Differential Evolution using Matlab

A report on,

Differential Evolution using Matlab Prepared in partial fulfillment of Study oriented seminar,

Course code: CE G514 Structural Optimization

Submitted by,

Mast. Mandar Pandurang Ganbavale (2013H143013H)

Under guidance and supervision of

Prof. A. Vasan

Birla Institute of Technology and Science, Pilani. Hyderabad Campus. (2013- 2014)

Differential Evolution Using Matlab

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CERTIFICATE This is certified that the project entitled

“Differential evolution using Matlab” Submitted by,

Mr. Ganbavale Mandar Pandurang (2013H143013H) In partial fulfillment of the requirements of course code CE G514 Study Oriented project, BITS Pilani, Hyderabad campus for the academic year 2013-2014. It is the record of their own work carried out under our supervision and guidance.

Prof. A. Vasan (Department of Civil Engineering) Birla Institute of Technology and Science, Pilani. Hyderabad Campus. 2013-14.

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ACKNOWLEDGMENT

Firstly, I would like to thank the curriculum of BITS, Pilani for giving me this opportunity in doing Study oriented project. I sincerely appreciate Prof. P. N. Rao for giving me opportunity to work on this topic. I am greateful for his support and guidance that have helped me to expand my horizon of thought and expression. I am also thankful to my friends for their help in sharing the information about various aspects of this project.

Mr. Mandar P. Ganbavale (2013H143013H)

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Outline of the Proposed Topic of Project work Name of Candidate

:

Mandar P. Ganbavale

ID No.

:

2013H143013H

Place of Research Work and Organization

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Department of Civil Engineering BITS Pilani, Hyderabad Campus

Proposed Supervisor’s Details Name

:

Dr. A. Vasan

Qualification

:

Ph. D

Designation

:

Professor

Organization

:

Department of Civil Engineering BITS Pilani, Hyderabad Campus

Project topic Differential Evolution using MATLAB.

Objective of the Proposed Project work 1. To study the multi-objective optimization technique Differential Evolution. 2. To study Application of this technique to civil engineering problems. 3. To apply this optimization technique to solve the problem having a 25 bar space truss to minimize weight, deflection and maximize its frequency.

Work done till now…….. - Read literatures regarding what is Differential Evolution algorithm is exactly and how it differs from GA.

- Read journals and papers regarding same. - Read application of this algorithm in Welding problem and 2D truss problem done in literatures.

- Read their MATLAB coding.

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Introduction Optimization problems are everywhere in academic research and real-world applications such as in engineering, finance, and scientific areas. Wherever resources like space, time and cost are limited, optimization problem arises. So, with no doubt, researchers and practitioners need an efficient and robust optimization approach to solve problems of different characteristics that are fundamental to their daily work, but at the same time, it is expected that solving a complex optimization problem itself should not be very difficult. In addition, an optimization algorithm should be able to reliably converge to the true optimum for a variety of different problems. Furthermore, the computing resources spent on searching for a solution should not be excessive. Thus, a useful optimization method should be easy to use, reliable and efficient to achieve satisfactory solutions. Differential Evolution (DE) is such an optimization approach that addresses these requirements. - Differential evolution algorithm is invented by Storn and Prince in 1995. - Differential Evolution has been shown to be a simple yet efficient optimization approach in solving a variety of benchmark problems as well as many real-world applications. - Differential evolution together with evolution strategies (ES’s) and evolutionary programing (EP) can be categorized into a class of population –based, Derivative free methods known as Evolutionary algorithm. - All this approaches mimic Darwinian evolution and evolve a population of individuals from one generation to another by analogous evolutionary operations such as mutation, crossover and selection. Steps involves differential evolution algorithm Initialization Mutation Cross-over Selection

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Convergence?

