricing to model calibration can be solved efficiently using modern optimization techniques. In this work, we restricted ourself to two application problems, namely, portfolio optimization and pricing tolling agreements, that are presented bellow in more details.

2.9.1.

OPTIMAL PORTFOLIO SELECTION

Financial portfolio selection is one of real world decision making problems with several, often conflicting, objectives which can be reduced to multi objective optimization. The question of optimal portfolio allocation has been of long-standing interest for academics and practitioners in finance. In 1950s Harry Markowitz published his pioneering work where he has proposed a simple quadratic program for selecting a diversified portfolio of securities [Markowitz, 1952]. His model for portfolio selection can be formulated mathematically either as a problem of maximization of expected return where risk, defined as variance of return, is (upper) bounded or as a problem of minimization of risk where expected return is (lower) bounded. The classical Markowitz approach to portfolio selection reduces the problem of two criteria optimization to a one criterion optimization where the second criterion is converted to a constraint. Reduction of a multi-criteria problem to one criterion problem not always is the best method to solve multi-criteria problems especially in the case of vague a priory comparability of criteria. In such problems some evaluation of a whole Pareto set is of interest. It is the case for a portfolio selection problem formulated as a multicriteria optimization problem where compromise between criteria crucially depends on the subjective priorities of a decision maker. Therefore Pareto set approximation is an important method facilitating

40


rational portfolio selection. To attack the problem of Pareto set approximation evolutionary optimization methods are claimed very promising; see e.g. [Li and Taiwo, 2006, Chang et al., 2000, Wang et al., 2006, Ehrgott et al., 2004, Xia et al., 2001, Mukerjee et al., 2002]. In subsequent years there has been considerable work on multiperiod and continuous-time models for portfolio management, pioneered by Merton [Merton, 1971]. The multi-period mean-variance optimization framework was extended to worst-case design with multiple rival return and risk scenarios in [Gülpinar and Rustem, 2007]. Ortobelli et al. [Ortobelli et al., 2009] proposed some stable Paretian models for optimal portfolio selection. They generalized the mean-variance analysis suggested by Li and Ng [Li and Ng, 2000], providing a three-parameter formulation of optimal dynamic portfolio selection. The extension of multi-horizon mean-variance portfolio is discussed in [Briec and Kerstens, 2009]. The mainstream computational model to solve recursive decision problems in the presence of uncertainty is currently provided by multi-stage stochastic programming in [Birge, 1997, Birge and Louveaux, 1997, Gülpinar et al., 2002]. In contrast to the mainstream stochastic programming approach to multi-period optimization, which has the drawback of being computationally intractable, it was proposed an approach in [Calafiore, 2008] based on a specific affine parameterization of the recourse policy, which allows to obtain a sub-optimal but exact and explicit problem formulation in terms of a convex quadratic program and efficiently attack multi-stage decision problems with many securities and periods. As an alternative to stochastic programming approaches the robust optimization methods are proposed in [Shen and Zhang, 2008, Bertsimasa and Pachamanovab, 2008, Topaloglou et al., 2008]. Since Markowitz [Markowitz, 1952] there have been proposed several modeling approaches allowing to identify portfolios with the highest expected returns for a given level of risk [Konno, 1990]. Other approaches model the stochastic nature of the problem directly as a stochastic program. In recent years, stochastic programming (SP) models have been increasingly used to address real life multi-period asset and liability management problems. A seminal contribution was made by Bradley and Crane [Bradley and Crane, 1972] who proposed a multi-stage model for bond portfolio management. More recently, Carino et al. [Carino et al., 1994] applied SP to the asset-liability management problem of the insurance industry, and Zenios et al. [Zenios et al., 1998] and Topaloglou et al. [Topaloglou et al., 2008] formulated models for a portfolio of fixed income securities. Authors of [Nielsen and Poulsen, 2004, Rasmussen and Clausen, 2007] proposed a multistage SP model for managing mortgage loans. [Balibek and Köksalan, 2010] have applied an optimization approach for debt management. Zhang and Zhang [Zhang and Zhang, 2009] improved Hibiki’s model [Hibiki, 2003] by adding CVaR and considering short sale constraints and proportional transaction costs simultaneously. In [Wei and Ye, 2007], the multi-period mean-variance stochastic markets model with bankruptcy control in stochastic environment is presented. Yu et al. [Yu et al., 2003] provide a bibliography of SP models in financial optimization.


41

Extensive collections of SP models for financial problems can also be found in Ziemba and Mulvey (1998) and Dupacova et al. [Dupacova et al., 2002]. In the specific context of financial allocation, a classical stochastic programming method based on Benders decomposition is proposed in [Dantzig and Infanger, 1993], and techniques for construction of scenario trees are discussed for instance in [Gulpinar et al., 2004, Pflug, 2001]. A possibly improved approach that uses scenario trees together with simulated paths has been recently studied in Hibiki [Hibiki, 2006]. Also, in [Rustem and Gülpinar, 2007] the authors extend the multi-period mean-variance model to deal with rival uncertainty scenarios, and propose a worstcase decision approach which uses a min-max technique in synergy with a stochastic optimization algorithm based on scenario trees. Scenario-based stochastic programming models aiming at maximizing expected portfolio value while taking into account cost variability have been recently proposed also in [Pinar, 2007, Takriti and Ahmed, 2004]. A survey with theoretical analysis of multiperiod models based on scenario trees is provided in [Steinbach, 2001]. The concept of scenarios is typically employed for modeling random parameters in multiperiod stochastic programming models. Scenarios can be constructed via a tree structure [Infanger, 2006, Carino et al., 1994]. This model is based on the expansion of the decision space, taking into account a conditional nature of the scenario tree. Conditional decisions are made at each node, subject to the modeling constraints. To ensure that the constructed representative set of scenarios covers the set of possibilities to a sufficient degree, the numbers of decision variables and constraints in the scenario tree may grow exponentially. This model is called a scenario tree model. We can generate sample paths associated with asset returns using a Monte Carlo simulation method. The advantage of the simulated path structure compared to the tree structure is to give a better accuracy to describe uncertainties of asset returns.

2.9.2.

PRICING TOLLING AGREEMENTS

Currently, there is plenty of methods developed for financial markets that can be used for pricing derivative contracts and managing risk. Notwithstanding electricity market is very different from financial one that’s why there is need for further investigations. Since energy markets are not sufficiently liquid and efficient, many commodity contracts require physical settlement which necessitates actual ownership of an asset. The valuation of electricity contracts differs from that of other financial contracts in that: • the underlying electricity is not a traded asset, meaning that it cannot be bought and hold; • electricity contracts often contain side constraints (e.g., various contract provisions)

42


on how financial payouts are derived from the underlying electricity or a physical asset generating electricity. Consequently, many energy trading firms expressed high interest in owning energy assets. The idea of a tolling agreement was introduced in order to circumvent the capital intensive side of the business [Ludkovski, 2005]. This contracts gives a right its buyer to run power plant when it is profitable, i.e. spark spread is large enough, subject to pre-specified exercise rules. A spark spread option is an option giving its holder at a specified time in the future the right to exercise a profit equal to a non-negative part of the difference between the price of energy and cost to produce it, i.e. the price of fuel multiplied on a coefficient called heat rate. Therefore, power plant owner has to find the optimal policy determined by the simple decision between continue in the same production regime and switch to the best alternative. In recent years a problem of scheduling flexibility of electricity generating facilities has attracted a lot of attention and interest in the academic literature [Deng and Oren, 2003, Deng and Xia, 2005, Ludkovski, 2005, Ryabchenko and Uryasev, 2011] due to increasing popularity of tolling agreement contracts. We will discuss literature with methods that have been applied in financial engineering practice for valuation of tolling contracts. Historically, tolling agreement contracts have become popular since de-regulation of energy markets in the 1970s. At the very beginning the discount cash flow method was applied by practitioners. In this approach, the asset value was estimated based on projections of future prices and proper weighing and discounting of possible cases. However it appeared to be not suitable due to high volatility of energy and fuel prices because of uncertainty elimination and disregard of dynamical behavior of prices. Markov Decision Processes approach became popular in 1980s. This approach is a tree-based versions of the stochastic control, where dynamic programming is used to solve the switching problem through backward recursion up the tree. The traditional variational approach to optimal switching was introduced by Brekke and Oksendal [Brekke and Oksendal, 1994] who considered a geometric Brownian motion for stochastic firm value (Xt ) and infinite horizon. Another similar work with explicit solution for a simple switching problem can be found in [Yushkevich, 2001]. However, this approach suffers from poor dimension-scaling properties, the computational complexity explodes for long horizons with many optionalities. The new widely used method by researchers for pricing tolling agreements is the strip of spark-spread options approach [Eydeland and Wolynec, 2003]. Here, a scheduling operational flexibility of energy generating facilities is represented as a sequence of so called spark spread options owned by a manager [Ryabchenko and Uryasev, 2011]. In many cases there are efficient approximations for pricing spread options that are very fast to compute and provide good bounds not just on the price. Application of this idea can be found in [Deng et al., 2001, Eydeland and Wolynec, 2003, Ryabchenko and Uryasev, 2011]. An overview of methods on spread options pricing is given in [Carmona and Durrleman, 2003]. Nevertheless, the method suffers from two major shortcomings. First,


43

it is inherently unsuitable for incorporating dynamic operational constraints. For example, there is no natural way of including switching costs. Second, the strip of options approach eliminates the time decay, making the resulting strategy completely stationary in time despite the finite horizon of the contract. Ludkovski [Ludkovski, 2005] proposed a framework based on recursive optimal stopping. Their developed method is based on Monte Carlo regressions and the scheme uses dynamic programming to simultaneously approximate the optimal switching times along all the simulated paths. Although newly appeared approaches based on stochastic dynamic are flexible in terms of incorporating various operational constraints, it is computationally inefficient when considering problems with relatively large horizons. To prevail this flaw, Ryabchenko and Uryasev [Ryabchenko and Uryasev, 2011] suggested a simple framework where the problem of finding the optimal exercise boundaries is then reduced to solving a simple linear programming problem. Our work is based on their proposed formulation that is shortly outlined in Chapter 5.

chapter 3 ALGORITHMS FOR UNIFORM PARETO SET APPROXIMATION One way to identify optimal solutions consists of two steps: first, generate a representative set of Pareto solutions; and second, select the most attractive one from this set. Unfortunately, it may be a case when the DM can find it undesirable or impractical to examine an huge number of Pareto solutions in order to choose the most eligible one, and the evaluation of a single Pareto solution can be highly expensive. To successfully apply this approach, the generated Pareto set must be truly representative of the complete Pareto front. In other words, the set must not over represent one region of the objective space, or neglect others. The uniform distribution of Pareto solutions facilitates the task of choosing the most desirable (final) solution from the set of Pareto solutions. Although, in previous chapters we discussed about the importance of the uniformity of Pareto front, we do provide the formal definition only in this chapter. In addition, we will describe the features that are in great demand to characterize the MO methods.

3.1.

CONCEPT OF UNIFORM DISTRIBUTION

In this research, we use the definition of uniform or even distribution given by Messac and Mattson [Messac and Mattson, 2004]: “A set of points is evenly distributed over a region if no part of that region is over or under represented in that set of points, compared to other parts.” The measure of distribution evenness is described below. We will define this in mathematical way. Let’s start from bi-criteria case. For each point, µi , in a set of solutions we draw two circles whose diameters joint two neighbor points. One cycle is the smallest with diameter dil that can interlink the nearest neighbor and the second one is the largest with maximum diameter diu that can be constructed from point µi to the farthest neighbor with no points inside the cycle as shown in Figure 3.1a.

45

3.1. CONCEPT OF UNIFORM DISTRIBUTION

i

du i

du

j

dl

~ ~

j

du

i

i

dl

i

dl (a) Bi-objective case

(b) Three-objective case

3.1 Fig.: Graphical explanation of distribution ˆ where dˆ and σd deAn estimate of the uniformity can be expressed as ξ = σd /d, note the mean and standard deviation of d, respectively; and where di = {dil , diu } and d = {di , . . . , d2np }, np is the number of generated Pareto points. A set of points is exactly evenly distributed when ξ = 0. Here, it is assumed that the objective space is normalized such that 0 ≤ µi ≤ 1, ∀i ∈ {1, 2, . . . , m}. In case of n dimensions, the diameter of a hyper-sphere is constructed instead of the circle shown in Figure 3.1a. An example of three criteria is given in Figure 3.1b, where the right side shows that points are (locally) evenly distributed when the diameter of the spheres are nearly equal, and that points are not evenly distributed (left side of figure) when the diameters of the spheres are not close in value. The uniform distribution is advantageous since it minimizes the size of not researched holes in the Pareto set, in other words it minimizes guaranteed distance between a potentially favorable solution and one of generated points. It is well known, however, that the uniform distribution of points is a difficult problem even for explicitly defined simple sets. Moreover, such distributions are not composite, i.e. the increase of the number of points by one implies complete relocation of points of the previous distribution. To achieve exactly evenly spaced Pareto set approximation practically it is not possible. Therefore in multi-objective optimization only the algorithms for quasi-uniform distribution of Pareto optimal solutions are considered. In order to generate the Pareto front, it is desirable that the method possess practical features [Messac et al., 2003, Messac and Mattson, 2004]: • generate an even distribution of Pareto points over the complete Pareto front, • guarantee to yield any Pareto point in the feasible design space, • be insensitive to design objective scaling, • be valid for an arbitrary number of design objectives, and

46

CHAPTER 3. ALGORITHMS FOR UNIFORM PARETO SET APPROXIMATION

• be relatively easy to implement.

3.2.

ALGORITHM OF ADJUSTABLE WEIGHTS

We investigated approximation of Pareto set by means of a finite set of points uniformly distributed in close vicinity of the Pareto set; the term “uniformly” is defined more precisely in the Section 3.1.. First method who attracted our attention was the widely used scalarization method of weighted criteria summation. Besides of the status of “classics” of multi objective optimization, an important argument to consider the method of weighted criteria summation is simplicity of its implementation. Since a multi criteria problem is converted into a single criterion optimization problem where the objective function is a weighted sum of the criteria functions, weights can be used to express the relative significance of different criteria. For some multi objective optimization problems it is convenient to take into account in this way the preferences of decision makers. However, we do not assume availability of information about importance of criteria, and for us weights play the role of parameters defining points in the Pareto set corresponding to the solutions of parametric single criteria problems. Assume that all fi (x) are convex functions; then for every point of Pareto set there exist weights wi , i = 1, . . . , m such that this Pareto point is a minimizer of f (x) defined by (2.6). The violation of the convexity assumption can imply not existence of weights implying location of the minimizer of the corresponding composite objective function f (x) in some subsets of Pareto set. However, in the problems of portfolio selection objectives fi (x) normally are defined by rather simple analytical expressions, and checking of the validity of convexity assumption is easy. The fact of existence of the correspondence between points in Pareto set and minimizers of the parametric minimization problem (2.6) theoretically justifies further investigation of properties of this correspondence. In many cases it is not an easy task to choose the set of weights defining solutions of (2.6) that would be well (in some sense uniformly) distributed over the Pareto set. To generate such a subset of the Pareto set by means of repeatedly solution of (2.6) with different weights the latter should be chosen in a special but a priory unknown way. We propose a branch and bound type method for iterative composing of a set of weights implying the desirable distribution of solutions of (2.6). The feasible region for weights Ω = (ω1 , . . . , ωm ) : 0 ≤ ωi ≤ 1, i = 1, . . . , m,

m ∑

ωi = 1,

(3.1)

i=1

is a standard simplex. Our idea is to partition Ω into sub simplices whose vertices are mapped to the Pareto set via (2.6). The sequential partition is controlled aiming to generate such new

3.2. ALGORITHM OF ADJUSTABLE WEIGHTS

47

3.2 Fig.: Graphical interpretation of simplex partitioning sub simplices that mapping of their vertices would generate points uniformly distributed in Pareto set. The partition procedure is arranged as a branching of a tree of nods corresponding to sub simplices. The original standard simplex is accepted as the root of the tree. Branching means partition of a simplex into two sub simplices where the new vertex is the midpoint of the favorable edge defined later. Simplicial partitioning procedure has been applied to construct algorithms for single criterion global optimization in [Steuer et al., 2008, Clausen and Zilinskas, 2002]; this procedure can be applied also in multi objective optimization if only preconditions for selection of a simplex for partition would be reconsidered. We aim to subdivide the original simplex into sub simplices in such a way that solutions of (2.6) corresponding to the vertices of these sub simplices would be well distributed over the Pareto set. Let us consider the current set of sub simplices in the space of weights, and points in the criteria space corresponding to the vertices of these simplices. The eligibility of a simplex to partition is defined by the longest distance between the points in the criteria space corresponding to the vertices of the considered simplex. Branching strategy is based on depth first search, and the selected simplices are partitioned until the longest distance between the corresponding points in criteria space is reduced up to predefined tolerance. Let us consider the current set of sub simplices in the space of weights, and points in the criteria space corresponding to the vertices of these simplices as illustrated in Figure 3.2. Generally, an algorithm can be described in the following steps: Step 1. Initialization. Perform single optimization of every function, then normalize function values to fit in range [0 1]. Calculate the distances for every edge in in criteria space of root simplex.

48


Step 2. Simplex division. Find the simplex with the longest edge and divide it into two sub simplices where the new vertex is the midpoint of the favorable edge. Step 3. Calculation of distances. Perform optimization with new weighs and calculate distances for new edges. Step 4. Termination. If stopping criteria are satisfied, then terminate. Otherwise, go to Step 2. Before calculation of the distance in criteria space, the normalized function values in range between 0 and 1 are used. The normalization is done according to (3.2) fi,norm =

fi − fi,min fi,max − fi,min

(3.2)

where fi,min , fi,max , and fi are minimum i-th function value, maximum i-th function value and i-th function value to be normalized, respectively. To calculate the length of k-th simplex edge, the Euclidean distance formula v u 3 u∑ dk = t (fij − fil )2

(3.3)

i=1

has been used. Several conditions can be considered as algorithm termination rule. The algorithm can terminate after the longest distance between the corresponding points in criteria space is reduced up to predefined tolerance. Other option may be to stop algorithm after a particular number of Pareto optimal solution is found. The pseudocode of AW algorithm is provided in Procedure 1. Here i is the number of s objective functions, S is number of produced simpleces, and s, s = 1, . . . , S; ωl,i is weighting factor of sth simplex lth vertex for ith objective function ; I is index of simplex selected for partitioning.

3.3.

ALGORITHM OF SUCCESSIVE LEXICOGRAPHIC GOAL PROGRAMMING

In this section we propose an approach based on preemptive or lexicographic GP (LGP) method aiming at uniformity of distribution of solutions in the Pareto set. In LGP different goals are classified into several levels of priorities. The higher priority is assigned to the more important goal. Therefore, the goals of first level are fulfilled before considering ones of second level. It is argued that this way is most practical despite some critics mentioned

3.3. ALGORITHM OF SUCCESSIVE LEXICOGRAPHIC GOAL PROGRAMMING

49

Procedure 1 AW 1: Initialization: ccrit = c, S = 1 2: Perform minimization of each single objective function fi , and identify fi,min and fi,max 3: Normalize functions values using (3.2) 4: Assign ωi and fi correspondingly to each vertex of a root simplex (see Fig. 3.2) 5: Calculate distances d1i 6: Set criteria C 1 = max d1i 7: Identify verteces k, l of longest edge; set I = 1, C = C 1 , 8: while C > ccrit do ω I +ω I 9: Compute ωi = l,i 2 k,i 10: Solve problem (2.6) 11: Normalize obtained functions values fi 12: Calculate distances di 13: Delete simplex I and create two new simpleces S = S + 1 14: Set C = maxs=1,...,S C s , I = s, and identify k, l 15: end while

later. At the beginning, the problem consisting of first level goals and constraints are solved. If this problem has multiple solution, another problem of second level priority is formulated including the goals of first level priority as hard constraints that obtained solution does not violate the goals of first level priority. This process is continued until a single solution is reached. All subsequent goals of higher level priorities are neglected. Let’s assume that the objective functions fi (·), i = 1, . . . , k are ranked giving a larger index to a less important objective. The term ’successive’ in the title means that a sequence of lexicographic goal programming problems is solved with different reference points producing different solutions. The algorithm is implemented as k − 1 nested loops. In i-th loop the i-th component of the reference vector g is selected, and i-th component of the solution is obtained; the mentioned components may coincide. If they do not coincide then the considered component of the solution vector is obtained by minimizing the under/over achievement. For the selected g the decision x∗ (a point in the feasible decision space) is computed while executing the most inner loop. A reference point g = (g1 , g2 , . . . , gk−1 ) is selected from the set of vertices of the cubic grid Nk−1 1 ), , . . . , gk−1 (g11 , . . . , g1N1 ) × (g21 , . . . , g2N2 ) × . . . × (gk−1 where gij = min fi (x) + j · si , j = 1, . . . , Ni , x∈X

si = (max fi (x) − min fi (x))/(Ni + 1), x∈X

x∈X

(3.4)

and it is assumed that minimizers and maximizers in (3.4) are unique. It is expected that

50


the distribution of solutions of (2.6) will inherit uniformity from the distribution of reference points. The following formulas describe the computation of the components of solution vector F jk−1 from i-th until k-th, i = 1, . . . , k − 1, where the reference vector is equal to (g1j1 , . . . , gk−1 )

Fi = fi (¯ x), − ¯− (¯ x, d¯+ arg min (d+ i + di ), i , di ) = − x∈X,d+ i ,di

− fi (x) − d+ = giji , i + di jm ¯− fm (x) − d¯+ m + dm = gm , m = 1, . . . , i − 1 − d+ ≥ 0, i , di

i = 1, . . . , k − 1, x∗ = arg minfk (x), Fk = fk (x∗ ), fm (x) −

d¯+ m

+

d¯− m

=

x∈X jm gm , m

= 1, . . . , k − 1.

The pseudo-code of the successive lexicographic goal programming (SLGP) algorithm is provided in Procedure 2. Procedure 2 SLGP 1: Sort the objective functions according to priority or importance 2: Define the set of reference vectors as in (3.4) N 3: for all g11 , . . . , g1 1 do ¯− 4: Compute F1 , d¯+ 1 , d1 5: for all g21 , . . . , g2N2 do ¯− 6: Compute F2 , d¯+ 2 , d2 7: ... Nk−1 1 8: for all gk−1 , . . . , gk−1 do + ¯ 9: Compute Fk−1 , dk−1 , d¯− k−1 10: Compute Fk 11: end for 12: ... 13: end for 14: end for It should be indicated that SLGP can yield dominated solutions like other methods based on lexicographic goal programming [Miettinen, 1999]. This disadvantage can be easily eliminated either using the Pareto filter [Messac et al., 2003] or a restoration method [Larbani and Aouni, 2010]. Here we applied the Pareto filter presented in [Messac et al., 2003] with small improvement. This filter provides a subset of a given set of points in objective space

3.3. ALGORITHM OF SUCCESSIVE LEXICOGRAPHIC GOAL PROGRAMMING

51

Procedure 3 Pareto filter 1: Initialization: i = 0, j = 0, k = 1, and m = number of generated solutions; m = f (mk ) 2: for all i = 1, . . . , m do 3: Set j = 0; 4: if i < m then 5: for all j = j + 1, . . . , m do 6: if i ̸= j then 7: if µi ̸= µj and (µi − µj )s ≥ 0, ∀s then 8: break; // µi is not global Pareto point 9: else 10: if j = m then 11: pk = µi ; // µi is global Pareto point 12: k = k + 1; 13: end if 14: end if 15: end if 16: end for 17: else 18: for all j = j + 1, . . . , m − 1 do 19: if µi ̸= µj and (µi − µj )s ≥ 0, ∀s then 20: break; // µi is not global Pareto point 21: else 22: if j = m − 1 then 23: pk = µi ; // µi is global Pareto point 24: k = k + 1; 25: end if 26: end if 27: end for 28: end if 29: end for

where none of the points is dominated by any one. Shortly, it eliminates all dominated points from the given set. The flaw of the original Pareto filter is not inclusion of the last point in the set. The pseudo-code of the Pareto filter is provided in Procedure 3.

The distribution of solutions crucially depends on the properties of the hyper surface defining the Pareto set. In some extend it can be controlled by choosing the scales of objectives and lengths of steps si (equivalently Ni ). SLGP is applicable to non convex objective functions. The other advantage of SLGP besides of applicability to non convex problems is simplicity of selection of a particular subset of the Pareto set intended to approximate; it is sufficient to form the corresponding grid of reference vectors.

52

3.4.


EXPERIMENTS DESCRIPTION

This section is devoted to describe the performed experiments. First, we will present the set of MOPs which were solved by different algorithms. Next, we will outline the performance metrics applied to measure the quality of generated Pareto subsets. Finally, we will depict performed experiments, discuss the obtained results and draw conclusions.

3.4.1.

TEST PROBLEMS

In this section we describe the different sets of both constrained and unconstrained problems. Before describing problems, it is important to say that some particular characteristics of the Pareto-optimal front like convexity or nonconvexity, discreteness, and nonuniformity can prevent MO algorithms from finding well distributed Pareto optimal solutions. Therefore, we selected problems which have been used in many studies in this area and possess attribute mentioned above. We thereby restricted ourselves to only two and three objectives in order to investigate the simplest case first. In our opinion, two and three objectives are sufficient to reflect essential aspects of multi-objective optimization. For this purpose, the problems ZDT1, ZDT2, ZDT4, Schaffer, Fonseca, Kursawe, Viennet2 and Viennet4 have been chosen from widely used benchmark [Zitzler et al., 2000]. We consider that the features of these problems make them meaningful enough for this research. Considered MOPs are given in Table 3.1. They have been grouped into categories for a better presentation of the results. The first group is composed of the bi-objective unconstrained problems Fonseca and Kursawe, as well as the problems ZDT1, ZDT2, and ZDT4, and constrained problem Tanaka. The test problem ZDT1, suggested by Zitzler et al. [Zitzler et al., 2000], has 30 decision variables and a convex true Pareto optimal front. The 30-decision variable problem ZDT2 has a concave Pareto optimal front. The 10-decision real-valued variable test problem ZDT4 is a multi-frontal convex problem having a large number of local Pareto optimal fronts and a single global Pareto optimal front. This is a difficult unconstrained test problem as it has 100 distinct Pareto optimal fronts, out of which only one is global. Next, we have included problems of more than two objectives, i.e., the second group comprises the problems Viennet2 and Viennet4. The first has three objectives and zero constraints, and the second one has three objectives and three constraints.

3.4.2.

ALGORITHMS SETUP

In order to know how competitive new algorithms are, we decided to compare them to six meta-heuristic algorithms which represent the state-of the-art. We have used algorithms implemented in Java using jMetal framework aimed at facilitating the development of meta-

53

3.4. EXPERIMENTS DESCRIPTION

3.1 Table: MOPs Name

Definitions

Constraints

Problem1

Min F = (f1 (x), f2 (x)) f1 (x) = x21 + x22 f2 (x) = (x1 − 2)4 + (x2 − 2)4

−105 ≤ x1,2 ≤ 105

Tanaka

Min F = (f1 (x, y), f2 (x, y)) f1 (x) = x f2 (x) = y

0 ≥ −x2 − y 2 + 1 +0.1· cos(16· arctan(x/y)) 0.5 ≥ (x − 0.5)2 + (y − 0.5)2 −π ≤ x, y ≤ π −5 ≤ xi ≤ 5

Kursawe

Min F = (f1 (x), f2 (x)) √ ∑ (−0.2∗ x2i +x2i+1 ) f1 (x) = ∑n−1 ) i=1 (−10e f2 (x) = ni=1 (|xi |0.8 + 5 sin x3i )

Fonseca

Min F = (f1 (x), f (x)) ∑ 2 − n (x − √1 )2 f1 (x) = 1 − e ∑i=1 i n − n (x + √1 )2 f2 (x) = 1 − e i=1 i n

−4 ≤ xi ≤ 4 i = 1, 2, 3

ZDT1

ZDT2

ZDT3

ZDT4

Viennet2

Viennet4

i = 1, 2, 3

Min F = (f1 (x), f2 (x)) f1 (x) = x1 f2∑ (x) = g· h m g(x) = 1 + 9 √ i=2 xi /(m − 1) h(f1 , g) = 1 − f1 /g

0 ≤ xi ≤ 1, i = 1, . . . , m m = 30

Min F = (f1 (x), f2 (x)) f1 (x) = x1 f2∑ (x) = g· h g(x) = 1 + 9 m i=2 xi /(m − 1) h(f1 , g) = 1 − (f1 /g)2

0 ≤ xi ≤ 1, i = 1, . . . , m m = 30

Min F = (f1 (x), f2 (x)) f1 (x) = x1 f2∑ (x) = g· h m g(x) = 1 + 9 √ i=2 xi /(m − 1) h(f1 , g) = 1 − f1 /g − (f1 /g)sin(10πf1 )

0 ≤ xi ≤ 1, i = 1, . . . , m m = 30

Min F = (f1 (x), f2 (x)) f1 (x) = x1 f2 (x) = g· h ∑ m 2 g(x) = 1 + 10(m √ − 1) + i=2 (xi − 10cos(4πxi )) h(f1 , g) = 1 − f1 /g

0 ≤ x1 ≤ 1 −5 ≤ xi ≤ 5, i = 2, . . . , m m = 10

Min F = (f1 (x, y), f2 (x, y), f3 (x, y)) 2 (y+1)2 f1 (x, y) = (x−2) + +3 2 13 (x+y−3)2 (−x+y+2)2 f2 (x, y) = + − 17 36 8 (x+2y−1)2 (2y−x)2 f3 (x, y) = + 17 − 13 175 Min F = (f1 (x, y), f2 (x, y), f3 (x, y)) 2 2 f1 (x, y) = (x−2) + (y+1) +3 2 13 (2y−x)2 (x+2y−1)2 + 17 − 13 f2 (x, y) = 175 2 (3x−2y+4)2 f3 (x, y) = + (x−y+1) + 15 8 27

−4 ≤ x, y ≤ 4, y < 4x + 4 x > −1 y >x−2 −4 ≤ x, y ≤ 4

54


heuristics for solving multi-objective optimization problems. They can be downloaded from [Durillo et al., 2010]. WS method, AW and SLGP algorithms were implememnted using software Matlab. All experiments were carried on a usual laptop with a processor of 2.1 Ghz. Next, we briefly describe the parameter settings of metaheuristics algorithms that are recommended by the authors [Eskandari and Geiger, 2008, Nebro et al., 2006b,a, Deb et al., 2005] and used in the subsequent experiments: • AbYSS. The size of P is 20, the subset generation method generates all pairwise combinations of individuals belonging to both Ref Set1 and Ref Set2 , the size of both Ref Set1 and Ref Set2 is 10, and the size of archive is 100. The crossover operator, used with probability1 in the solution combination method of the scatter search is SBX and distribution index is equal to 20. • CellDE. The population size is 100 individuals. The binary tournament is applied for parents selection, and the differential evolution is used for recombination with parameters CR = 0.5, F = 0.5. Feedback is 20% of the population (20 individuals). • FastPGA. Both an initial and maximum population size are 100 individuals. SBX crossover and polynomial mutation are done with probability 1.0 and distribution indexes of 20. • MOCell. The size of population and parents selection is the same as in CellDE. Recombination is simulated binary,A crossover probability of pc = 0.9 and polynomial mutation with probability pm = 1/n (where n is the number of decision variables) is performed. An archive size is 100 individuals with feedback 20% of the population. • NSGA-II. We have used the real-coded version. The operators for crossover and mutation are SBX and polynomial mutation, with distribution indexes of ηc = 20 and ηm = 20, respectively. A crossover probability of pc = 0.9 and a mutation probability pm = 1/n are used. The population size is 100 individuals. • SPEA2. We have used the following values for the parameters: both the population and the archive have a size of 100 individuals, and the crossover and mutation operators are the same as those used in NSGA-II, using the same values concerning their application probabilities and distribution indexes.

3.4.3.

TESTS RESULTS

To present results in more systematic and organized way, we combine results of WS, AW ans SLGP methods (first group) into one table, and results of meta-heuristics algorithms

55


(second group) into other one for every performance metric. To evaluate meta-heuristic algorithms, we performed two series of experiments. First we ran all of them for 25,000 function evaluations and then repeated, this time with the execution of 50,000 function evaluations as the stopping condition. We have made 100 independent runs of each experiment, and the results obtained are shown in following tables: estimates of GD are presented in Table 3.5, IGD in Tables 3.2 and 3.6, HV in Tables 3.3 and 3.7, and Spread in Tables 3.4 and 3.8. The values included in the tables with results of meta-heuristic algorithm are the mean (Avg.) and the standard deviation (Std.), while WS method and proposed ones have single value of performance metrics for every problem. As WS and AW methods are applicable only to convex problem, the cells in the table of metrics of nonconvex problems (ZDT2, ZDT3, Fonseca, Kursawe, and Tanaka) are left blank. The best estimate of performance metrics for each problem are written in bold for every methods group. The best one among both group have a gray background. 3.2 Table: Estimates of inverted generational distance (IGD) metric obtained by WS, AW and SLGP Problem

WS

AW

SLGP

Prob1 ZDT1 ZDT4 ZDT2 ZDT3 Fonseca Kursawe Tanaka

3.29E-3 9.14E-5 1.83E-4 -

3.29E-3 7.65E-5 5.70E-5 -

3.29E-3 5.52E-5 1.12E-4 2.73E-5 1.27E-4 4.67E-05 8.59E-05 1.69E-04

Viennet2 Viennet4

4.88E-4 2.35E-4

7.63E-5 2.34E-4

1.40E-4 1.75E-4

At the beginning we will start with estimates obtained using first group methods: WS, AW and SLGP. We do not provide table of GD estimates, because WS and AW are applied only to solve convex problems that have single local and global solution; and we use Pareto filter for SLGP in nonconvex problems. Therefore, we get only non-dominated solutions and GD estimate is equal to zero. Table 3.2 shows that all three methods have the same IGD metric of Problem1. AW got the lowest value of IGD in 3 out of 5 convex problems, while SLGP is superior in 2 out of 5 ones. The same positions of AW and SLGP algorithms are kept in accordance with hypervolume metric and spread, i.e., AW method is advantageous in Problem1, ZDT4, and Viennet2, while SLGP is superior in ZDT1 and Viennet4. This test of convex problems

56


3.3 Table: Estimates of hypervolume (HV) metric obtained by WS, AW and SLGP Problem

WS

AW

SLGP


9.29E-1 6.53E-1 6.52E-1 -

9.30E-1 6.54E-1 6.62E-1 -

9.26E-1 6.61E-1 6.61E-1 3.28E-1 5.11E-1 3.12E-1 3.99E-1 2.92E-1

Viennet2 Viennet4

9.25E-1 8.80E-1

9.25E-1 8.63E-1

9.18E-1 8.81E-1

3.4 Table: Estimates of spread (∆) metric obtained by WS, AW and SLGP Problem

WS

AW

SLGP


9.37E-1 7.34E-1 7.40E-1 -

4.66E-1 6.47E-1 2.09E-1 -

7.38E-1 2.78E-1 2.84E-1 2.26E-1 5.11E-1 2.96E-1 5.22E-1 5.44E-1

Viennet2 Viennet4

9.34E-1 8.30E-1

5.21E-1 6.70E-1

1.13E-0 4.85E-1

points out that there is no single winer and depending on problem description either AW or SLGP algorithm can be employed. Let’s take a look to performance metrics obtained by meta-heuristic algorithms. Table 3.5 shows that MOCell obtains the lowest (best) values for the GD metric in thirteen out of twenty tests with lowest standard deviation in almost all the cases. This means that the resulting Pareto fronts from MOCell are closer to the optimal Pareto fronts than those computed by the rest. We would like to highlight the suitability of MOCell for solving the first group of MOPs (bi-objective ones), in which this algorithm obtains the closest fronts for twelve out of sixteen cases (eight problems with 25,000 and 50,000 function evaluations). The second place according to lower GD value is shared by CellDE and SPEA2, which demonstrated best results in three out of twenty cases. The rest of meta-heuristic algorithms,


57

except NSGA-II, did not show their advantages in getting best results. Surprisingly, AbYSS couldn’t cope with Problem1 and exhibited worse metrics values. From all problems, the particular accuracy was achieved by CellDE in ZTD2 problem, i.e, the GD value is equal to 6.83E-6. It should be noted that the differences in term of GD in Viennet2 and Viennet4 problem are not very large, thus indicating a similar ability of the all algorithms to compute Pareto fronts approximation in three-objective problems. However, in two-objective problems this difference is noticeable. The reported results in Table 3.6 shows that MOCell keeps his winner position in thirteen out of twenty tests just with slight changes, i.e. its provided solutions in terms with IGD metric on average are the best in all bi-criteria problems except ZDT3 in both test and Problem1, when 25000 evaluations were performed. CellDE is a leader for solving threecriteria problems, but there is no significant difference among all average values of IGD. NSGAII and SPEA2 obtain competitive results of problem ZDT3 with 25000 and 50000 function evaluations, respectively. The results of spread metric ∆ given in Table 3.8 indicate that MOCell outperforms the other algorithms concerning the diversity of the obtained Pareto fronts in the bi-objective MOPs (it yields the lowest values in seven out of eight bi-objective problems). Here, it reports average ∆ values noticeably lower than other meta-heuristic algorithms, especially for Fonseca, ZDT1 and ZDT2 problems. In group of three-objective problems, the best spread of Viennet2 and Viennet4 solutions is provided by CellDE and SPEA2, respectively. AbYSS, FastPGA, and NSGA-II cannot display superiority in solutions spread-out. Hypervolume metric is used as a measure of both convergence and diversity, thus it should prove the results of the GD, IGD and ∆. The obtained HV values for bi-objective problems (see Table 3.7), with some exceptions, are significantly lower than ones for three-objective problems. When we look at the two objective MOPs, we can see that differences in some cases diminish and it is not easy to decide about a clear winner. According to average HV values, SPEA2 is good as MOCell with regards to Tanaka, ZDT3 and ZDT4 (with 50000 function evaluations), and CellDE yields the same values as MOCell for Fonseca problem. However, the reported results of HV confirm MOCell being advantageous in fourteen of sixteen tests of bi-objective problems. AbYSS having superiority in solving three-objective problems. If we further analyze the HV values of three-objective problems, we observe that CellDE yields the best values. In general, making more function evaluations lightly improves the obtained Pareto fronts in most of the experiments. Pareto frontier of all problems obtained by WS, AW, SLGP and meta-heuristic algorithms are presented in Figures 3.3 - 3.22. In these figures black points denote Pareto optimal solutions, while grey crosses mark dominated solutions.

Avg.

702 3.18E-3 3.23E-3 5.93E-3 5.39E-2 2.59E-4 2.31E-4 3.94E-4

9.62E-4 5.26E-3

Avg.

174 3.97E-4 1.15E-3 1.26E-3 1.13E-2 1.84E-4 1.38E-4 2.85E-4

9.60E-4 5.47E-3


Viennet2 Viennet4

F=50000


Viennet2 Viennet4

3.49E-4 1.95E-4

649 5.85E-4 2.19E-3 2.57E-3 4.73E-2 2.37E-5 3.83E-5 5.31E-5

Std.

3.27E-4 2.89E-4

1950 2.11E-3 3.79E-3 3.52E-3 7.44E-2 3.19E-5 7.98E-5 7.81E-5

Std.

AbYSS

F=25000

Problem

1.02E-3 4.61E-3

3.21E-2 1.81E-5 6.83E-6 4.62E-5 4.25E-1 9.33E-5 8.04E-5 2.02E-4

Avg.

1.01E-3 4.56E-3

3.81E-4 3.42E-4 6.91E-5 7.01E-4 3.77E-1 1.19E-4 1.24E-4 2.62E-4

Avg.

2.85E-4 1.50E-4

2.78E-1 9.55E-6 3.40E-6 3.21E-5 6.82E-1 2.83E-5 2.49E-5 2.56E-5

Std.

2.45E-4 1.52E-4

2.66E-3 1.94E-4 7.50E-5 3.78E-4 3.51E-1 2.79E-5 2.91E-5 3.16E-5

Std.