Yes

1. Sampling the search space at multiple, randomly chosen initial points i.e. a population of individual vectors. 2. Since, Differential evolution is in nature a derivative-free continuous function optimizer, it encodes parameters as floating-point numbers and manipulates them with simple arithmetic operations such as addition, subtraction, and multiplication and generates new points that are the perturbations/mutations of existing points. For this differential evolution mutates a (parent) vector in the population with a scaled difference of other randomly selected individual vectors.

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3. The resultant mutation vector is crossed over with the corresponding parent vector to generate a trial or offspring vector. 4. Then, in a one-to-one selection process of each pair of offspring and parent vectors, the one with a better fitness value survives and enters the next generation. This procedure repeats for each parent vector and the survivors of all parent-offspring pairs become the parents of a new generation in the evolutionary search cycle. The evolutionary search stops when the algorithm converges to the true optimum or a certain termination criterion such as the number of generations is reached.

Background of Differential evolution algorithm 1. Evolutionary algorithm -

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Follows the principle of natural evolution and survival of the fittest that are described by the Darwinian Theory. The major applications of evolutionary algorithms are in optimization, although they have also been used to conduct data mining, generate learning systems, and build experimental frameworks to validate theories about biological evolution and natural selection, etc. EAs differ from traditional optimization techniques in that they usually evolve a population of solutions or individual points in the search space of decision variables, instead of starting from a single point. At each iteration, an evolutionary algorithm generates new solutions which is also called as offspring by mutating and/or recombining current or parent solutions and then conducts a competitive selection to weeds out poor solutions. In comparison with traditional optimization techniques, such as calculus-based nonlinear programming methods in, evolutionary algorithms are usually more robust and achieve a better balance between the exploration and exploitation in the search space when optimizing many real-world problems. Different main streams of evolutionary algorithms have evolved over the past forty years. The majority of the current implementations descend from three strongly related but independently developed branches: o Evolution strategies o Genetic algorithms, and o Evolutionary programming. These approaches are closely related to each other in terms of their underlying principles, while their exact operations and usually applied problem domains differ from one approach to another. They are better suited for discrete optimization because the decision variables are originally encoded as bit strings and are modified by logical operators. Evolution strategies and evolutionary programming, however, concentrate on mutation, although evolution strategies may also incorporate crossover or recombination as an operator. Evolutionary strategies is a continuous function optimizer in nature because it encodes parameters as floating-point numbers and manipulates them by arithmetic operations.

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Although differences exist among these evolutionary algorithms, they all rely on the concept of a population of individuals or solutions, which undergo such probabilistic operations as mutation, crossover and selection to evolve toward solutions of better fitness in the search space of decision variables. - The mutation introduces new information into the population by randomly generating variations to existing individuals. - The crossover or recombination typically performs an information exchange between different individuals in the current population. - The selection imposes a driving force towards the optimum by preferring individuals of better fitness. - The fitness value may reflect the objective function value and/or the level of constraint satisfaction. - These operations compose a loop, and evolutionary algorithms usually execute a number of generations until the obtained best-so-far solution is satisfactory or other termination criterion is fulfilled. 2. Differential evolution Similar to other evolutionary algorithms, differential evolution is a population based, derivative-free function optimizer. It usually encodes decision variables as floatingpoint numbers and manipulates them with simple arithmetic operations such as addition, subtraction, and multiplication. The initial population {𝑋1,𝑖,0 = (𝑥1,𝑖,0 , 𝑥2,𝑖,0 , 𝑥3,𝑖,0 , … … . . , 𝑥𝐷,𝑖,0 )|𝑖 = 1,2,3, … . . , 𝑁𝑃} is randomly generated according to a normal or uniform distribution 𝑥𝑗𝑙𝑜𝑤 ≤ 𝑥𝑗,𝑖,0 ≤ 𝑥𝑗𝑢𝑝 𝑓𝑜𝑟 𝑗 = 1,2,3,4,5, … . 𝐷 Where, 𝑁𝑃 = Population size D = Dimension of the problem 𝑥𝑗𝑙𝑜𝑤 = Lower limit of jth vector component 𝑥𝑗𝑢𝑝 = Upper limit of jth vector component After initialization, DE enters a loop of evolutionary operations: mutation, crossover and selection. 2.1.Mutation At each generation g, this operation creates mutation vectors 𝑣𝑖,𝑔 based on the current parent population {𝑋1,𝑖,0 = (𝑥1,𝑖,0 , 𝑥2,𝑖,0 , 𝑥3,𝑖,0 , … … . . , 𝑥𝐷,𝑖,0 )|𝑖 = 1,2,3, … . . , 𝑁𝑃} the following are different mutation strategies frequently used in the literature, DE/rand/1