CellDE

7.02E-1 2.99E-5 3.01E-5 1.97E-5 2.23E-4 3.71E-5 2.47E-5 2.63E-5

Std.

1.14E-1 6.07E-5 4.32E-5 1.74E-5 6.54E-5 3.16E-5 2.64E-5 2.40E-5

Std.

9.27E-4 3.60E-4 5.68E-3 2.11E-4

1.58E-2 1.31E-4 1.11E-4 5.39E-5 8.94E-5 3.00E-4 1.82E-4 1.68E-4

Avg.

8.68E-4 3.16E-4 5.66E-3 2.14E-4

1.65E-1 1.39E-4 1.20E-4 6.21E-5 3.56E-4 2.98E-4 1.82E-4 1.76E-4

Avg.

FastPGA

1.61E-0 5.04E-5 2.21E-5 3.71E-5 1.68E-4 1.07E-5 1.03E-5 2.52E-5

Std.

8.90E-4 5.58E-3

5.73E-3 2.04E-5 1.30E-5 1.64E-5 4.39E-5 5.88E-5 3.35E-5 9.41E-5

Avg.

3.12E-4 1.27E-4

2.95E-2 3.84E-5 5.95E-5 4.72E-5 3.50E-5 7.61E-6 7.83E-6 1.72E-5

Std.

8.34E-4 3.28E-4 5.58E-3 1.48E-4

3.06E-1 9.14E-5 3.76E-5 5.62E-5 2.78E-4 8.91E-5 5.73E-5 1.56E-4

Avg.

MOCell

2.49E-2 5.86E-5 5.00E-5 1.83E-5 6.59E-5 3.68E-5 2.71E-5 2.78E-5

Std.

4.10E-4 2.30E-4

9.20E-0 5.17E-5 3.73E-5 2.16E-5 2.83E-4 3.68E-5 3.18E-5 2.65E-5

Std.

8.87E-4 3.73E-4 5.77E-3 2.24E-4

7.95E-3 1.43E-4 1.24E-4 6.53E-5 8.94E-5 3.20E-4 2.00E-4 1.84E-4

Avg.

8.79E-4 5.77E-3

1.97E-0 1.80E-4 1.60E-4 8.27E-5 4.64E-4 3.25E-4 2.03E-4 1.80E-4

Avg.

NSGA-II

25.2E-0 7.23E-5 3.65E-5 3.27E-5 8.48E-3 2.45E-5 2.30E-5 3.12E-5

Std.

7.44E-2 2.80E-5 3.35E-5 2.24E-5 5.20E-5 2.43E-5 1.73E-5 2.20E-5

Std.

9.06E-4 1.72E-4 4.49E-3 1.63E-4

2.01E-2 6.51E-5 5.92E-5 3.64E-5 8.10E-5 1.74E-4 1.09E-4 9.87E-5

Avg.

9.21E-4 1.64E-4 4.44E-3 1.91E-4

3.63E-0 1.64E-4 1.71E-4 9.30E-5 1.42E-3 1.75E-4 1.19E-4 1.51E-4

Avg.

SPEA2

3.5 Table: Average and standard deviation of generational distance (GD) metric of metaheuristic algorithms

58 CHAPTER 3. ALGORITHMS FOR UNIFORM PARETO SET APPROXIMATION

Avg.

6.98E-0 3.42E-4 3.80E-4 4.49E-4 5.25E-3 4.92E-5 3.91E-5 8.88E-5

8.77E-5 1.59E-4

Avg.

1.55E-0 5.95E-5 7.01E-5 1.04E-4 1.12E-3 4.43E-5 3.31E-5 6.08E-5

8.71E-5 1.66E-4


Viennet2 Viennet4

F=50000


Viennet2 Viennet4

7.90E-6 1.19E-5

5.45E-0 1.16E-4 1.98E-4 1.81E-4 1.20E-3 9.87E-7 4.75E-6 9.35E-6

Std.

6.88E-6 1.04E-5

18.5E-0 2.82E-4 6.68E-4 2.71E-4 2.88E-3 3.46E-6 8.16E-6 1.80E-5

Std.

AbYSS

F=25000

Problem

6.06E-5 1.36E-4

2.86E-3 2.90E-5 2.64E-5 2.33E-5 3.69E-2 4.35E-5 3.28E-5 5.06E-5

Avg.

6.13E-5 1.37E-4

1.15E-4 3.81E-5 2.74E-5 5.48E-5 3.32E-2 4.46E-5 3.46E-5 6.24E-5

Avg.

Std.

2.09E-6 5.49E-6

2.38E-2 3.52E-7 2.59E-7 8.33E-7 2.45E-2 1.08E-6 2.29E-6 6.52E-6

Std.

1.91E-6 5.94E-6

4.03E-4 7.39E-6 1.61E-6 2.36E-5 1.91E-2 1.14E-6 2.67E-6 1.03E-5

CellDE

9.30E-5 1.61E-4

1.56E-3 3.46E-5 3.14E-5 3.78E-5 6.75E-5 5.73E-5 3.92E-5 4.86E-5

Avg.

9.03E-5 1.64E-4

1.55E-2 3.47E-5 3.13E-5 7.30E-5 8.14E-5 5.75E-5 3.90E-5 5.49E-5

Avg.

9.35E-6 1.04E-5

2.76E-3 1.18E-6 1.07E-6 8.37E-5 2.87E-6 2.02E-6 1.42E-6 2.52E-6

Std.

8.23E-6 1.12E-5

5.90E-2 1.27E-6 1.03E-6 1.97E-4 1.75E-5 1.99E-6 1.60E-6 6.08E-6

Std.

FastPGA

8.72E-5 1.84E-4

1.12E-3 2.71E-5 2.47E-5 2.91E-5 5.40E-5 4.05E-5 2.90E-5 3.97E-5

Avg.

8.66E-5 1.87E-4

1.58E-2 2.78E-5 2.49E-5 7.52E-5 6.63E-5 4.11E-5 2.95E-5 4.95E-5

Avg.

8.81E-6 1.38E-5

2.94E-3 1.90E-7 1.21E-7 6.49E-5 8.39E-7 2.06E-7 2.43E-7 4.50E-6

Std.

7.23E-6 1.44E-5

1.05E-1 2.96E-7 2.00E-7 1.73E-4 1.25E-5 2.82E-7 3.73E-7 8.81E-6

Std.

MOCell

9.92E-5 1.64E-4

1.75E-3 3.78E-5 3.41E-5 2.79E-5 7.44E-5 6.13E-5 4.23E-5 5.29E-5

Avg.

9.83E-5 1.63E-4

3.52E-2 3.81E-5 3.45E-5 2.83E-5 1.03E-4 6.18E-5 4.24E-5 5.68E-5

Avg.

1.08E-5 1.14E-5

2.92E-3 2.17E-6 1.49E-6 1.37E-6 3.88E-6 2.53E-6 1.63E-6 2.98E-6

Std.

1.06E-5 1.07E-5

1.25E-1 1.93E-6 1.97E-6 1.64E-6 6.90E-5 2.39E-6 2.09E-6 5.74E-6

Std.

NSGA-II

6.50E-5 1.38E-4

3.13E-3 2.87E-5 2.61E-5 2.23E-5 5.63E-5 4.46E-5 3.21E-5 4.04E-5

Avg.

6.51E-5 1.40E-4

7.01E-2 3.01E-5 4.07E-5 5.52E-5 4.34E-4 4.46E-5 3.25E-5 5.37E-5

Avg.

2.73E-6 5.22E-6

6.66E-3 3.47E-7 3.73E-7 4.77E-7 1.83E-6 9.74E-7 8.31E-7 4.03E-6

Std.

2.03E-6 9.51E-6

3.17E-1 5.52E-7 8.49E-5 1.32E-4 4.87E-4 8.78E-7 1.06E-6 9.05E-6

Std.

SPEA2

3.6 Table: Average and standard deviation of inverted generational distance (IGD) metric of metaheuristic algorithms


59

Avg.

4.07E-2 6.13E-1 2.87E-1 4.35E-1 6.12E-1 3.10E-1 4.01E-1 3.02E-1

9.22E-1 8.26E-1

Avg.

1.76E-1 6.59E-1 3.25E-1 5.10E-1 2.34E-3 3.11E-1 4.02E-1 3.04E-1

9.22E-1 8.27E-1


Viennet2 Viennet4

F=50000


Viennet2 Viennet4

1.14E-3 1.14E-3

2.91E-1 5.56E-3 1.11E-2 4.26E-3 1.56E-2 2.26E-4 2.70E-4 6.87E-4

Std.

1.01E-3 1.18E-3

1.44E-1 2.20E-2 5.20E-2 3.03E-2 5.28E-2 3.21E-4 5.13E-4 9.95E-4

Std.

AbYSS

F=25000

Problem

1.31E-1 1.23E-4 5.30E-5 8.86E-4 3.28E-2 1.22E-4 1.75E-4 4.02E-4

Std.

3.78E-4 6.88E-4

5.04E-2 2.70E-3 1.02E-3 8.93E-3 1.56E-2 1.56E-4 2.64E-4 4.92E-4

Std.

9.25E-1 3.22E-4 8.31E-1 6.84E-4

9.09E-1 6.62E-1 3.28E-1 5.14E-1 7.34E-3 3.12E-1 4.02E-1 3.05E-1

Avg.

9.25E-1 8.31E-1

9.24E-1 6.57E-1 3.27E-1 4.99E-1 2.34E-3 3.12E-1 4.02E-1 3.04E-1

Avg.

CellDE

9.21E-1 8.25E-1

8.05E-1 6.61E-1 3.28E-1 5.15E-1 6.60E-1 3.09E-1 4.01E-1 3.05E-1

Avg.

9.21E-1 8.25E-1

5.24E-1 6.60E-1 3.27E-1 5.14E-1 6.56E-1 3.09E-1 4.01E-1 3.04E-1

Avg.

9.38E-4 1.75E-3

2.28E-1 2.18E-4 1.86E-4 4.72E-4 9.85E-4 3.70E-4 2.01E-4 1.82E-4

Std.

9.89E-4 1.48E-3

3.67E-1 2.35E-4 2.33E-4 5.22E-3 3.14E-3 3.83E-4 2.01E-4 2.98E-4

Std.

FastPGA

9.22E-1 8.26E-1

8.54E-1 6.62E-1 3.29E-1 5.16E-1 6.61E-1 3.12E-1 4.03E-1 3.06E-1

Avg.

9.22E-1 8.26E-1

6.92E-1 6.61E-1 3.28E-1 5.15E-1 6.58E-1 3.12E-1 4.03E-1 3.05E-1

Avg.

7.25E-4 1.03E-3

1.87E-1 2.79E-5 2.45E-5 4.89E-4 5.32E-4 6.81E-5 4.42E-5 1.23E-4

Std.

8.25E-4 1.07E-3

3.09E-1 1.94E-4 3.05E-4 9.65E-4 2.34E-3 1.10E-4 9.34E-5 3.36E-4

Std.

MOCell

9.20E-1 8.24E-1

7.91E-1 6.60E-1 3.27E-1 5.15E-1 6.59E-1 3.08E-1 4.01E-1 3.04E-1

Avg.

9.20E-1 8.24E-1

4.40E-1 6.59E-1 3.26E-1 5.15E-1 6.54E-1 3.08E-1 4.01E-1 3.04E-1

Avg.

1.37E-3 1.62E-3

2.44E-1 3.19E-4 2.28E-4 9.16E-5 9.68E-4 4.31E-4 2.42E-4 2.18E-4

Std.

1.41E-3 1.63E-3

3.66E-1 3.65E-4 3.97E-4 1.61E-4 3.95E-3 4.08E-4 2.57E-4 2.96E-4

Std.

NSGA-II Avg.

9.25E-1 8.30E-1

7.18E-1 6.616E-1 3.28E-1 5.16E-1 6.61E-1 3.11E-1 4.02E-1 3.06E-1

Avg.

9.25E-1 8.30E-1

4.22E-4 7.42E-4

2.48E-1 1.10E-4 9.82E-5 7.43E-5 7.73E-4 2.02E-4 1.31E-4 2.61E-4

Std.

3.62E-4 8.08E-4

3.45E-1 2.87E-4 4.03E-3 7.81E-4 1.22E-2 2.12E-4 1.64E-4 5.00E-4

Std.

SPEA2

4.08E-1 6.60E-1 3.26E-1 5.14E-1 6.48E-1 3.11E-1 4.02E-1 3.05E-1

3.7 Table: Average and standard deviation of hypervolume (HV) metric of metaheuristic algorithms

60 CHAPTER 3. ALGORITHMS FOR UNIFORM PARETO SET APPROXIMATION

Avg.

1.01E-0 1.65E-1 2.24E-1 7.66E-1 5.46E-1 1.06E-1 4.26E-1 7.57E-1

3.04E-1 5.24E-1

F=50000


Viennet2 Viennet4

3.75E-2 2.86E-2

1.44E-1 7.53E-2 1.58E-1 8.60E-2 3.23E-1 1.30E-2 9.78E-3 4.33E-2

Std.

3.46E-2 2.63E-2

3.34E-1 5.36E-1

Viennet2 Viennet4

Std.

9.83E-1 1.11E-1 6.00E-1 1.54E-1 5.61E-1 2.20E-1 9.03E-1 6.98E-2 1.02 1.90E-1 1.36E-1 1.45E-2 4.42E-1 1.55E-2 8.96E-1 4.96E-2

Avg.

AbYSS


F=25000

Problem

8.08E-2 1.23E-2 1.27E-2 7.91E-3 8.45E-2 1.14E-2 1.21E-2 2.89E-2

Std.

1.74E-2 1.83E-2

3.82E-2 2.21E-2 1.58E-2 2.51E-2 8.04E-2 1.26E-2 1.25E-2 3.22E-2

Std.

1.55E-1 1.45E-2 4.50E-1 2.01E-2

5.15E-1 1.69E-1 1.71E-1 7.36E-1 7.70E-1 1.62E-1 4.71E-1 7.26E-1

Avg.

1.66E-1 4.48E-1

5.03E-1 2.17E-1 1.89E-1 7.61E-1 8.21E-1 1.78E-1 4.82E-1 7.69E-1

Avg.

CellDE

4.28E-1 5.86E-1

6.64E-1 3.18E-1 3.22E-1 7.28E-1 3.18E-1 3.36E-1 5.19E-1 7.47E-1

Avg.

4.18E-1 5.82E-1

9.27E-1 3.14E-1 3.16E-1 7.30E-1 3.05E-1 3.41E-1 5.21E-1 7.76E-1

Avg.

4.76E-2 3.61E-2

2.08E-1 2.70E-2 2.78E-2 1.09E-2 2.72E-2 3.07E-2 2.09E-2 2.32E-2

Std.

4.47E-2 4.24E-2

2.31E-1 2.75E-2 3.10E-2 1.71E-2 2.62E-2 3.32E-2 2.44E-2 3.02E-2

Std.

FastPGA

2.47E-1 5.08E-1

4.29E-1 5.89E-2 6.43E-2 7.03E-1 6.87E-2 6.29E-2 4.11E-1 6.66E-1

Avg.

2.54E-1 5.07E-1

8.01E-1 7.66E-2 7.93E-2 7.05E-1 1.11E-1 7.62E-2 4.13E-1 7.22E-1

Avg.

3.42E-2 3.03E-2

2.72E-1 1.16E-2 1.40E-2 1.55E-2 1.08E-2 9.08E-3 2.46E-3 2.18E-2

Std.

3.66E-2 2.86E-2

2.76E-1 1.10E-2 1.17E-2 3.71E-3 2.19E-2 9.93E-3 4.86E-3 3.07E-2

Std.

MOCell

2.15E-1 3.31E-2 3.12E-2 1.57E-2 3.58E-2 3.25E-2 2.50E-2 2.71E-2

Std.

1.90E-1 3.28E-2 3.38E-2 1.44E-2 3.79E-2 3.76E-2 2.19E-2 2.71E-2

Std.

4.54E-1 4.68E-2 6.00E-1 3.87E-2

7.31E-1 3.76E-1 3.81E-1 7.45E-1 4.12E-1 4.00E-1 5.67E-1 7.87E-1

Avg.

4.59E-1 4.70E-2 6.07E-1 4.03E-2

1.02E-0 3.73E-1 3.78E-1 7.48E-1 3.98E-1 3.99E-1 5.66E-1 8.09E-1

Avg.

NSGA-II

3.8 Table: Average and standard deviation of spread (∆) metric of metaheuristic algorithms

2.23E-1 1.41E-2 4.54E-2 1.60E-2 1.49E-1 1.41E-2 1.09E-2 3.26E-2

Std.

2.57E-1 1.29E-2 1.39E-2 3.30E-3 1.26E-2 1.47E-2 9.89E-3 2.21E-2

Std.

2.00E-1 1.99E-2 4.30E-1 2.04E-2

9.47E-1 1.46E-1 1.39E-1 7.04E-1 1.20E-1 1.38E-1 4.35E-1 6.85E-1

Avg.

2.03E-1 2.22E-2 4.32E-1 2.20E-2

1.10E-0 1.50E-1 1.61E-1 7.12E-1 2.92E-1 1.37E-1 4.40E-1 7.51E-1

Avg.

SPEA2


61

CHAPTER 3. ALGORITHMS FOR UNIFORM PARETO SET APPROXIMATION Pareto Set

SLGP

0.8

0.8

0.6

0.6 f2

1

f2

1

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

f1

0.6

0.8

1

0.6

0.8

1

f1

AW

WS

0.8

0.8

0.6

0.6 f2

1

f2

1

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

f1

f1

3.3 Fig.: Pareto set and solutions of problem ZDT1 obtained by WS, AW and SLGP

AbYSS

CellDE

0.8

0.8

0.6

0.6 f2

1

f2

1

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

0.8

1

0

0.2

0.4

0.8

1

0

0.2

0.4

0.8

1

f1 FastPGA

0.6

0.6

0.6 f1 MOCell

f2

1 0.8

f2

1 0.8

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

0

1

f1 NSGAII 1

0.8

0.8

0.6

0.6

0.6 f1 SPEA2

f2

1

f2

62

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6 f1

0.8

1

0

0.6 f1

3.4 Fig.: Solutions of problem ZDT1 obtained by metaheuristic algorithms

63


Pareto Set

SLGP

0.8

0.8

0.6

0.6 f2

1

f2

1

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

f1

0.6

0.8

1

f1

3.5 Fig.: Pareto set and solutions of problem ZDT2 obtained by SLGP

AbYSS

CellDE

0.8

0.8

0.6

0.6

dominated optimal

f2

1

f2

1

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

0.8

1

0

0.2

0.4

0.8

1

0

0.2

0.4

0.8

1

f1 FastPGA 0.8

0.8

0.6

0.6 f2

1

f2

1

0.6 f1 MOCell

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

0

1

f1 NSGAII 0.8

0.8

0.6

0.6 f2

1

f2

1

0.6 f1 SPEA2

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6 f1

0.8

1

0

0.6 f1


64


SLGP 1

0.5

0.5

0

0

f2

f2

Pareto Set 1

−0.5 −1

−0.5

0

0.2

0.4

0.6

0.8

−1

1

dominated optimal 0

0.2

0.4

f1

0.6

0.8

1

f1

3.7 Fig.: Pareto set and solutions of problem ZDT3 obtained by SLGP

CellDE 1

0.5

0.5

0

0

f2

f2

AbYSS 1

−0.5 −1

−0.5

0

0.2

0.4

0.6

0.8

−1

1

0

0.2

0.4

0.8

1

0

0.2

0.4

0.8

1

0

0.2

0.4

0.8

1

1

1

0.5

0.5

0

0

f2

f2

f1 FastPGA

−0.5 −1

−0.5

0

0.2

0.4

0.6

0.8

−1

1

1

1

0.5

0.5

0

0

f2

f2

f1 NSGAII

−0.5 −1

0.6 f1 MOCell

0.6 f1 SPEA2

−0.5

0

0.2

0.4

0.6 f1

0.8

1

−1

0.6 f1


65


0.5

f2

0.5

0

f2

SLGP 1

0

0.5 f1 AW

0

1

1

1

0.5

0.5

0

f2

f2

Pareto Set 1

0

0.5 f1

0

1

0

0.5 f1 WS

1

0

0.5 f1

1

3.9 Fig.: Pareto set and solutions of problem ZDT4 obtained by WS, AW, and SLGP AbYSS

CellDE

0.8

0.8

0.6

0.6 f2

1

f2

1

0.4

0.4

0.2

0.2

0

0

0.2

0.4 f1 FastPGA

0.6

0

0.8

0.8

0.8

0.6

0.6

0.2

0.4

0.6 f1 MOCell

0.8

1

0

0.2

0.4

0.8

1

0

0.2

0.4

0.8

1

f2

1

f2

1

0

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6 f1 NSGAII

0.8

0

1

0.8

0.8

0.6

0.6 f2

1

f2

1

0.6 f1 SPEA2

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6 f1

0.8

1

0

0.6 f1


66


SLGP 30

25

25

20

20 f2

f2

Pareto Set 30

15

15

10

10

5

5

0

0

2

4 f1

6

0

8

0

2

30

25

25

20

20

15

10

5

5 0

2

4 f1

8

6

8

15

10

0

6

WS

30

f2

f2

AW

4 f1

6

0

8

0

2

4 f1

3.11 Fig.: Pareto set and solutions of problem Problem1 obtained by WS, AW, and SLGP AbYSS

CellDE

20

20

dominated optimal

f2

30

f2

30

10 0

10

0

2

4 f1 FastPGA

6

0

8

20

20

2

4 f1 MOCell

6

8

0

2

4 f1 SPEA2

6

8

0

2

4 f1

6

8

f2

30

f2

30

0

10 0

10

0

2

4 f1 NSGAII

6

0

8

20

20 f2

30

f2

30

10 0

10

0

2

4 f1

6

8

0

3.12 Fig.: Solutions of problem Problem1 obtained by metaheuristic algorithms

67


SLGP 0

−2

−2

−4

−4

−6

−6

f2

f2

Pareto Set 0

−8

−8

−10

−10

−12 −20

−18

−16

−14

−12 −20

−18

f1

−16

−14

f1

3.13 Fig.: Pareto set and solutions of problem Kursawe obtained by SLGP

CellDE 0

−2

−2

−4

−4

−6

−6

f2

f2

AbYSS 0

−8

−8

−10

−10

dominated optimal

−12 −20

−18

−12 −20

−18

−16

−14

0

0

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10

−12 −20

−18

−16

−12 −20

−14

−18

0

0

−2

−2

−4

−4

−6

−6

−8

−8

−10

−10 −18

−16 f1

−14

−16

−14

−16

−14

f1 SPEA2

f2

f2

f1 NSGAII

−12 −20

−16 f1 MOCell

f2

f2

f1 FastPGA

−14

−12 −20

−18 f1

3.14 Fig.: Solutions of problem Kursawe obtained by metaheuristic algorithms

68


Pareto Set

SLGP

0.8

0.8

0.6

0.6 f2

1

f2

1

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

0

1

0

0.2

f1

0.4

0.6

0.8

1

f1

3.15 Fig.: Pareto set and solutions of problem Fonseca obtained by SLGP

AbYSS

CellDE

0.8

0.8

0.6

0.6 f2

1

f2

1

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

0

1

dominated optimal 0

0.2

0.4

0.8

1

0

0.2

0.4

0.8

1

0

0.2

0.4

0.8

1

f1 FastPGA 0.8

0.8

0.6

0.6 f2

1

f2

1

0.6 f1 MOCell

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

0

1

f1 NSGAII 0.8

0.8

0.6

0.6 f2

1

f2

1

0.6 f1 SPEA2

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6 f1

0.8

1

0

0.6 f1

3.16 Fig.: Solutions of problem Fonseca obtained by metaheuristic algorithms

69


SLGP

1

1

0.8

0.8

0.6

0.6

f2

f2

Pareto Set

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6 f1

0.8

0

1

dominated optimal 0

0.2

0.4

0.6 f1

0.8

1

3.17 Fig.: Pareto set and solutions of problem Tanaka obtained by SLGP

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 0

0.2

0.4

0.6 0.8 f1 FastPGA

0

1

1

1

0.8

0.8

0.6

0.6

f2

f2

f2

1

0

0.4

0.4

0.2

0.2

0

f2

CellDE

0

0.2

0.4

0.6 f1 NSGAII

0.8

0

1

1

1

0.8

0.8

0.6

0.6

f2

f2

AbYSS

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6 f1

0.8

1

0

dominated optimal 0

0.2

0.4

0.6 f1 MOCell

0.8

1

0

0.2

0.4

0.8

1

0

0.2

0.4

0.8

1

0.6 f1 SPEA2

0.6 f1

3.18 Fig.: Solutions of problem Tanaka obtained by metaheuristic algorithms

70


WS

−12

−12

−12.5

−12.5

f3

f3

Pareto Set

−13 −16 −16.5 f2

−17

3

4

3.5 f1

−13 −16

4.5

−16.5 f2

−17

−12

−12

−12.5

−12.5

−13 −16 −16.5 f2

−17

3

4

4.5

SLGP

f3

f3

AW

3

3.5 f1

4

3.5 f1

−13 −16

4.5

−16.5 f2

−17

3

3.5 f1

4

4.5

3.19 Fig.: Solutions of problem Viennet2 obtained by WS, AW, and SLGP

AbYSS CellDE dominated optimal

−12

−12.5

f3

f3

−12

−13 −16 −16.5 f2

−17 3

3.5 f1

4

−12.5 −13 −16

4.5

−16.5 f2

−12

−12

−12.5

−12.5

−13 −16 −16.5 f2

−17 3

3.5 f1

4

−16.5 f2

−12.5

−12.5

f3

f3

−12

−13 −16 f2

4.5

−17 3

3.5 f1

4

4.5

SPEA2

−12

−17 3

4

−13 −16

4.5

NSGAII

−16.5

3.5 f1

MOCell

f3

f3

FastPGA

−17 3

3.5 f1

4

4.5

−13 −16 −16.5 f2

−17 3

3.5 f1

4

4.5

3.20 Fig.: Solutions of problem Viennet2 obtained by metaheuristic algorithms

71


3.21 Fig.: Pareto set and solutions of problem Viennet4 obtained by WS, AW, and SLGP

AbYSS

CellDE dominated optimal

20

f3

f3

25 20

15 10 −11 −12 f2

−13

4

6

8

10 −11 −12

f1

f2

6 −13

20

10 −11 −12 f2

−13

4

6

8

10 −11 −12 f2

f1

−13

f3

f3

10 −11 −12 −13

4

6

8

f1

SPEA2

20

f2

f1

20

NSGAII

4

4 MOCell

f3

f3

FastPGA

8

6 f1

8

20

10 −11 −12 f2

−13

4

6

8

f1

3.22 Fig.: Solutions of problem Viennet4 obtained by metaheuristic algorithms

72


Figures 3.3 and 3.9 show, that Pareto frontiers obtained by AW and SLGP algorithms are well distributed and confirms that uniformly changing weights in WS methods do not produce an uniformly distributed Pareto frontier. In Figures 3.4 and 3.10 we can observe that solutions generated by meta-heuristic algorithms are non-dominating but low GD and IGD metric values state that their Pareto frontiers are close to the true one. In addition, the quality of solutions obtained by AbYSS, FastPGA, and NSGA-II are lower than ones of CellDE, MOCell, and SPEA2. As ZDT2 problem is concave, only composite Pareto frontier and solutions generated by SLGP are given in Figure 3.5. Although, Pareto frontier obtained by SLGP looks well distributed, the MOCell algorithm provided slightly better values of ∆ and IGD. ZDT3 problem is neither convex nor concave. The best distribution of solutions of this problem is achieved by SLGP, but MOCell is the vinner in term of HV and IGD. We can therefore conclude from these experiments that our proposed AW and SLGP algorithms are able to generate more or less uniformly distributed Pareto frontier. SLGP algorithm proved to be efficient to solve not only convex problems, but non-convex ones as well. Although, MOCell and CellDE algorithms demonstrated theirs good performance in solving bi-objective and three-objective problems, respectively, they require a large number of function evaluations and generates a large number of non-dominating solutions.

3.4.4.

QUALITY OF PARETO SET APPROXIMATIONS

In order to investigate an influence of generated solutions on uniformity of distribution, we solve two problems by WS, AW and SLGP several time with different step size/number of solutions and calculate two performance metrics, HV and ∆. First, 16 solutions are obtained by each method for both problems. Then 25, 49, and 81 solutions are calculated and computational results of problem ZDT1 and Viennet2 are provided in Table 3.9. Looking at the estimates of problem Viennet2 we can notice that values of HV obtained by all three methods improve while increasing number of solutions. This fact is obvious and can be justified by the mathematical equation defining this metric. However, the estimates of spread metric do not give so clear pattern. While ∆ values of Viennet2 problem solved by AW method is the lowest when having 25 solutions and highest with 16 solutions, then SLGP gets the best spread-out of 49 solutions and the worse of 16 solutions. However, SLGP keeps almost the same ∆ value (around 0.27) with regards of all different number of solutions. This means that distribution of obtained Pareto subset do not depend on number of solutions composing this subset. This is confirmed by the Figures 3.23 and 3.24, that point out that quality of the Pareto frontiers obtained by AW and SLGP are not significantly influenced by increasing solution amount. After interpreting given data and figures we can conclude that quality of uniformity of generated Pareto subset do not depend on amount of generated solutions and increasing

73


3.9 Table: Estimates of performance metrics obtained with different step size/number of solutions of problem ZDT1 and Viennet2 ZDT1

Viennet2

N = 16

HV

∆

WS AW SLGP

0.5821 0.6278 0.6272

0.6487 0.1770 0.2753

N = 25

HV

∆

WS AW SLGP

0.6063 0.6443 0.6441

0.7405 0.1892 0.2767

N = 49

HV

∆

WS AW SLGP

0.6397 0.6540 0.6559

0.7134 0.3659 0.2774

N = 81

HV

∆

WS AW SLGP

0.6481 0.6540 0.6596

0.6955 0.6475 0.2776

HV

∆

0.9034 1.1067 0.9078 0.8878 0.7856 1.1104 HV

∆

0.9116 1.1529 0.9131 0.5475 0.8434 0.9626 HV

∆

0.9259 1.1184 0.9218 0.6663 0.9021 0.8810 HV

∆

0.9302 0.9989 0.9299 0.6795 0.9190 0.7595

number of solutions not necessarily will results in better spread-out of obtained points.

74


0.5

0.5 f1 WS(25)

0

1

0

0.5 f1 AW(25)

0

1

1

0.5

0.5

0.5

0

0.5 f1 WS(49)

0

1

f2

1

f2

1

0

0.5 f1 AW(49)

0

1

1

1

0.5

0.5

0.5

0

0

0.5 f1 WS(81)

0

1

f2

1

f2

0

0.5 f1 AW(81)

0

1

1

1

1

0.5

0.5

0.5

0

0

0.5 f1

1

0

f2

f2

0

f2

0.5

f2

0.5

0

f2

SLGP(16) 1

0

f2

AW(16) 1

f2

f2

WS(16) 1

0

0.5 f1

1

0

0

0.5 f1 SLGP(25)

1

0

0.5 f1 SLGP(49)

1

0

0.5 f1 SLGP(81)

1

0

0.5 f1

1

3.23 Fig.: Distribution of Pareto subsets composed by different number of ZDT1 solutions obtained by WS, AW, and SLGP

75


AW(16) −12

−12.5

−12.5

−12.5

−16.5 f2 −17 3

4

−13 −16

5

−16.5 f2 −17 3

f1

4

−13 −16

5

−16.5 f2 −17 3

f1

AW(25) −12

−12

−12.5

−12.5

−12.5

−16.5 f2 −17 3

4

f3

−12

−13 −16

−13 −16

5

−16.5 f2 −17 3

f1

WS(49)

4

−13 −16

5

−16.5 f2 −17 3

f1

AW(49)

−12.5

−12.5

−12.5

−16.5 f2 −17 3

4

f3

−12

f3

−12

−13 −16

5

−16.5 f2 −17 3

f1

WS(81)

4

−16.5 f2 −17 3

f1

−12

−12.5

−12.5

−12.5

f3

−12

4 f1

5

−13 −16 −16.5 f2 −17 3

5

f1

4

5

f1

SLGP(81)

−12

−16.5 f2 −17 3

4

−13 −16

5

AW(81)

−13 −16

5

SLGP(49)

−12

−13 −16

4 f1

SLGP(25)

f3

f3

WS(25)

f3

f3

−12

−13 −16

f3

SLGP(16)

−12 f3

f3

WS(16)

4 f1

5

−13 −16 −16.5 f2 −17 3

4

5

f1

3.24 Fig.: Distribution of Pareto subsets composed by different number of Viennet2 solutions obtained by WS, AW, and SLGP

76

3.5.


CHAPTER CONCLUSIONS

In this chapter, we proposed two approaches to generate well distributed Pareto sets. The adjustable weights method is advantageous because it is free of weights selection different to weighted sum method; they are iteratively adopted within the algorithm. In addition, unlike priori methods, it does not incorporate information about importance of criteria because weights play the role of parameters defining points in the Pareto set corresponding to the solutions of parametric single criteria problems. The successive lexicographic goal programming algorithm demonstrated its efficiency and suitability to provide well distributed Pareto set approximations. Its main superiority over adjustable weights algorithm is capability to cope with non-convex objective functions. Moreover, it is more flexible because we can choose not whole decision space, but some particular range of interest for approximation. Unlike meta-heuristic algorithms, both developed algorithms for multi-objective problems provide only Pareto optimal solutions. Moreover, they are simple to implement.

chapter 4 OPTIMAL PORTFOLIO SELECTION 4.1.

INTRODUCTION

Risk plays an important role in modern finance, including risk management, capital asset pricing and portfolio optimization. Managing financial risk requires to make decisions about risks that are acceptable versus those that are not. Organizations manage financial risk using a variety of strategies and products. It is important to understand how these products and strategies work to reduce risk within the context of the organization’s risk tolerance and objectives [Horcher, 2005]. It is crucial to select a correct measures of risk, because they have a crucial role in optimization under uncertainty, especially in coping with the losses that might be incurred in finance or the insurance industry [Rockafellar and Uryasev, 2002]. The original mean-risk portfolio formulation introduced by Markowitz [Markowitz, 1952] has provided the fundamental basis for the development of a large part of the modern financial theory applied to the single-period portfolio optimization problem (also known as buy-and-hold portfolio strategy). Markowitz’s original model only considers an investment in one period: the investor decides on his portfolio at the beginning of the investment period (for instance, a year) and patiently waits without intermediate intervention. However, a drawback of this basic approach is that it is tuned to a single period, and it can therefore provide shortsighted strategies of investment, if applied repeatedly over many subsequent periods. To overcome this issue, one may formulate from the beginning the allocation problem over an horizon composed of multiple periods (T -1 periods), with the goal of minimizing the total risk over the investment path, while satisfying constraints on the portfolio composition and on desired expected return at all the intermediate stages. Seminal contributions to multi-stage decisions in finance have been given in Merton [Merton, 1971], where an approach based on continuous-time dynamic programming is proposed. However,

78

CHAPTER 4. OPTIMAL PORTFOLIO SELECTION

the dynamic programming approach is impractical for actual numerical implementation, due to the curse of dimensionality. It was reported in [Brandt, 1999, Brennan et al., 1997] that incorporating more than a few state variables in the dynamic programming formulation leads to unworkable problem size. This can be a reason of most multiperiod models encountered in the literature to be two periods models with only a few securities. For example a meanvariance discrete-time problem is reduced to a control problem with only one state variable in [Infanger, 2006], under the hypotheses of no transaction costs, no composition constraints and serially independent returns. It should be noted that the presence of constraints on portfolio composition and/or of transaction costs makes the problem harder from the computational viewpoint. In this chapter we experimentally investigate several evolutionary multi objective optimization methods and compare their efficiency in problems of portfolio selection with the efficiency of specially tailored method of adjustable weights and SLGP. Test problems are based on standard portfolio quality criteria. We do not concern here in match between analytical properties of the criteria functions and such properties favorable for the considered methods; we believe, however that general (global) structure of multi-criteria portfolio selection problem will be invariant with respect to switching from criteria defined by simple analytical formula to criteria defined by complicated numerical methods. This chapter is organized as follows. Section 4.2. presents measures of risk and weights their merits and demerits. In section 4.3. we discuss about transaction costs modeling, their impact on portfolio diversification and rebalancing. Section 4.4.1. presents the single period portfolio approach, while next section outlines more sophisticated multiperiod stochastic portfolio optimization and future trends. Finally, the obtained results of portfolio optimization problems and conclusions are given in Section 4.5. and 4.6., respectively.

4.2.

RISK MEASURES

Risk is defined as a possibility of suffering harm, loss or hazard or as the effect of uncertainty on objectives. Thus, risk is something that usually can be controlled whereas uncertainty is beyond any control. Risk management is the process of achieving a desired return/profit taking into considerations of risks, through a particular strategy. Most often the financial risk management is defined as a process dealing with the uncertainties resulting from financial markets. According to Tapiero [Tapiero, 2004] risk management consists of altering the states a system may reach in a desirable manner (financial, portfolio, cash flow etc.), and their probabilities or reducing their consequences to planned or economically tolerable levels. In the pre-Markowitz era financial risk was considered as a correcting factor of expected return, and risk-adjusted returns were defined on an ad-hoc basis [Szegö, 2002]. The main

79

4.2. RISK MEASURES

innovation introduced by Markowitz is to measure the risk of a portfolio via the joint distribution of returns of all assets [Markowitz, 1952]. Since then, several other measures for optimal portfolio selection have been proposed. For the case of returns distributed as discrete random variables, these models have the relevant advantage to be LP solvable [Guastaroba et al., 2009]. Konno and Yamazaki [Konno and Yamazaki, 1991] proposed mean absolute deviation as LP computable measure which can be seen as approximations to the variance. Recently, the shortfall or quantile measures are gaining more popularity in various financial applications [Mansini et al., 2003, 2005]. Examples of such measures are the CVaR introduced by Rockafellar and Uryasev [Rockafellar and Uryasev, 2000] and the Worst Realization analyzed by Young [Young, 1998]. The other type of measures that have found application in various financial optimization problems is safety measures [Guastaroba et al., 2009]. Mansini et al. [Mansini et al., 2005] introduced some portfolio optimization models based on the use of multiple CVaR risk measures to allow a better modeling of the risk aversion. Let’s us introduce the most popular risk measures: volatility, value at risk (VaR), and conditional value at risk (CVaR). Financial risk has been traditionally associated with the statistical uncertainty on the final outcome. The traditional measure of risk associated to a given investment is volatility. The portfolio volatility σp is the standard deviation defined as follows: σp =

√∑ ∑ i

ωi ωj σi σj ρij

(4.1)

j

where ρij is the correlation between asset i and j, ωi is the weighting of component asset i and σi is the covariance of asset i. However, the fact that risks are often described using volatility is based on the idea that distribution of price changes is Gaussian. This measure has several major drawbacks. First, here losses and profit play symmetric roles, but it comes out of notion of risk. Second, the presence of extreme events in the financial time series can results in a very bad empirical determination of the variance. From both a fundamental point of view and for a better financial risk control, other measure of risk is necessary. In the context of portfolio optimization, risk is a characteristic of the portfolio return distribution that is disliked by any rational investor. The introduction of the notation of Value at Risk (VaR) filled a gap in the financial industry’s demand to estimate the extreme risk. VaR is a statistical measure of the possible portfolio loss over a predefined period. The main idea of VaR is to provide the information about possible portfolio losses implied by the left hand side tail of the return distribution, especially when this distribution is not normal. The definition of VaR is closely related to the probability distribution of portfolio returns at the end of the investment period t. If portfolio change in value over this interval is ∆V t ,

80


then with chosen confidence level β, VaRtβ is defined as P(∆V t < −VaRtβ ) = 1 − β

(4.2)