𝑣𝑖,𝑔 = 𝑋𝑟0 ,𝑔 + 𝐹𝑖 (𝑋𝑟1,𝑔 − 𝑋𝑟2 ,𝑔 )

DE/current-to-rest/1

𝑣𝑖,𝑔 = 𝑋𝑖,𝑔 + 𝐹𝑖 (𝑋𝑏𝑒𝑠𝑡,𝑔 − 𝑋𝑖,𝑔 ) + 𝐹𝑖 (𝑋𝑟1 ,𝑔 − 𝑋𝑟2,𝑔 )

DE/best/1

𝑣𝑖,𝑔 = 𝑋𝑏𝑒𝑠𝑡,𝑔 + 𝐹𝑖 (𝑋𝑟1,𝑔 − 𝑋𝑟2 ,𝑔 )

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Where, 𝑟0 , 𝑟1 , 𝑟2 = distinct integers uniformly chosen from the set {1,2, … . . , 𝑁𝑃}\{𝑖} , 𝑋𝑟1,𝑔 − 𝑋𝑟2,𝑔 = difference vector to mutate the parent, 𝑋𝑏𝑒𝑠𝑡,𝑔 = best vector at the current generation 𝐹𝑖 = the mutation factor which usually ranges on the interval (0, 1+). In classic DE algorithms, Fi = F is a single parameter used for the generation of all mutation vectors, while in many adaptive DE algorithms each individual i is associated with its own mutation factor Fi. The above mutation strategies can be generalized by implementing multiple difference vectors other than 𝑋𝑟1 ,𝑔 − 𝑋𝑟2 ,𝑔 . The resulting strategy is named as ‘DE/–/k’ depending on the number k of difference vectors adopted. The mutation operation may generate trial vectors whose components violate the predefined boundary constraints. Possible solutions to tackle this problem include resetting schemes, penalty schemes, etc. A simple method is to set the violating component to be the middle between the violated bound and the corresponding components of the parent individual, i.e. 𝑣𝑗,𝑖,𝑔

(𝑥𝑗𝑙𝑜𝑤 + 𝑥𝑗,𝑖,𝑔 ) = 𝑖𝑓 𝑣𝑗,𝑖,𝑔 < 𝑥𝑗𝑙𝑜𝑤 2

𝑣𝑗,𝑖,𝑔 =

(𝑥𝑗𝑢𝑝 + 𝑥𝑗,𝑖,𝑔 ) 2

𝑖𝑓 𝑣𝑗,𝑖,𝑔 > 𝑥𝑗𝑢𝑝

where 𝑣𝑗,𝑖,𝑔 and 𝑥𝑗,𝑖,𝑔 are the j-th components of the mutation vector 𝑣𝑖,𝑔 and the parent vector 𝑋𝑖,𝑔 at generation g, respectively. This method performs well especially when the optimal solution is located near or on the boundary. 2.2.Cross-over After mutation, a ‘binomial’ crossover operation forms the final trial vector 𝑢𝑖,𝑔 = (𝑢1,𝑖,𝑔 , 𝑢2,𝑖,𝑔 , … … … . , 𝑢𝐷,𝑖,𝑔 ) 𝑢𝑗,𝑖,𝑔 = {