Here P(a) is the probability of occurrence for the event a. In risk management applications, β has usually value of 95% or 99%. It can be easily seen that VaR is nothing else than the maximum portfolio loss incurred if we exclude the worst 1 − β cases. The use of VaR as the measure of risk become de facto norm in risk management. However, it suffers from being unstable and difficult to work with numerically when losses are not ’normally” distributed. Moreover, it doesn’t satisfy the ’coherent risk measures” properties, i.e. it lacks convexity [Artzner et al., 1999]. As a result, anyone who uses this measure will not have the initiative to diversify their portfolio. Finally, it has a very serious shortage, i.e. it provides no handle on the extent of the losses that might be suffered beyond the threshold amount indicated by this measure. As an alternative measure to VaR hat does quantify the losses that might be encountered in the tail, the Conditional Value at Risk (CVaR) was introduced by Rockafellar and Uryasev [Rockafellar and Uryasev, 2000]. It is an appropriate measure of risk fulfilling the properties that a good non-symmetric risk measure should have, i.e. properties like coherence or convexity[Doege et al., 2006]. It has gained increasing acceptance that goes further than simple mean-variance framework. Let the random variable l(x, Y) denote the loss function of a decision variable x ∈ Rn , which can be seen as a portfolio, and a given random vector Y ∈ Rm representing the future values of stochastic variables such as e.g., spot price with distribution p(Y). Let VaRβ denote the β - quantile of the induced loss & profit distribution. The risk measure CVaR with confidence level β is defined by the conditional expectation for any x ∈ X CVaRβ (x) = E[l(x, Y)|l(x, Y) ≥ VaRβ ] ∫ −1 = (1 − β) l(x, Y)p(Y)dY

(4.3)

l(x,Y)≥VaRβ (x)

Its graphical interpretation is given in Figure (4.1). CVaR is nothing else than the expectation of the portfolio loss, when the expectation is calculated conditional on the loss exceeding the β-VaR of the portfolio. CVaR goes one step beyond in describing the tail of the distribution; it measures the first moment of the distribution behind the VaR. It is proved in [Rockafellar and Uryasev, 2000] that this CVaR formulation can be characterized by the convex optimization problem [ −1

]

∫

CVaRβ (x) = minα∈R α + (1 − β)

[l(x, Y) − α] p(Y)dY +

Y∈Rn

(4.4)

81

4.2. RISK MEASURES

4.1 Fig.: Graphical interpretation of CVaR for given x ∈ X Convexity guarantees that if a minimum CVaR portfolio can be found, then this portfolio is optimal. In addition, an investor minimizing CVaR would minimize the VaR, because CVaR is equal or greater than VaR [Sarykalin et al., 2008] CVaRβ (x) ≥ VaRβ (x)

(4.5)

If we generate J scenarios, j = 1, . . . , J with realizations ω1 , . . . , ωJ of the random variable Y, the function in Eq. (4.4) can be approximated by a convex piecewise linear function in α: [ ] J ∑ 1 CVaRβ (x) = minα∈R α + [l(x, ωj ) − α]+ (4.6) J(1 − β) j=1 Due to its all advantages CVaR has been chosen as a measure of risk in portfolio optimization in [Rockafellar and Uryasev, 2000, Doege et al., 2006]. For more details about VaR and CVaR, their advantages and disadvantages we point to [Sarykalin et al., 2008]. In a field of finance the risk can be controlled by using heading to offset losses that can occur and by risk reduction using diversification to reduce exposure risks [Liu and Wu, 2007]. Diversification is an important tool in managing financial risks. A diversified portfolio composed of different assets with small mutual correlations is less risky because the gains of some of the assets more or less compensate the loss of the others [Bouchaud and Potters, 2000]. Although the risk of loss still exists, diversification may reduce the opportunity for large adverse outcomes. Diversification, typically, reduces the frequency of best-case outcomes as well. Usually, to measure the diversification the covariance structure of investors’ portfolios is employed. To estimate the extent of diversification, normalized portfolio variance (NV), deviation from the market portfolio and the total number in the portfolio are used in [Goetzmann and Kumar, 2008]. The other not surprising way to increase the diversity of portfolio is

82


to minimize the maximum investment proportion weight [Steuer et al., 2008] or to maximize the entropy [Fang et al., 1997, Jana et al., 2009] that is given as follows S(x) = −

∑

xi logxi

(4.7)

i

The topic of diversification measures and their validity is very well delighted in [Robins and Wiersema, 2003].

4.3.

TRANSACTION COSTS

Standard asset-allocation models generally ignore transaction costs and other costs related to portfolio and allocation revisions. However, the effect of transaction costs is far from insignificant. On the contrary, if transaction costs are not taken into consideration, they can eat a significant part of the returns. It is known that a consumer who ignores realistic transaction costs, and trades continuously, would end up bankrupt. Whether transaction costs are handled efficiently or not by the portfolio or fund manager can therefore make all the difference in attempting to outperform the peer group or a particular benchmark [Fabozzi et al., 2007]. Transaction costs refer to any costs incurred while implementing an investment decision. The inclusion of transaction costs in the portfolio selection problem may present a challenge to the portfolio manager, but is an important practical consideration [Fabozzi et al., 2007]. Therefore, it is to be expected that the inclusion of transaction costs in asset-allocation models will result in a reduced amount of trading and rebalancing. Due to sudden changes in the market trend an investor might prefer to rebalance her/his portfolio composition to possibly reduce losses or better exploit returns growth. Such adjustments might be desirable even if implying additional costs [Guastaroba et al., 2009]. When transaction costs are present, or additional constraints are imposed, the optimal policy is very difficult to compute [Skaf and Boyd, 2008]. Portfolio optimization problems with transaction costs have been widely studied by various authors, e.g., [Akian et al., 2001, Davis and Norman, 1990, Morton and Pliska, 1995, Pliska and Suzuki, 2004, Taksar et al., 1988]. Taksar et al. [Taksar et al., 1988] formulate portfolio optimization as a singular control problem. Their study is extended by Akian et al.[Akian et al., 2001] to a multi-dimensional market model case. Singular control is also applied to portfolio optimization problems in [Davis and Norman, 1990] and many other papers. Morton and Pliska [Morton and Pliska, 1995] study portfolio optimization with fixed transaction costs formulated as an impulsive control problem, while reference [Tamura, 2006] consider both fixed and proportional transaction costs. Breuer and Jandacka [Breuer and Jandacka, 2008] solved portfolio selection problem with transaction costs under expected

83

4.3. TRANSACTION COSTS

shortfall constraints. In [Muthuraman and Kumar, 2004], the portfolio optimization problem in the presence of both transaction costs has been converted to a sequence of fixed boundary problems and an innovative update procedure was used to solve this problem. Since some transaction costs as brokerage commissions, fees and taxes are observable directly, let’s assume that the transaction costs are the sum of the transaction costs associated with each trade n ∑ C(ω) = cj (ωj ), (4.8) j=1

where cj is the transaction cost function for asset j. The transaction costs can be modelled in different ways. One way to model transaction costs is a linear one, with the costs for each transaction proportional to the traded amount  y c j sell ci (ωi ) = z c j buy

if we are selling asset j

(4.9)

if we are buying asset j

where yj ≥ 0 and zj ≥ 0. Then transaction cost function Cj is represented as cj (ωj ) = yj csell + zj cbuy

(4.10)

In practice, transactions often involve both fixed and proportional costs. The introduction of fixed transaction costs in addition to proportional transaction costs makes the optimal portfolio selection problem more complicated. In this case, due to the presence of a fixed transaction fee irrespective of the size of transaction, the optimal control strategy is discontinuous as opposed to the case with proportional transaction costs only [Zakamouline, 2006]. For example, both the constant and fixed transaction costs can be introduced in the minimum required return constraint as shown in eq. (4.11) to guarantee a minimum net return rate of the portfolio. N ∑ j=1

(rj − cj )qj ωj −

N ∑ j=1

fj z j ≥ µ 0

N ∑

qj ω j

(4.11)

j=1

where qj is defined as the quotation of security j, qj ωj is the amount invested in security j, fj is the fixed transaction cost incurred by the investor when selecting asset j, µ0 is a parameter representing the minimum required portfolio rate of return, rj is the mean rate of return for security j, and zj is a binary variable, which takes value 1 if ωj > 0. Both modelling alternatives may be interesting to a decision maker. On the contrary, the transaction costs may be included in the budget constraint (4.12) depending or whether the capital is used to buy the assets and to pay the transaction costs

84


like in [Wu et al., 2009]. N ∑

((1 + cj )qj ωj + fj zj ) = capital

(4.12)

j=1

In practice, transaction costs are not convex functions of the amount traded, indeed, the costs for either buying or selling are likely to be concave [Lobo et al., 2007].

4.4.

PROBLEM FORMULATION

The problem of portfolio selection can be formulated as the problem to find an optimal strategy for allocating wealth among a number of securities (investment) and to obtain an optimal risk-return trade-off. The portfolio optimization problem may be formulated in various ways depending on the selection of the objective functions, the definition of the decision variables, and the particular constraints underlying the specific situation [Mukerjee et al., 2002, Stummer and Sun, 2005, Ehrgott et al., 2006]. Beyond the expected return and variance of return, like in Markowitz portfolio model [Markowitz, 1952], the additional objective function can include number of securities in a portfolio, turnover, amount of short selling, dividend, liquidity, excess return over of a benchmark random variable and other [Mukerjee et al., 2002]. In the bank portfolio management, the additional criteria such as the prime rate, processing cost, expected default rate, probability of unexpected losses, quantity of the long-term and short-term can be considered [Stummer and Sun, 2005]. For example, the multi-objective portfolio selection problem can include the following objectives [Ehrgott et al., 2006]: (to be maximized) portfolio return, dividend, growth in sales, liquidity, portfolio return over that of a benchmark, and (to be minimized) deviations from asset allocation percentages, number of securities in portfolio, turnover (i.e., costs of adjustment), maximum investment proportion weight, amount of short selling. In single-period portfolio optimization theory one basic implication of Markowitz model is that investors hold well diversified portfolios of securities. Nevertheless, in practice, investors typically select portfolios consisting in a small number of securities. There are many reasons for this fact, the most relevant of which is the presence of transaction costs. In financial markets, any movement of money among assets incurs in a transaction cost. In many papers the emphasis has been put on properly modelling transaction costs. This issue is addressed in next subsection, which is followed by the multi-stage stochastic portfolio problem.

4.4.1.

SINGLE PERIOD PORTFOLIO OPTIMIZATION

In this subsection we start with single-period Markowitz model [Markowitz, 1952] that is modelled as a mean-risk bicriteria optimization problem where the expected return is maximized and the variance as a scalar risk measure is minimized as shown in (4.13). Here

85

4.4. PROBLEM FORMULATION

N is the number of assets the investor can choose from, ωi is a share of ith asset in the portfolio, i.e. it represents the fractions of capital invested in the asset i, ri is expected ith asset return, and σi is a covariance of ith asset returns, and ρij is correlation between assets i and j. The imposed constraint on ωi to be non-negative means that short selling of assets is not allowed. N ∑ ris ωi max min

i=1 N ∑ N ∑

ωi ωj σi σj ρij

i=1 j=1

s.t.

N ∑

(4.13)

ωi = 1

i=1

ωi ≥ 0,

i = 1, . . . , N

The bi-objective model 4.13 is extended to three criteria model by adding function of dividends maximization as a third objective function.

4.4.2.

MULTI-STAGE STOCHASTIC PORTFOLIO OPTIMIZATION

Institutional investors must manage their strategic asset mix over time to achieve favorable returns, in the presence of uncertainties and subject to various legal constraints, policies, and other requirements. A multi-period portfolio optimization model can be used in order to determine an optimal asset mix. In this section we formulate the multi-asset multi-period portfolio optimization problem as a stochastic control problem with three criteria, accounting for the profit maximization and both risk and transaction cost minimization. In multistage stochastic programming models, decision variables, and constraints are divided into groups corresponding to time periods, or stages t = 0, . . . , T . It is essential to define what is known at each stage in advance. In this model we assume that there are N assets with random prices ρt,i and returns rt,i at the end of each time period. At initial moment t = 0 we have cash and we want to invest a given amount W0 in order to maximize our amount at the end of investment planning period T . Young [Young, 1998] proposed to select a portfolio based on the maximization of the worst portfolio realization, i.e. maxM (ω) = { min µt }. Instead of the worst case scenario, t=1,...,T

we may maximize the mean of a specified size (quantile) of worst realizations, i.e. the CVaR for a given confidence level 0 < α ≤ 1. As it was mentioned above, the third objective function is the minimization of the transaction costs. While we can rebalance the portfolio at period t, we do not withdraw or add cash to it, everything what is earn is invested in the next period. It is assumed that

86


transaction costs are paid by cash. Therefore, we consider an approach which consists of a sequence of short 1-month periods. This design accounts for the short term dynamics of the CVaR model in the first few years T and the possibility of frequent rebalancing as shown in Figure 4.2.

4.2 Fig.: Investment horizon The situation can be described by a stochastic data process ξ = {ξ1 , . . . , ξT } repreξ senting the realizations of asset prices ρξt,i and returns rt,i , and decision processes ω(ξ) = ξ1 ξT ξ1 ξT {ω0 , ω1 , . . . , ωT }, y(ξ) = {y0 , y1 , . . . , yT }, and z(ξ) = {z0 , z1ξ1 , . . . , zTξT }, where ωi,t , yi,t , and zi,t correspond to the units held, units sold and units bought at each stage for each assets while buying or selling assets the linear trading costs cbuy and csell are sustained. Suppose that there are a total of S distinct scenarios for assets prices. We formulated problem in a following way: S ∑

max

min

ps

T −1 ∑ N ∑

s=1

t=0 i=1

S ∑

T −1 ∑ N ∑

ps

s=1

s rt,i ωt,i ρt,i

yt,i ρst,i csell + zt,i ρst,i cbuy

t=0 i=1

min CVaR s.t.

N ∑ i=1 N ∑

ω0,i ρs0,i = W0

(4.14)

s (1 + rt−1,i )ωt−1,i ρst−1,i =

i=1

ω0,i = z0,i ,

N ∑

ωt,i ρst,i

t = 1, . . . , T − 1, s = 1, . . . , S

i=1

y0,i = 0, i = 1, . . . , N

ωt,i = ωt−1,i − yt,i + zt,i t = 1, . . . , T − 1, i = 1, . . . , N yt,i zt,i = 0,

i = 1, . . . , N

ωt,i , yt,i , zt,i ≥ 0,

t = 0, . . . , T − 1, i = 1, . . . , N

Here ps is the probability of scenario s occurrence and the initial price ρs0,i of asset i is the same for all scenarios. The limits on the positions set to ωt,i ≥ 0 means that short positions are not allowed. The first constraint in 4.14 guarantees that the total initial investment equals the initial wealth. Next constraint states that the wealth accumulated at the end of

4.5. DISCUSSION ON EXPERIMENTAL RESULTS

87

tth period over all n assets under scenario s has to be equal to amount after rebalancing at the beginning of period t + 1. This constraint can be modified to include costs or return goals. The third constraint depict that at initial period we can only buy some assets and this amount is set equal to holding amounts, while selling amounts is set to zero. The fourth constraints ensures selling, holding and buying balance for every asset i. The nonlinear constraint yt,i zt,i = 0 can be omitted since simultaneous buying and selling of the same asset i can never be optimal. Thus, decision variables as an investment units are the same for all scenarios and they can obtain only nonnegative values. If any number of convex transaction costs and convex constraints are combined, the resulting problem is convex. Linear transaction costs, as well as all the portfolio constraints described above, are convex. The model can be enhanced further by, for instance, imposing constraints on CVaR value for every period or requiring that the portfolio proportions do not deviate too much.

4.5.

DISCUSSION ON EXPERIMENTAL RESULTS

4.5.1.

CASE 1: DETERMINISTIC PORTFOLIO OPTIMIZATION

4.5.1.1.

TWO CRITERIA PROBLEM

The first deterministic problem is based on a simple two objectives portfolio model including the standard deviation of the returns and mean of the returns, where the returns are monthly returns of stocks; returns mean percentage change in value. The second problem included three objectives, where annual dividend yield is added to two above mentioned objectives. For the experiment we used a data set of 10 Lithuanian companies’ stock data from Lithuanian market. The AW method was implemented for two and three criteria cases in MATLAB using fminconstr() for solving minimization problem (4.14). To illustrate the advantage of the proposed algorithm of adjustable weights over the standard weighted sum method both methods were applied for the construction of Pareto set of two criteria portfolio optimization with the data mentioned above. Figure 4.3 shows the distributions of the Pareto points found by both methods. In the weighted sum method weights have been changed with the step for 0.05. The tolerance for branching of the algorithm of adjustable weights have been set aiming at generation of similar number of points as generated by weighting method. In this experiment the number of points generated by the weighted sum method was equal to 21, and that generated by our algorithm of adjustable weights was equal to 26. The AW and SLGP algorithms have been implemented in MATLAB. Experiments were

88


2.5 uniform adjustable

Return %

2

1.5

1

1

2

3

4 Risk %

5

6

7

4.3 Fig.: Solutions in Pareto set generated by the WS method and AW algorithm

performed to compare performance of the selected meta-heuristic methods, WS, AW and SLGP. The parameters of FastPGA, MOCELL, AbYSS, and NSGAII are used the same as in previous chapter. The evolutionary methods are randomized, but the algorithms of adjustable weights and successive lexicographic goal programming are deterministic. Because of this difference the experimental results in tables below are presented correspondingly. The averages (Avg.) and standard deviations (Std.) of all performance measures for each meta-heuristic algorithm are calculated from the samples of 100 runs. The best results in Tables 4.1 and 4.2 are printed in boldface. Although, computational time is one of the most important performance measures in comparison of optimization methods, it has not been included in our study. This can be justified only by the severe differences in implementations of the algorithms (e.g. in Java and in MATLAB) making direct comparison of running times unfair. Among deterministic approaches, a clear winner is adjustable weights algorithm according to IGD, HV, and spread metrics that are reported in the Table 4.1. It can be noticed that GD value of all these methods is equal to zero; this means that all solutions lie precisely in Pareto set. It follows that MOCeLL outperforms the other meta-heuristic methods in both cases. The differences in performance weaken with increase of number of evaluations as it is well illustrated by the results of the last block of results in the Table 4.1. In general, it can be observed that AW is superior in term of HV, GD and IGD (when F=25000), and it is outperformed by MOCell only with regards to diversity. SLGP provides slightly worse results than AW, but they demonstrate its efficiency and suitability to solve this problem. The upper part of the Pareto set shown in Fig.4.3 is most difficult to reveal for all

89


4.4 Fig.: Solutions found by all algorithms 4.1 Table: Performance metrics for bi-objective portfolio problem Method

GD

IGD

HV

∆

WS AW SLGP

0.0 0.0 0.0

1.12E-03 6.17E-05 1.13E-04

0.7423 0.7881 0.7872

1.4117 0.2096 0.4399

F=25000

Avg.

Std.

Avg.

Std.

Avg.

Std.

Avg.

Std.

AbYSS FastPGA MOCeLL NSGAII SPEA2 CellDE

5.55E-4 3.35E-4 1.46E-4 3.26E-4 3.83E-4 1.90E-4

2.80E-4 1.59E-4 4.78E-5 3.65E-5 3.66E-4 8.32E-4

1.15E-3 3.21E-4 7.14E-5 8.80E-5 2.06E-3 2.29E-4

1.57E-3 6.09E-4 8.36E-5 4.62E-5 1.58E-3 5.06E-4

0.7610 0.7812 0.7865 0.7838 0.7477 0.7840

3.54E-2 1.06E-2 1.03E-3 5.86E-4 3.50E-2 7.59E-3

0.3667 0.3772 0.1449 0.3991 0.6647 0.4842

1.79E-1 6.79E-2 3.21E-2 3.03E-2 1.08E-1 6.42E-2

F=50000

Avg.

Std.

Avg.

Std.

Avg.

Std.

Avg.

Std.

AbYSS FastPGA MOCeLL NSGAII SPEA2 CellDE

2.85E-4 2.91E-4 6.86E-5 3.21E-4 2.50E-4 1.71E-4

6.67E-5 3.56E-5 1.04E-5 4.37E-5 4.40E-5 8.34E-4

1.25E-4 7.65E-5 5.83E-5 8.31E-5 9.15E-5 2.28E-4

3.99E-4 3.13E-6 8.44E-7 4.71E-6 8.28E-6 5.05E-4

0.7839 0.7847 0.7877 0.7840 0.7834 0.7843

7.65E-3 3.71E-4 9.21E-5 4.50E-4 6.91E-4 7.60E-3

0.1751 0.3345 0.1178 0.4011 0.4958 0.4759

5.68E-2 2.69E-2 1.64E-2 3.38E-2 1.79E-2 6.45E-2

evolutionary methods. Approximation of the Pareto set with a curve of changing curvature shows that the curve is flattening at its upper part. Similar dependence between flatness of the Pareto set and decrease of quality of its approximation using meta-heuristic methods

90


Pareto Set

SLGP

2

2 Return

2.5

Return

2.5

1.5

1

1.5

0

10

20 30 Risk

1

40

0

10

AW

20 30 Risk

40

WS

2

2 Return

2.5

Return

2.5

1.5

1

1.5

0

10

20 30 Risk

40

1

0

10

20 30 Risk

40

4.5 Fig.: Solutions of bi-objective portfolio problem generated by WS, AW, and SLGP

is mentioned also by the other authors. For illustration of this phenomenon the points generated by all methods in the mentioned part of Pareto set are presented in Figure 4.4; maximal number of function evaluations was fixed equal to 25000. It can be noticed that the method of adjustable weights does not suffer from flattening of the Pareto frontier. The whole Pareto frontiers generated by deterministic and meta-heuristic algorithms are displayed in Figures 4.5 and 4.6. In summary, we can conclude that AW and SLGP are competitive with meta-heuristic algorithms in solving bi-objective portfolio problem. AW algorithm proved to be superior and more efficient than SLGP and MOCell, the best one among other meta-heuristic algorithms. The latter requires to perform more than 25,000 function evaluations in order to outperform AW in term of inverted generational distance for two criteria portfolio selection problem.

4.5.1.2.

THREE CRITERIA PROBLEM

The experimental results for three criteria portfolio problem are given in Table 4.2. These results show that the winner among the weighted sum and proposed algorithms according to all performance measure is the AW algorithm. However, it is not trivial to decide about its superiority when looking at the approximated Pareto sets by AW and SLGP that are presented in Figure 4.7. From these pictures, it seems that the quality of Pareto subset generated by SLGP is higher, but the performance metrics confirm opposite.

91


2

1.5

0

10

20 30 Risk FastPGA

1.5

1

40

2.5

2.5

2

2

Return

Return

Return

2

1

1.5

1

Return

CellDE 2.5

0

10

20 30 Risk NSGAII

2.5

2.5

2

2

1.5

1

0

10

20 30 Risk

40

dominated optimal 0

10

20 30 Risk MOCell

40

0

10

20 30 Risk SPEA2

40

0

10

20 30 Risk

40

1.5

1

40

Return

Return

AbYSS 2.5

1.5

1

4.6 Fig.: Solutions of bi-objective portfolio problem generated by meta-heuristic algorithms

When 25000 function evaluations were performed in meta-heuristic algorithms, CellDE demonstrates its better performance with respect to IGD and HV than the other metaheuristic algorithms. In case of 50000 function evaluation, the obtained results are different and shows that SPEA2 is advantageous according to GD, IGD and spread measures. CellDE keeps its position with respect to HV in both cases. The best spread value is obtained by SPEA2 among all algorithms, however only the later and MOCell provide better spread value than AW and SLGP. The approximations of Pareto sets obtained by meta-heuristics methods from one run with 25000 function evaluations are presented in Figure 4.8. Metaheuristic algorithms provide both dominating (black points) and dominated (grey crosses) solutions, while WS, AW as well as SLGP obtained only Pareto optimal solutions. From figures 4.7 and 4.8 it can be seen how much the quality of approximation of Pareto set by evolutionary methods is behind of that of AW and SLGP; this indicates the perspective of

92


4.2 Table: Performance metrics for three criteria portfolio problem; F denotes the number of function evaluations by the meta-heuristic methods

WS AW SLGP F=25000

GD

IGD

HV

S

0.0 0.0 0.0

1.46E-3 2.32E-4 2.61E-4

0.4136 0.494 0.4935

0.859 0.525 0.531

Avg.

Std.

AbYSS 2.46E-3 5.76E-4 CellDE 5.87E-3 2.93E-3 FastPGA 1.33E-3 3.60E-4 MOCell 1.29E-3 2.92E-4 NSGAII 1.37E-3 3.29E-4 SPEA2 1.35E-3 2.15E-4 F=50000

Avg.

AbYSS 2.06E-3 CellDE 7.32E-3 FastPGA 1.30E-3 MOCell 1.30E-3 NSGAII 1.42E-3 SPEA2 1.29E-3

Avg. 1.37E-3 1.63E-4 2.01E-4 2.04E-4 2.10E-4 2.34E-4

Std.

Avg.

4.02E-4 1.57E-3 3.24E-4 3.03E-4 3.71E-4 2.63E-4

7.14E-4 1.59E-4 1.97E-4 2.05E-4 2.05E-4 1.57E-4

Std.

Avg.

Std.

Avg.

Std.

4.31E-4 0.398 0.027 0.701 0.057 7.99E-6 0.493 0.003 0.542 0.076 1.22E-5 0.486 0.002 0.565 0.045 1.44E-5 0.487 0.002 0.492 0.039 1.37E-5 0.484 0.002 0.584 0.040 1.63E-4 0.486 0.005 0.431 0.037 Std.

Avg.

Std.

4.02E-4 0.448 0.018 7.61E-6 0.495 0.002 1.30E-5 0.488 0.002 1.42E-5 0.489 0.002 1.29E-5 0.486 0.002 6.60E-6 0.492 0.002

Avg.

Std.

0.634 0.580 0.567 0.495 0.593 0.420

0.058 0.044 0.052 0.038 0.044 0.028

developed algorithms. Finally, we can summarize, that AW and SLGP justify their efficiency in solving three criteria portfolio selection problem. Although, the diversity of solutions obtained by MOCell and SPEA2 is better than AW and SLGP with respect to ∆ metric (only in case of 25000 function evaluations), but most of them are non-dominating in both cases (GD values are 1.30E-3 and 1.29E-3). Therefore, taking into consideration these facts, we can state that AW and SLGP perform better.

4.5.2.

CASE 2: STOCHASTIC PORTFOLIO OPTIMIZATION

4.5.2.1.

DATA AND SAMPLE PATHS GENERATION

Here we consider a stochastic model for changes of the underlying asset prices of all the instruments in a portfolio. We selected four asset classes: equity, US government bonds, high-yield bonds and emerging market bonds. Historical market data have been obtained from the Yahoo finance server. To represent these classes we chose data of five market indexes from 2000 October till 2010 June:

93

4.5. DISCUSSION ON EXPERIMENTAL RESULTS WS

10

5

0 2.5

40

Annual dividend yield


Pareto Set

20

2 1.5 Monthly return

1

0

10

5

0 2.5

40 2

Risk

10

5

0 2.5

40 2

20 1.5 Return

1.5 Return

1

0

Risk

SLGP



AW

20

1

0

10

5

0 2.5

Risk

40 2

20 1.5 Return

1

0

Risk

4.7 Fig.: Pareto set and solutions of three-objective portfolio problem generated by WS, AW, and SLGP

• HYB: The New America High Income Fund Inc. • TEI: Templeton Emerging Markets Income Fund • FLXBX: Flex-funds US Government Bond • DHY: Credit Suisse High Yield Bond Fund • DJI: Dow Jones Industrial Average The average historic daily returns and sigmas of selected indexes are presented in Table 4.3. Asset returns are simulated as the proportional increments of constant drift, constant volatility stochastic processes, thereby approximating continuous-time geometric Brownian motion according to equation (4.15). √ dS = µdt + σdz = µdt + σϵ dt (4.15) S where S is the asset price, µ is expected asset return, σ is the volatility of the asset price and ϵ represents a random drawing from a standartized normal distribution.

94

CHAPTER 4. OPTIMAL PORTFOLIO SELECTION CellDE

10

5

0 2.5

40 2

1.5 Return

20 1

0



AbYSS

dominated optimal

10

5

0 2.5

Risk

1

0

20 Risk

2 1.5 1 Monthly return SPEA2

0

20 Risk


0

20 Risk


40

FastPGA

5

0 2.5


1

0

20 Risk

40



MOCell 10

10 5 0 2.5

10

5

0 2.5


1

0

20 Risk

40



NSGAII

40

10

5

0 2.5

1

40

4.8 Fig.: Solutions of three-objective portfolio problem generated by meta-heuristic algorithms

The number of scenarios of selected assets returns were generated using Monte Carlo simulation of correlated asset returns. For this purpose, function portsim() from Matlab 7.5 Financial Toolbox was employed. The example of ten sample paths of daily HYB assets price for two years is shown in Figure 4.9.

4.3 Table: Average daily returns×(10e-4) and sigma×(10e-2) of assets

Returns Sigma Ann. Returns

HYB 3.575 1.731 9%

TEI 6.514 1.549 16.4%

FLXBX 1.399 0.306 3.5%

DHY 3.333 1.994 8.4%

DJI 0.801 1.311 2%

95


4.9 Fig.: Scenarios of HYB prices for two years 4.4 Table: Proportional trading costs to trading volume Case Case 1 Case 2 Case 3

4.5.2.2.

HYB 0% 0.95 % 3.80 %

TEI 0% 1.0 % 4.0%

FLXBX 0% 0.90 % 3.60 %

DHY 0% 0.90 % 3.60 %

DJI 0% 0.85 % 3.40 %

RESULTS

To examine impact of the transaction costs, we solved problem with the different transaction costs, that are given in Table 4.4. In case 1, we have only two-objective problem because the transaction costs are equal to zero. In case 2, the highest cost is 1% of traded amount, while in case 3, the highest incurred costs are 4% of traded amount of assets. The initial portfolio contained only cash in amount of 100,000 U.S. dollars. The composite Pareto frontiers obtained by WS, AW and SLGP algorithms in case 1 with different levels of α is given in Figure 4.10. It can be noticed, that the biggest profit that can be earn is $36894.86, but with 0.1 probability we can lose $30,040.53, with 0.05 probability it is possible to sustain losses of $82,726.87, and with 0.01 probability we can suffer losses of $413,634.35. The performance metrics of stochastic optimal portfolio problem solved by WS, AW and SLGP algorithms are provided in Table 4.5. Here the calculation were performed with confidence level α = 0.95 for CVaR. Since the similar results were obtained in cases with α equal to 0.9 and 0.99, they are not shown. These results revealed that the quality of solutions generated by WS is behind of ones found by AW and SLGP with respect to IGD, HV and spread metrics. Although, the values of performance metrics obtained by SLGP are better than ones of AW in two cases, however they are very similar. Therefore, it would be

96

CHAPTER 4. OPTIMAL PORTFOLIO SELECTION 4

4

0.99

x 10

3.5

Returns

3 2.5 2 1.5 0.90 0.95 0.99

1 0.5

0

1

2

3

4

CVaR

5 4

x 10

4.10 Fig.: Case 1. Pareto solutions obtained with different risk level α = 0.9, 0.95, 0.99 4.5 Table: Performance metrics for two and three criteria stochastic portfolio problems with different risk level

Method

IGD

HV

∆

Case 1

WS AW SLGP

0.0047 0.4972 1.2215 2.53E-4 0.5928 0.1309 1.42E-4 0.5763 0.1098

Case 2

WS AW SLGP

0.0088 0.0041 0.0035

0.3152 1.3053 0.4545 0.5391 0.4945 0.5481

Case 3

WS AW SLGP

0.0096 0.0054 0.0055

0.3154 1.2962 0.4815 0.5853 0.4945 0.5482

inadvertent to state that one of them is superior. The Figure 4.11 shows Pareto approximations found by SLGP algorithm using costs of Case 3. It presents the portfolio configurations for different risk levels. From Figure 4.12, we can see how much the transaction cost lowers the return. The transaction cost does not lower the return linearly. Since it is incorporated into the optimization, it can also affect the choice of stocks. Obtained results confirms the fact that transaction cost is an important factor for an investor to take into consideration in portfolio selection. It is not trivial enough

97

4.6. CHAPTER CONCLUSIONS

0.90 0.95 0.99

4000 3900 3800 3700 3600 3500 3400 4 3 4

2

x 10

1 Returns

0

3

2

1

0

7

6

5

4

4

x 10

CVaR

4.11 Fig.: Case 3: Solutions obtained by SLGP with different risk level α = 0.9, 0.95, 0.99

1000

4000 3900

Costs

3800

Costs

950

900

3700 3600 3500

850 4

3400 4 3 4

x 10

2 1 Returns

0

2

3

4 CVaR

5

3

7

6

4

x 10

2 1

4

x 10

Returns

(a) Case 2

0

2

3

4

5

7

6

4

x 10

CVaR

(b) Case 3

4.12 Fig.: Pareto frontiers with risk level α = 0.95 to be neglected, and the optimal portfolio depends upon the total costs of transaction.

4.6.

CHAPTER CONCLUSIONS

MOCeLL is a leader among meta-heuristic algorithms with respect to all performance metrics in the deterministic bi-objective portfolio optimization problem. The AW provides better results in comparison with the SLGP and is highly competitive to the meta-heuristic ones. From the results of the experiments for the three criteria portfolio optimization there is no clear winner among six considered meta-heuristic algorithms. With respect to two,

98


spread and inverted generational distance, of four performance metrics, SPEA is superior over others, but in terms of the hypervolume is slightly outperformed by CellDE. AW and SLGP are competitive with meta-heuristic algorithms in solving bi-criteria and three-criteria portfolio problems. In bi-objective problem, AW algorithm proved to be superior and more efficient than SLGP and MOCell. The latter one requires to perform more than 25,000 function evaluations in order to outperform AW in term of inverted generational distance. In three-criteria problem, the diversity of solutions obtained by MOCell and SPEA2 is better than AW and SLGP with respect to ∆ metric (only in case of 25000 function evaluations), but most of them are non-dominating in both cases (GD values are 1.30E-3 and 1.29E-3). The results of stochastic portfolio optimization problem confirmed the efficiency of our proposed algorithms, that provided similar values of performance metrics. However they did not revealed which one, AW or SLGP, is a leader in solving this problem.

chapter 5 PRICING TOLLING AGREEMENTS 5.1. 5.1.1.

PROBLEM FORMULATION SINGLE OBJECTIVE

The unique physical characteristics of electricity make its price the most volatile one among all commodity prices. Noting the extremely high price volatility, power market participants are especially wary of the price risk associated with business transactions and they resort to customized (most likely long-term) business transactions to hedge their respective unique risk profiles thus making the bilateral and multilateral power supply contracts ubiquitous. We concentrate on a tolling agreement for a gas-fired power plant in a de-regulated market. The fuel and electricity prices fluctuate and the target is to obtain the maximum value from the plant. This is achieved by optimizing the dispatching policy, i.e. deciding when the plant is running and when it is offline. In case of running the power plant, the natural gas has to be bought and converted into electricity and sold on the market. The problem of pricing tolling agreements depends to the class of multiple optimal stopping problems and is extremely difficult to solve from computational prospective [Ryabchenko and Uryasev, 2011]. The commodity prices are modelled as stochastic processes and the owner manages the plant by using optimal policy. The optionality of running the plant comes only from startup/shutdown decisions. In this work we focus on the problem to find a single optimal stationary exercise boundary using grid variables that was presented by Ryabchenko and Uryasev in [Ryabchenko and

100

CHAPTER 5. PRICING TOLLING AGREEMENTS

Uryasev, 2011]. They proposed a simple linear programming formulation that is given below:

max ξi,j

NS ∑ N 1 ∑ j e−r(1−i) (ξij Mij − νij (Cs,i + Mij ) − µji Cd ) NS j=1 i=1

(5.1)

j s.t. νij ≥ ξij − ξi−1 , νij ≥ 0, i = 1, . . . , N, j = 1, . . . , NS , j µji ≥ ξi−1 − ξij , µji ≥ 0, i = 1, . . . , N, j = 1, . . . , NS ,

ξi,j ≤ ξi+1,j , i = 1, . . . , (NV − 1), j = 1, . . . , NH ,

(5.2)

ξi,j ≤ ξi,j+1 , i = 1, . . . , NV , j = 1, . . . , (NH − 1), 0 ≤ ξi.j ≤ 1, i = 1, . . . , NV , j = 1, . . . , NH , where NS and N denotes number of sample paths and number of period, respectively, ¯ j − HG ¯ j ) − K, ¯ Q(P j − HGj ) − K) Mij = max(Q(P i i i i

(5.3)

j is the spark spread, Cs,i = CS + LGji is the startup cost, L = QH is the total amount of consumed gas in MMBtu, Cd is shutdown costs, CS startup costs, K is renting costs, Pij and Gji are price processes for energy and gas, r is one period risk free interest rate, H is heat rate, Q - generating capacity, ξij is the renter’s switching decisions (1 means ”on", 0 means ”off"), NV - number of vertical nodes in the grid, NH - number of horizontal nodes, j j µji = [ξi−1 − ξij ]+ and νij = [ξij − ξi−1 ]+ are auxiliary variables.

It should be noted that ξi,j and ξij are different variables, i.e. ξi,j denotes the grid variables at coordinates i and j, and ξij denotes the renter’s switching decision in sample path j at period i. Each ξij corresponds to one of the grid points or variables. In this linear program the maximal number of variables and constraints it does not depend on the number of time periods or sample paths, it only depends on the size of the grid. The grid is built uniformly in logarithmic (lnG,lnP ) plane as shown in Figure (5.1). More detail on grid construction and variables interpolation on a grid can be found in [Ryabchenko and Uryasev, 2011]. While the profit of generating electricity comes from the positive spark spread between generated electricity and the input fuel, it is clear that a power plant would only lose money when the spark spread becomes negative possibly due to too low an electricity price or too high a fuel cost. In times of the spark spread turning so negative that a temporary shutdown of the power generating unit is justified, the operator has to turn off the unit and restart it later when the profit of generating electricity becomes positive again. The solution of optimal power plant operations on an 80 × 80 grid is presented in Figure (5.2). Here light points represent the power plant state of shut down and dark points mean that it is profitable to run the power plant. However, frequent restarts are detrimental to a generation unit since a restart reduces the

5.1. PROBLEM FORMULATION

101

5.1 Fig.: Interpolation on a grid on a logarithmic plane

unit’s lifetime and increases the likelihood of a forced outage [Deng and Xia, 2005]. Due to this fact, there is usually a provision specifying the maximum number of restarts allowed in a tolling contract. Sometimes this constraint is implemented through imposing an extremely high penalty charge on each restart beyond certain threshold on the cumulative number of restarts in the contract effectively capping the total number of restarts at the threshold level. As a result, a tolling contract holder cannot order to shut down the plant at will whenever the electricity spot price is lower than the heat rate adjusted generating fuel cost. Consequently, the value of a tolling agreement is affected by such a constraint.

5.1.2.

BI-OBJECTIVE FORMULATION

We extended pricing tolling agreements problem by adding the CVaR as a second objective function to the formulation given above (5.1), i.e. the problem has the following form NS ∑ N 1 ∑ j max e−r(1−i) (ξij Mij − νij (Cs,i + Mij ) − µji Cd ) ξi,j NS j=1 i=1 (5.4) max CVaRα subject to (5.2) In this case we will maximize the CVaR of revenue, i.e. it can be treated as worst case profit maximization. The graphical interpretation is given in Figure 5.3. If scenarios are used, the CVaR of probability distribution can be calculated solving the following optimization

102


5.2 Fig.: Optimal power plant operation on 80 × 80 grid in (lnG, lnP ) plane

5.3 Fig.: CVaR of revenue

problem: NS ∑ 1 CVaRα = max πi η i ξ,ηi (1 − α)Ns i=1

s.t. − prof iti + ξ − ηi ≤ 0, ηi ≥ 0,

i = 1, . . . , N s

(5.5)

i = 1, . . . , N s

where πi is the probability of achieving a profit equal to prof iti , and ξ is an auxiliary variable whose optimal value is equal to the VaR. Once the optimization problem is solved, for those scenarios with a profit lower than the VaR, the value of the auxiliary variable ηi represents the difference between the VaR and the value of the profit in scenario i; for other scenarios, ηi is zero.

103

5.2. EXPERIMENTAL RESULTS

In tolling agreements usually owner of plants put some restriction on who the power plant can be operated, for example, there can be some restriction on a number of switching off/on the power plant [Deng and Xia, 2005]. Other recommend to consider a restricted situation where a fixed upper bound is put on the total number of switches allowed.

5.2. 5.2.1.

EXPERIMENTAL RESULTS SCENARIOS GENERATION

The risk-neutral dynamics of energy and gas price processes are considered being the Ornstein-Uhlenbeck processes modeled according to equations: d ln Pt = ae (be − ln Pt )dt + σe dWt1 , d ln Gt = ag (bg − ln Gt )dt + σg dWt2 ,

(5.6)

dWt1 dWt2 = ρdt, where a, b, and σ are positice constants. be and bg may be interpreted as long term levels, and ae and ag as mean reversion rates for electricity and gas prices, respectively; σe and σg are volatility; ρ is the instantaneous correlation coefficient between the two price processes, Wti (i = 1, 2) are standard Brownian motions. To simulate prices at times 0 = t0 < t1 < · · · < tn we used an algorithm relying on the explicit form of the solution to the stochastic differential equation: √ ln Pti+1 = ln Pti e−ae (ti+1 −ti ) + be (1 − e−ae (ti+1 −ti ) ) + σe √ ln Gti+1 = ln Gti e−ag (ti+1 −ti ) + bg (1 − e−ag (ti+1 −ti ) ) + σg

1 − e−2ae (ti+1 −ti ) N1 (0, 1), 2ae 1 − e−2ag (ti+1 −ti ) N2 (0, 1), 2ag

(5.7)

with N1 (0, 1), N2 (0, 1) correlated random sample from standard normal distribution, where the two sequences of random numbers with given correlation ρ are generated in two steps: 1. Two sequences of uncorrelated normal distributed random numbers X1 , X2 are generated. √ 2. A new sequence is defined as Y1 = ρX1 + 1 − ρ2 X2 . Contrary to the Euler scheme that entails some discretization errors, this simulation algorithm is free of convergence problems associated with the size of the discretization step. For more details on processes simulation we refer to [Glasserman, 2004]. The example of one generated sample path of gas and electricity price, when they follow mean-reverting process is given in Figure (5.4).

104


5.4 Fig.: Electricity and gas prices, $ for 1-year We conducted all our experiments on a laptop PC equipped with Intel(R) Core(TM) 2 Duo CPU 2.10 GHz and 1.00 GB of RAM. The code of this problem was written in C++ and the CPLEX (R) 11.0 solver was used to solve the linear model.

5.2.2.

CASE STUDY

We consider short-term case, where for every calculation we used 500 sample paths of gas and energy price. The scenario generation parameters is given in Table 5.1. In this study we consider setup parameters values given in Table 5.2. The initial prices of electricity and natural gas are sampled from the historical data as $34.7 per MWh and $3 per MMBtu, ¯ and H values. respectively [Deng and Xia, 2005]. This problem was solved with different H

5.1 Table: Parameters used for short-term study ae 0.0651

be 3.5527

σe 0.1507

ag 0.0087

bg 1.3638

σg 0.0468

ρ 0.177

5.2 Table: Setup parameters used in short-term study N 730

¯ MWh Q, 100*18=1800

Q, MWh 1800*0.2=360

Cs , $ 2000

r,% 5

Cd , $ 1000

This problem is one of depending to large-scale problems, because the number of different pairs of variables on 80x80 grid on average is equal to 23,600. At the beginning, we solved

105


only single objective problem, i.e. maximized the profit and then with obtained solution we calculated the CVaR value with different confidence level. The prices of tolling agreement and CVaR values using a grid 80 × 80 of 30 runs are presented in Table 5.3. It should be mentioned that in this case we had on average 23,600 different pairs of variables on the grid. Here, positive CVaR value indicates the profit of power plant runner in the worst case, and negative values mean losses with α = 0.95. From obtained results we can notice, that in case of profit maximization, in worst case with 5% of probability we can incur losses despite of the heat rate. 5.3 Table: Prices (in $) and CVaR of 1-year tolling agreement contracts calculated on 80×80 grid ¯ H 7.5 8.5 10.5 13.5

Avg. price 12,374,565 9,564,613 4,108,135 171,522

Std. 396,861 321,187 173,253 27,331

Avg. CVaR0.95 -7,666 -32,705 -291,433 -294,132

Std. 50,093 24,731 105,554 59,213

The results of maximization of CVaR with confidence level α = 0.95 are shown in Table 5.4. In this table, we present VaR value, because it is proved that while maximizing/minimizing CVaR, the VaR value will be maximized/minimized as well. It can be noticed, that in worst case a small profit profit can be produced or at least there is no loss experienced. If we compared the prices in both cases, i.e. profit maximization and CVaR maximization, we could observe not very big difference between the contract prices. 5.4 Table: Prices, CVaR0.95 and VaR0.95 of 1-year tolling agreement contracts calculated on 80 × 80 grid ¯ H 7.5 8.5 10.5 13.5

Avg. price 12,349,965 9,523,238 4,075,820 93,229

Std. 407,196 325,707 1,328 22,398

Avg. CVaR0.95 794,855 500,999 5,189 0

Std. 201,285 136,211 3,720 0

Avg. VaR0.95 1,837,970 1,089,162 78,693 0

Std. 316,120 230,313 12,832 0

The solutions of one generated sample path found by WS, AW and SLGP algorithms are ¯ = 8.5 and confidence presented in Figure 5.5. Here the calculations were performed with H level α = 0.95. In the traditional weighted sum method, the weights were changed by step equal to 0.05, however this method found only four Pareto optimal solutions. It should be noted that here we are maximizing both of our objective functions, because CVaR means our profit in worst case. Unfortunately, we discovered that the mataheuristic algorithms requires

106


a lot of recurses and are costly to solve this large scale problem. Due to these reasons we couldn’t solve this problem by means of metaheuristic algorithms.

5.5 Fig.: Solutions generated by WS, AW and SLGP algorithms The average performance metric values of 30 runs are reported in Table 5.5. This shows that the WS method is not able to generate well-distributed Pareto set. In contrary, our proposed algorithms demonstrate their efficiency to cope with this problem. On average the algorithm of adjustable weights is superior over successive lexicographic goal programming algorithm with regards to the ∆, but it steps aside for SLGP in accordance with HV and IGD. 5.5 Table: The average values of performance metrics of tolling agreement problem, CVaR0.95 Method WS AW SLGP

IGD 0.0300 0.0045 0.0042

HV 0.3447 0.4840 0.4868

∆ 0.3358 0.1732 0.1906

In order to check the contracts price sensitivity to confidence level, we performed CVaR maximization with different α values. The results of CVaR with confidence level α = 0.99 are shown in Table 5.6. ¯ in The averages of maximum CVaR values of one-year agreement with different α and H ¯ the difference between CVaR Figure (5.6). We can notice that with increasing value of H ¯ value the CVaR values are equal to zero. However, values decreases while with the highest H ¯ = 7.5 we see that profit in the worst case with α = 0.95 looking at the CVaR values with H

107


5.6 Table: Prices, CVaR0.99 and VaR0.99 of 1-year tolling agreement contracts ¯ H 7.5 8.5 10.5 13.5

Avg. price 11437785 9100257 3812026 95041

Std. 80670 981712 112056 21474

Avg. CVaR0.99 87923 42176 427 0

Std. 78244 42881 372 0

Avg. VaR0.99 233930 114865 1009 0

Std. 133767 80939 637 0

is much bigger (approximately equal to $ 0.79 millions) while with α = 0.99 it is only $ 0.088 millions.

5.6 Fig.: Average of maximum CVaR values ($) with different confidence levels

¯ = 8.5 5.7 Fig.: Sensitivity to different confidence levels, when H

108

5.3.


CHAPTER CONCLUSIONS

In this chapter, we solved a problem of pricing tolling agreement contracts formulated as linear stochastic problem using optimal exercise boundary. Later we extended this single objective formulation to bi-objective problem by adding maximization of revenue CVaR as a second objective function. The bi-criteria problem has been solved using the simple weighted sum method and the algorithms of adjustable weights and successive lexicographic goal programming. The obtained results confirmed the efficiency and suitability of proposed algorithms to generate well distributed solutions over Pareto sets and their advantages against WS. However we cannot distinguish a clear winner according to all performance metrics. The analysis of confidence level changes impact on short-term agreements prices was performed as well.

chapter 6 MAIN RESULTS AND CONCLUSIONS This research allows to present obtained results and make following conclusions: 1. There are proposed two effective multi-objective optimization algorithms to find uniformly distributed Pareto set approximation for multi-criteria problems, namely, adjustable weights and successive lexicographic goal programming algorithms. 2. The experiments proved the adjustable weights algorithm is advantageous against weighted sum method, i.e. it provided 13 (out of 15) better values of HV, IGD and . Its advantage is that the weights are selected adaptively while running the algorithm. Moreover, unlike preemptive methods, it does not incorporate information about importance of criteria because weights play the role of parameters defining points in the Pareto set corresponding to the solutions of parametric single criteria problems. Moreover, unlike preemptive methods, it does not incorporate information about importance of criteria because weights play the role of parameters defining points in the Pareto set corresponding to the solutions of parametric single criteria problems. 3. The proposed algorithm of successive lexicographic goal programming is superior to adjustable weights algorithm due to its applicability not only to convex problems. In addition, it is possible to select a particular region of the criteria space intended to approximate; it is sufficient to form the corresponding grid of reference vectors. 4. The experiments and comparison with meta-heuristic algorithms revealed that proposed algorithms are competitive and provides good results. They provide only Pareto optimal solutions on the contrary to meta-heuristic algorithms that perform many function evaluations in order to generate solutions. 5. The performed calculations showed that increasing number of generated solutions does

110

CHAPTER 6. MAIN RESULTS AND CONCLUSIONS

not influence the uniformity of found Pareto approximation, i.e. the quality of spread uniformity along Pareto frontier not necessarily improves by generating more solutions. 6. The adjustable weights and successive lexicographic goal programming algorithms proved to be suitable to solve optimal portfolio selection problems as well as pricing tolling agreements problem.

References A. Abraham and L. Jain. Evolutionary multiobjective optimization. Evolutionary Multiobjective Optimization, pages 1–6, 2005. T. Aittokoski and K. Miettinen. Efficient evolutionary method to approximate the Pareto optimal set in multiobjective optimization. In EngOpt 2008 - International Conference on Engineering Optimization, 2008. M. Akian, A. Sulem, and M.I. Taksar. Dynamic optimization of long-term growth rate for a portfolio with transaction costs and logarithmic utility. Math Finance, 11:153–188, 2001. J. Andersson. A survey of multiobjective optimization in engineering design. Technical Report LiTH-IKP-R-1097, Department of Mechanical Engineering, Linköping University, 2000. B. Aouni and O. Kettani. Goal Programming Model: A Glorious History and a Promising Future. European Journal of Operation Research, 33:225–231, 2001. M. Arenas-Parra, A. Bilbao-Terol, and M.V. Rodriguez-Uria. A Fuzzy Goal Programming Approach to Portfolio Selection. European Journal of Operation Research, 133:287–297, 2001. M. Arenas-Parra, A. Bilbao-Terol, B. Perez-Gladish, and M.V. Rodriguez-Uria. A new approach of Romero’s extended lexicographic goal programming: fuzzy extended lexicographic goal programming. Soft Computing - A Fusion of Foundations, Methodologies and Applications, 2009. doi: 10.1007/s00500-009-0533-y. P. Artzner, F. Delbaen, J.M. Eber, and D. Heath. Coherent measures of risk. Mathematical Finance, 9:203–228, 1999. R. Azmi and M. Tamiz. A Review of Goal Programming for Portfolio Selection. New Developments in Multiple Objective and Goal Programming, 638:15–33, 2010. E. Balibek and M. Köksalan. A multi-objective multi-period stochastic programming model for public debt management. European Journal of Operational Research, xxx:xxx–xxx, 2010.

112

REFERENCES

D. Bertsimasa and D. Pachamanovab. Robust multiperiod portfolio management in the presence of transaction costs. Computers & Operations Research, 35:3–17, 2008. N. Beume and G. Rudolph. Faster s-metric calculation by considering dominated hypervolume as klee’s measure problem. In Computational Intelligence, pages 233–238, 2006. J.R. Birge. Stochastic programming computation and applications. INFORMS Journal on Computing, 9(2):111–133, 1997. J.R. Birge and F. Louveaux. Introduction to stochastic programming. Springer Verlag, 1997. J.-P. Bouchaud and M. Potters. Theory of Financial Risks: From Statistical Physics to Risk Management. Cambridge University Press, 2000. S.P. Boyd and L. Vandenberghe. Convex optimization. Cambridge Univ Pr, 2004. S.P. Bradley and D.B. Crane. A dynamic model for bond portfolio management. Management Science, 19:139–151, 1972. M.W. Brandt. Estimating portfolio and consumption choice: A conditional Euler equations approach. Journal of Finance, 54(5):1609–1645, 1999. K.A. Brekke and B. Oksendal. Optimal switching in an economic activity under uncertainty. SIAM Journal on Control and Optimization, 32(4):1021–1036, 1994. M.J. Brennan, E.S. Schwartz, and R.M. Lagnado. Strategic asset allocation. Journal of Economic Dynamics and Control, 21:1377–1403, 1997. T. Breuer and M. Jandacka. Portfolio selection with transaction costs under expected shortfall constraints. Computational Management Science, 5:305–316, 2008. W. Briec and K. Kerstens. Multi-horizon Markowitz portfolio performance appraisals: Ageneral approach. Omega, 37:50–62, 2009. L. Bui, H. Abbass, D. Essam, and D. Green. Performance analysis of evolutionary multiobjective optimization methods in noisy environments. In Proceedings of the 8th Asia Pacific Symposium on Intelligent and Evolutionary Systems, pages 29–39, 2004. G.C. Calafiore. Multi-period portfolio optimization with linear control policies. Automatica, 44:2463–2473, 2008. D.R. Carino, T. Kent, D.H. Myers, C. Stacy, M. Sylvanus, A.L. Turner, K. Watanabe, and W.T. Ziemba. The Russell Yasuda Kasai model: An asset/liability model for a Japanese Insurance Company using multistage stochastic programming. Interfaces, 24:29–49, 1994.

REFERENCES

113

ACC Carlos. Recent Trends in Evolutionary Multiobjective Optimization: Evolutionary Multiobjective Optimization: Theoretical Advances And Applications, 2005. R. Carmona and V. Durrleman. Pricing and hedging spread options 45(4). SIAM Review, 45:627–685, 2003. C.-T. Chang. Efficient structures of achievement functions for goal programming models. Asia-Pacific Journal of Operational Research, 24:755–764, 2007. T.-J. Chang, N. Meade, J.E. Beasley, and Y.M. Sharaiha. Heuristics for cardinality constrained portfolio optimisation. Computers and Operations Research, 27:1271–1302, 2000. A. Charnes and W.W. Cooper. Management models and industrial applications of linear programming. Wiley, New York, 1961. A. Charnes, W.W. Cooper, and R. Ferguson. Optimal estimation of executive compensation by linear programming. Management Science, 1:138–151, 1955. S.-J. Chen and C.-L. Hwang. Fuzzy multiple attribute decision making: Methods and applications. Springer, 1992. W. Chen, M.M. Wiecek, and J. Zhang. Quality utility: a Compromise Programming approach to robust design. Journal of mechanical design, 121(2):179–187, 1999. J. Clausen and A. Zilinskas. Global Optimization by Means of Branch and Bound with simplex Based Covering. Computers and Mathematics with Applications, 44:943–955, 2002. C.A.C. Coello. Evolutionary Multiobjective Optimization: Theoretical Advances and Applications, chapter Recent Trends in Evolutionary Multiobjective Optimization. Advanced Information and Knowledge Processing. Springer Verlag, pages 7–32, 2004. Y. Collette and P. Siarry. Multiobjective Optimization Principles and Case Studies. SpringerVerlag, Berlin, Germany, 2004. G.B. Dantzig and G. Infanger. Multi-stage stochastic linear programs for portfolio optimization. Annals of Operations Research, 45:59–76, 1993. I. Das. On characterizing the “knee” of the Pareto curve based on Normal-Boundary Intersection. Structural and Multidisciplinary Optimization, 18(2):107–115, 1999. I. Das and JE Dennis. A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Structural and Multidisciplinary Optimization, 14(1):63–69, 1997.

114

REFERENCES

I. Das and J.E. Dennis. Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems. SIAM Journal on Optimization, 8(3):631–657, 1998. M.H.A. Davis and A.R. Norman. Portfolio selection with transaction costs. Mathematics of Operations Research, 15:676–713, 1990. K. Deb. Multi-objective optimization using evolutionary algorithms. Wiley, 2001. K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan. A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2):182–197, 2005. S.-J. Deng and Z. Xia. Pricing and Hedging Electricity Supply Contracts: a Case with Tolling Agreements. Technical report, School of Industrial and Systems Engineering, Georgia Institute of Technology, 2005. S.J. Deng and S.S. Oren. Incorporating operational characteristics and startup costs in option-based valuation of power generation capacity. Probability in the Engineering and Informational Sciences (PEIS), 17:155–181, 2003. S.J. Deng, B. Johnson, and A. Sogomonian. Exotic electricity options and the valuation of electricity generation and transmission assets. Decision Support Systems, 30(3):383–392, 2001. J. Doege, P. Schiltknecht, and H.-J. Luthi. Risk management of power portfolios and valuation of flexibility. OR Spectrum, 28:267–287, 2006. Y. Donoso and R. Fabregat. Multi-Objective Optimization in Computer Networks Using Metaheuristics. Taylor & Francis Group, 2007. J. Dupacova, J. Hurt, and J. Stepan. Stochastic Modelling in Economics and Finance. Kluwer Academic Publishers, Dordrecht., 2002. J. Durillo, A. Nebro, F. Luna, and E. Alba. Solving three-objective optimization problems using a new hybrid cellular genetic algorithm. Parallel Problem Solving from Nature–PPSN X, pages 661–670, 2008. J.J. Durillo, A.J. Nebro, and E. Alba. The jmetal framework for multi-objective optimization: Design and architecture. In CEC 2010, pages 4138–4325, Barcelona, Spain, July 2010. J. Eddy and K. Lewis. Effective generation of pareto sets using genetic programming. In Proceedings of ASME Design Engineering Technical Conference. Citeseer, 2001.

REFERENCES

115

M. Ehrgott and M. Wiecek. Mutiobjective programming. Multiple criteria Decision Analysis: State of the art surveys, pages 667–708, 2005. M. Ehrgott, K. Klamroth, and C. Schwehm. Decision aiding an MCDM approach to portfolio optimization. European Journal of Operational Research, 155:752–770, 2004. M. Ehrgott, C. Waters, R.N. Gasimov, and O. Ustun. Multiobjective Programming and Multiattribute Utility Functions in Portfolio Optimization. 2006. T. Erfani and S.V. Utyuzhnikov. Directed Search Domain: A Method for Even Generation of Pareto Frontier in Multiobjective Optimization. Engineering Optimization, 00:1–17, 2010. H. Eskandari and C.D. Geiger. A Fast Pareto Genetic Algorithm Approach for Solving Expensive Multiobjective Optimization Problems. Journal of Heuristics, 14(3):203–241, June 2008. A. Eydeland and K. Wolynec. New developments in modeling, pricing and hedging. John Wiley & Sons, Hoboken, NJ„ 2003. F.J. Fabozzi, P.N. Kolm, and D. Pachamanova. Robust Portfolio Optimization and Management. Frank J. Fabozzi series. Wiley finance series. John Wiley & Sons, Inc., 2007. S.-C. Fang, J.R. Rajasekera, and H.-S.J. Tsao. Entropy optimization and mathematical programming, volume 8 of International series in operations research & management science. Springer, 1997. L.J. Fogel. Autonomous Automata. Industrial Research, 4:14–19, 1962. L.J. Fogel, A.J. Owens, and M.J. Walsh. Artificial Intelligence through Simulated Evolution. John Wiley, 1966. P. Glasserman. Monte Carlo methods in financial engineering. Springer-Verlag New York, Inc., 2004. W.N. Goetzmann and A. Kumar. Equity Portfolio Diversification. Review of Finance, 12: 433–463, 2008. T.F. Gonzalez, editor. Handbook of Approximation Algorithms and Metaheuristics. Taylor & Francis Group, LLC, 2007. C. Grosan and A. Abraham. Generating Uniformly Distributed Pareto Optimal Points for Constrained and Unconstrained Multicriteria Optimization. In INFOS2008, Cairo-Egypt, March 27-29 2008.

116

REFERENCES

C. Grosan and A. Abraham. Approximating Pareto frontier using a hybrid line search approach. Information Sciences, 180:2674–2695, 2010. G. Guastaroba, R. Mansini, and M.G. Speranza. Models and Simulations for Portfolio Rebalancing. Computational Economics, 33:237–262, 2009. N. Gülpinar and B. Rustem. Worst-case robust decisions for multi-period mean-variance portfolio optimization. European Journal of Operational Research, 183:981–1000, 2007. N. Gülpinar, B. Rustem, and R. Settergren. Multistage stochastic programming in computational finance. In Decision making, economics and finance: Optimization models, pages 33–47. Kluwer Academic Publishers, 2002. N. Gulpinar, B. Rustem, and R. Settergren. Optimization and simulation approaches to scenario tree generation. Journal of Economics, Dynamics and Control, 28(7):1291–1315, 2004. N. Hibiki. A hybrid simulation/tree stochastic optimization model for dynamic asset allocation. In B. Scherer, editor, Asset and Liability Management Tools: A Handbook for Best Practice, pages 269–294. Risk Books, 2003. N. Hibiki. Multi-period stochastic optimization models for dynamic asset allocation. Journal of Banking & Finance, 30:365–390, 2006. M. Hintermüller and I. Kopacka. Mathematical Programs with Complementarity Constraints in Function Space: C- and Strong Stationarity and a Path-Following Algorithm. SIAM Journal on Optimization, 20:868–902, 2009. J.H. Holland. Adaptation in natural and artificial systems. 1975. Ann Arbor MI: University of Michigan Press, 1975. K.A. Horcher. Essentials of financial risk management, volume 32 of Essentials series. John Wiley and Sons, 2005. J.P. Ignizio. A review of Goal Programming: A Tool for Muiltiobjective Analysis. Journal of Operational Research Society, 29:1109–1119, 1978. J.P. Ignizio. A Note on Computational Methods in Lexicographic Linear Goal Programmming. Journal of Operational Research Society, 32(6):539–542, 1983. G. Infanger. Handbook of asset liability management, chapter Dynamic asset allocation strategies using a stochastic dynamic programming approach. Elsevier, 2006.

REFERENCES

117

P. Jana, T.K. Roy, and S.K. Mazumder. Multi-objective possibilistic model for portfolio selection with transaction cost. Journal of Computational and Applied Mathematics, 228: 188–196, 2009. D. Jones and M. Tamiz. Practical Goal Programming. Springer, 2010. B Kaminski, M. Czupryna, and T. Szapiro. Multiobjective Programming and Goal Programming: Theoretical Results and Practical Applications, volume 618 of Lecture Notes in Economics and Mathematical Systems, chapter On Conditional Value-at-Risk Based Goal Programming Portfolio Selection Procedure, pages 243–252. Springer-Verlag Berlin Heidelberg, 2009. E.M. Kasprzak and K. Lewis. An approach to facilitate decision tradeoffs in Pareto solution sets. Journal of Engineering Valuation and Cost Analysis, 3(2):173–187, 2000. I.Y. Kim and OL De Weck. Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation. Structural and Multidisciplinary Optimization, 31(2):105–116, 2006. J.D. Knowles and D.W. Corne. Approximating the nondominated front using the Pareto Archived Evolution Strategy. Evolutionary Computation, 8(2):149–172, 2000. H. Konno. Piecewise linear risk function and portfolio optimization. Journal of the Operations Research Society of Japan, 33(2):139–156, 1990. H. Konno and H. Yamazaki. Mean-absolute deviation portfolio optimization model and its application to Tokyo stock market. Management Science, 37:519–531, 1991. D. Kuhn, P. Parpas, and B. Rustem. Threshold Accepting Approach to Improve Boundbased Approximations for Portfolio Optimization. Computational Methods in Financial Engineering, Part-I:3–26, 2008. S. Kukkonen and J. Lampinen. GDE3: The third evolution step of generalized differential evolution. In The 2005 IEEE Congress on Evolutionary Computation, 2005, pages 443– 450, 2005. M. Larbani and B. Aouni. On the pareto optimality in goal programming. In ,Proceedings of ASAC 2007: the Annual Conference of the Administrative Sciences Association of Canada, 2007. M. Larbani and B. Aouni. A new approach for generating efficient solutions within the goal programming model. Journal of the Operational Research Society, 2010.

118

REFERENCES

D. Li and W.L. Ng. Optimal dynamic portfolio selection: multiperiod mean-variance formulation. Mathematical Finance, 10:387–406, 2000. J. Li and S. Taiwo. Enhancing Financial Decision Making Using Multi-Objective Financial Genetic Programming. In Proceedings of IEEE Congress on Evolutionary Computation 2006, pages 2171–2178, 2006. C.-M. Lin and M. Gen. Multiobjective resource allocation problem by multistage decisionbased hybrid genetic algorithm. Applied Mathematics and Computation, 187:574–583, 2007. C.-M. Lin and M. Gen. Multi-criteria human resource allocation for solving multistage combinatorial optimization problems using multiobjective hybrid genetic algorithm. Expert Systems with Applications, 34:2480–2490, 2008. M. Liu and F.F. Wu. Portfolio optimization in electricity markets. Electric Power Systems Reseach, 77:1000–1009, 2007. M.S. Lobo, M. Fazel, and S. Boyd. Portfolio optimization with linear and fixed transaction costs. Annals of Operations Research, 152:341–365, 2007. M. Ludkovski. Optimal Switching with Applications to Energy Tolling Agreements. PhD thesis, Princeton University, 2005. R. Mansini, W. Ogryczak, and M.G. Speranza. LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics, 14:187–220, 2003. R. Mansini, W. Ogryczak, and M.G. Speranza. Conditional value at risk and related linear programming models for portfolio optimization. Annals of Operations Research, 152:227– 256, 2005. H. Markowitz. Portfolio selection. Journal of Finance, 7:77–91, 1952. R.T. Marler and J.S. Arora. Survey of multi-objective optimization methods for engineering. Structural and Multidisciplinary Optimization, 26:369–395, 2004. R.T. Marler and J.S. Arora. The weighted sum method for multi-objective optimization: new insights. Structural and Multidisciplinary Optimization, 41:853–862, 2010. doi: 10.1007/s00158-009-0460-7. R. Merton. Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3:373–413, 1971.

REFERENCES

119

A. Messac. From dubious construction of objective functions to the application of physical programming. AIAA journal, 38(1):155–163, 1900. A. Messac. Physical programming: Effective optimization for computational design. AIAA journal, 34(1):149–158, 1996. A. Messac and A. Ismail-Yahaya. Required relationship between objective function and Pareto frontier orders: Practical implications. AIAA journal, 39(11):2168–2174, 2001. A. Messac and A. Ismail-Yahaya. Multiobjective robust design using physical programming. Structural and multidisciplinary optimization, 23(5):357–371, 2002. A. Messac and C.A. Mattson. Generating well-distributed sets of Pareto points for engineering design using physical programming. Optimization and Engineering, 3(4):431–450, 2002. A. Messac and C.A. Mattson. Normal constraint method with guarantee of even representation of complete pareto frontier. AIAA journal, 42(10):2101–2111, 2004. A. Messac, C. Puemi-Sukam, and E. Melachrinoudis. Aggregate objective functions and pareto frontiers: Required relationships and practical implications. Optimization and Engineering, 1(2):171–188, 2000. A. Messac, CP Sukam, and E. Melachrinoudis. Mathematical and pragmatic perspectives of physical programming. AIAA journal, 39(5):885–893, 2001. A. Messac, A. Ismail-Yahaya, and C.A. Mattson. The normalized normal constraint method for generating the Pareto frontier. Structural and Multidisciplinary Optimization, 25:86– 98, 2003. K. Miettinen. Nonlinear multiobjective optimization. Springer, 1999. A. Morton and S.R. Pliska. Optimal portfolio management with fixed transaction costs. Math Finance, 5:337–356, 1995. A. Mukerjee, R. Biswas, K. Deb, and A.P. Mathur. Multi-objective evolutionary algorithm for the risk-return trade-off in bank loan management. International Transactions in Operational Research, 9:583–597, 2002. K. Muthuraman and S. Kumar. Multi-dimensional Portfolio Optimization with Proportional Transaction Costs, October 2004. URL SSRN: http://ssrn.com/abstract=563944. A.J. Nebro, J.J. Durillo, F. Luna, B. Dorronsoro, and E. Alba. A Cellular Genetic Algorithm for Multiobjective Optimization. In Proceedings of NICSO 2006, pages 25–36, 2006a.

120

REFERENCES

A.J. Nebro, F. Luna, E. Alba, A. Beham, and B. Dorronsoro. AbYSS Adapting Scatter Search for Multiobjective Optimization. Tech Rep ITI-2006-2, Departamento de Lenguajes y Ciencias de la Computacin, University of Malaga, 2006b. S.S. Nielsen and R. Poulsen. A two-factor stochastic programming model of Danish mortgage-backed securities. Journal of Economic Dynamics and Control, 28:1267–1289, 2004. S. Ortobelli, S.T. Rachev, and F.J. Fabozzi. Risk management and dynamic portfolio selection with stable Paretian distributions. Journal of Empirical Finance, xxx:xxx–xxx, 2009. G.Ch. Pflug. Scenario tree generation for multiperiod financial optimization by optimal discretization. Mathematical Programming, 89:251–271, 2001. M. Pinar. Robust scenario optimization based on downside-risk measure for multi-period portfolio selection. OR Specturm, 29:295–309, 2007. S.R. Pliska and K. Suzuki. Optimal tracking for asset allocation with fixed and proportional transaction costs. Quantitative Finance, 4:233–243, 2004. K.M. Rasmussen and J. Clausen. Mortgage loan portfolio optimization using multi-stage stochastic programming. Journal of Economic Dynamics & Control, 31:742–766, 2007. I. Rechenberg. Cybernetic solution path of an experimental problem. Library translation, 1122, 1965. J.A. Robins and M.F. Wiersema. The Measurement of Corporate Portfolio Strategy: Analysis of the Content Validity of Related Diversification Indexes. Strategic Management Journal, 24:39–59, 2003. R.T. Rockafellar and S. Uryasev. Optimization of conditional value-at-risk. Journal of Risk, 2(3):21–41, 2000. R.T. Rockafellar and S. Uryasev. Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26:1443–1471, 2002. C. Romero. Extended lexicographic goal programming: a unifying approach. Omega, 29: 63–71, 2001. C. Romero. A general structure of achievement function for a goal programming model . European Journal of Operational Research, 153(3):675–686, March 2004.

REFERENCES

121

B. Roy. Classement et choix en prsence de points de vue multiples (la mthode ELECTRE). la Revue d’Informatique et de Recherche Oprationelle (RIRO), 8:57–75, 1968. B. Roy. Methodologie Multicritiere d’Aide a la Decision. Econometrica, Paris, France, 1985. B. Rustem and N. Gülpinar. Worst-case optimal robust decisions for multiperiod portfolio optimization. European Journal of Operational Research, 2007. S. Ruzika and M.M. Wiecek. Approximation methods in multiobjective programming. Journal of Optimization Theory and Applications, 126(3):473–501, 2005. V. Ryabchenko and S. Uryasev. Pricing Energy Derivatives by Linear Programming: Tolling Agreement Contracts. The Journal of Computational Finance, 2011. J.H. Ryu, S. Kim, and H. Wan. Pareto Front Approximation with Adaptive Weighted Sum Method in Multiobjective Simulation Optimization. In M.D. Rossetti, R.R. Hill, B. Johansson, A. Dunkin, and R.G. Ingalls, editors, Proceedings of the 2009 Winter Simulation Conference, 2009. S. Sarykalin, G. Serraino, and S. Uryasev. Value-at-Risk vs. Conditional Value-at-Risk in Risk Management and Optimization. Tutorials in Operation Research Informs, pages 270–294, 2008. S. Shan and G.G. Wang. An efficient Pareto set identification approach for multiobjective optimization on black-box functions. Journal of Mechanical Design, 127:866, 2005. R. Shen and S. Zhang. Robust portfolio selection based on a multi-stage scenario tree. European Journal of Operational Research, 191:864–887, 2008. P.K. Shukla. Computational Science - ICCS 2007, volume Lecture Notes in Computer Science, chapter On the Normal Boundary Intersection Method for Generation of Efficient Front, pages 310–317. Springer Berlin / Heidelberg, 2007. J.D. Siirola, S. Hauan, and A.W. Westerberg. Computing Pareto fronts using distributed agents. Computers & Chemical Engineering, 29(1):113–126, 2004. J. Skaf and S. Boyd. Multi-Period Portfolio Optimization with Constraints and Transaction Costs. Technical report, Working Paper, Stanford University, 2008. M.C. Steinbach. Markowitz revisited: Mean-variance models in financial portfolio analysis. SIAM Review, 43(1):31–85, 2001. R. E. Steuer, Y. Qi, and M. Hirschberger. Portfolio Selection in the Presence of Multiple Criteria. In C. Zopounidis, M. Doumpos, and P.M. Pardalos, editors, Handbook of Financial Engineering. Springer, 2008.

122

REFERENCES

C. Stummer and M. Sun. New Multiobjective Metaheuristic Solution Procedures for Capital Investment Planning. Journal of Heuristics, 11:183–199, 2005. G. Szegö. Measures of risk. Journal of Banking & Finance, 26:1253–1272, 2002. S. Takriti and S. Ahmed. On robust optimization of two-stage systems. Mathematical Programming, 99:109–126, 2004. M. Taksar, M.J. Klass, and D. Assaf. A diffusion model for optimal portfolio selection in the presence of brokerage fees. Mathematics of Operations Research, 13:277–294, 1988. T. Tamura. Maximizing the Growth Rate of a Portfolio with Fixed and Proportional Transaction Costs. Applied Mathematics and Optimization, 54:95–116, 2006. C. Tapiero. Risk and Financial management: Mathematical and Computational Methods. John Wiley & Sons, Ltd, 2004. N. Topaloglou, H. Vladimirou, and S.A. Zenios. A dynamic stochastic programming model for international portfolio management. European Journal of Operational Research, 185 (3):1501–1524, 2008. P.K. Tripathi, S. Bandyopadhyay, and S.K. Pal. Multi-objective particle swarm optimization with time variant inertia and acceleration coefficients. Information Sciences, 177(22):5033– 5049, 2007. S.V. Utyuzhnikov, P. Fantini, and M.D. Guenov. A method for generating a well-distributed Pareto set in nonlinear multiobjective optimization. Journal of Computational and Applied Mathematics, 223(2):820–841, 2009. ˘ ˘ A. Zilinskas and J. Zilinskas. Global Optimization Based on a Statistical Model and Simplicial Partitioning. Computers and Mathematics with Applications, 44:957–967, 2002. D.A. Van Veldhuizen and G.B. Lamont. Multiobjective Evolutionary Algorithm Research: A History and Analysis. Tech. Rep. TR-98-03, Dept. Elec. Comput. Eng., Graduate School of Eng., Air Force Inst. Technol, Wright-Patterson, AFB, OH, 1998. M.A. Villalobos-Arias, G.T. Pulido, and C.A. Coello Coello. A Proposal to use stripes to maintain diversity in a multi-objective particle swarm optimizer. In Proceedings of Swarm Intelligence Symposium, 2005. S.-M. Wang, J.-C. Chen, H. M. Wee, and K. J. Wang. Non-linear Stochastic Optimization Using Genetic Algorithm for Portfolio Selection. International Journal of Operations Research, 3(1):16–22, 2006.

REFERENCES

123

S.-Z. Wei and Z.-X. Ye. Multi-period optimization portfolio with bankruptcy control in stochastic market. Applied Mathematics and Computation, 186:414–425, 2007. Z. Wu, Z. Ni, C. Zhang, and L. Gu. A Novel PSO For Multi-stage Portfolio Planning. In International Conference on Artificial Intelligence and Computational Intelligence, 2009. Y. Xia, S. Wang, and X. Deng. Theory and methodology: a compromise solution to mutual funds portfolio selection with transaction costs. European Journal of Operation Research, 134:564–581, 2001. M.R. Young. A minimax portfolio selection rule with linear programming solution. Management Science, 44:673–683, 1998. L.Y. Yu, X.D Ji, and S.Y Wang. Stochastic programming models in financial optimization: A survey. Advanced Modelling and Optimization, 5(1):75–201, 2003. A. Yushkevich. Optimal switching problem for countable Markov chains: average reward criterion. Mathematical methods of operations research, 53:1–24, 2001. L.A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. V.I. Zakamouline. European option pricing and hedging with both fixed and proportional transaction costs. Journal of Economic Dynamics & Contro, 30:1–25, 2006. S.A. Zenios, M.R. Holmer, R. McKendall, and C. Vassiadou-Zeniou. Dynamic models for fixed income portfolio management under uncertainty. Journal of Economic Dynamics and Control, 22:1517–1541, 1998. X.-L. Zhang and K.-C. Zhang. Using genetic algorithm to solve a new multi-period stochastic optimization model. Journal of Computational and Applied Mathematics, 231:114–123, 2009. A. Zhou, Y. Jin, Q. Zhang, B. Sendhoff, and E. Tsang. Combining model-based and geneticsbased offspring generation for multi-objective optimization using a convergence criterion. 2006 IEEE Congress on Evolutionary Computation, pages 3234–3241, 2006. E. Zitzler. Evolutionary algorithms for multiobjective optimization: Methods and applications. PhD thesis, Swiss Federal Institute of Technology Zurich, 1999. E. Zitzler and L. Thiele. Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Study. In A.E. Eiben, editor, Conference on Parallel Problem Solving from Nature, pages 292–301. Springer-Verlag, Amsterdam, 1998.

124

REFERENCES

E. Zitzler, K. Deb, and L. Thiele. Comparison of Multi Objective Evolutionary Algorithms: Empirical Results. Evolutionary Computation, 8(2):173–195, 2000. E. Zitzler, M. Laumanns, L. Thiele, et al. SPEA2: Improving the strength Pareto evolutionary algorithm. EUROGEN 2001. Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems, pages 95–100, 2002. E. Zitzler, D. Brockhoff, , and L. Thiele. The Hypervolume Indicator Revisited: On the Design of Pareto-compliant Indicators Via Weighted Integration . In Evolutionary MultiCriterion Optimization . Springer Berlin / Heidelberg, 2007. C. Zopounidis and M. Doumpos. Multi-criteria decision aid in financial decision making: methodologies and literature review. Journal of multi-criteria decision analysis, 11(4-5): 167–186, 2002.

algorithms for uniform distribution of solutions over pareto set, and

algorithms for uniform distribution of solutions over pareto set, and

Suggest Documents

Uniform Distribution Uniform Distribution Uniform Distribution ...

Approximating the Set of Pareto-Optimal Solutions in ... - IEEE Xplore

Reliability and Quantile Analysis of Pareto Distribution

The Effect of Non-uniform Temperature Distribution over the Charge

Algorithms for Nash and Pareto Equilibria for Resource ... - arXiv

for the generalized Pareto distribution - Semantic Scholar

Maximum Likelihood Estimation for Generalized Pareto Distribution ...

Inferences for New Weibull-Pareto Distribution

The New Weibull-Pareto Distribution

Weibull-Pareto Distribution and Its Applications

The Double Pareto-Lognormal Distribution - Mathematics and ...

The Generalized Pareto Distribution and Threshold ...

learning dnf over the uniform distribution using a quantum example ...

learning dnf over the uniform distribution using a quantum example

Evaluation of Pareto Optimal Solutions in Intermodal

Approximate Pareto Optimal Solutions of Multi

Constant factor Approximation Algorithms for Uniform Hard ...

The Pareto or Truncated Pareto Distribution? - Semantic Scholar

A Convergent Approximation of the Pareto Optimal Set for Finite

Uniform Distribution - Khadi Uniform Clarification.pdf - Google Drive

uniform distribution theory

uniform distribution theory - Boku

uniform distribution theory

uniform distribution theory