𝑣𝑗,𝑖,𝑔 … … … … … … . . 𝑖𝑓 𝑟𝑎𝑛𝑑𝑗 (0,1) ≤ 𝐶𝑅𝑖 𝑜𝑟 𝑗 = 𝑗𝑟𝑎𝑛𝑑 𝑥𝑗,𝑖,𝑔 … … … … … … … … … … … … … … … … . . 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Where, 𝑟𝑎𝑛𝑑𝑗 (𝑎, 𝑏)= uniform random number on the interval (a, b) and newly generated for each j, 𝑗𝑟𝑎𝑛𝑑 = 𝑗𝑟𝑎𝑛𝑑𝑖𝑎𝑛𝑡 (1, D) = integer randomly chosen from 1 to D and newly generated for each i, the crossover probability, 𝐶𝑅𝑖 ∈ [0,1], roughly corresponds to the average fraction of vector components that are inherited from the mutation vector. In classic DE, 𝐶𝑅𝑖 = CR is a single parameter that is used to generate all trial vectors, while in many adaptive DE algorithms, each individual I is associated with its own crossover probability 𝐶𝑅𝑖 . 5

2.3.Selection The selection operation selects the better one from the parent vector 𝑋𝑖,𝑔 and the trial vector 𝑢𝑖,𝑔 according to their fitness values f (・). For example, if we have a minimization problem, the selected vector is given by 𝑋𝑖,𝑔+1 = {

𝑢𝑖,𝑔 … … … … … … . . 𝑖𝑓 𝑓(𝑢𝑖,𝑔 ) < 𝑋𝑖,𝑔 𝑋𝑖,𝑔 … … … … … … . … … . . 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

and used as a parent vector in the next generation. The above one-to-one selection procedure is generally kept fixed in different DE algorithms, while the crossover may have variants other than the binomial operation in. Thus, a DE algorithm is historically named, for example, DE/rand/1/bin connoting its DE/rand/1 mutation strategy and binomial crossover operation.

Application of DE for problem Single objective problem The welded beam shown in Figure is designed for minimum cost subject to constraints on shear stress in weld (t ), bending stress in the beam (s), buckling load on the bar (P), end deflection of the beam (d), and side constraints.

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Instructions for using the program Variable definition n = population size itermax = maximum number of iterations F = Scaling factor Cr = Crossover rate Lb = Lower bound Ub = Upper bound d = no. of variables Sol = randomly generated population Best = best member fmin = minimum value of the objective function V = mutated vector g = constraints Steps during the program 1. Initialize all the parameters required to run the code i.e., n, Lb, Ub, itermax, d, etc. 2. After initialization has been done a random population is generated between the upper and lower bounds. 3. The constraints are evaluated and the objective function obtained for the entire population 4. The main part of the program consists of the mutation, crossover and selection which is part of the code and it runs to give the best member of the iteration and the best global member also. 5. During the main program two random vectors are added together with a scaling factor F and this is added to the best member vector calculated and a bound check is imposed. 6. After the mutation process takes place the crossover of the vectors to produce a trial vector starts. 7. The trial vector is obtained and is compared with the present best member and if found to be better it is selected in to the new population for next and further iterations. 8. Finally the results of the best member and the function value is obtained and tabulated. Results of the program The following table is generated for 500 iterations:

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We see that the value obtained for F = 0.5 and CR = 0.95 is the lowest i.e. fmin = 1.7251

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Figure 2: Showing fmin for F = 0.5 and CR= 0.8

Problem 2 Multi-objective optimization The optimal design of the three-bar truss shown in Figure is considered using two different objectives with the cross-sectional areas of members 1 (and 3) and 2 as design variables. ASSUMING VALUE OF H AS 1,

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Instructions for using program Variable definition n = population size itermax = maximum number of iterations F = Scaling factor Cr = Crossover rate Lb = Lower bound Ub = Upper bound d = no. of variables Sol1 = randomly generated population for population 1 Best1 = best member for population 1 fmin1 = minimum value of the objective function for population 1 V1 = mutated vector for population 1 g1 = constraints for population 1 Sol2 = randomly generated population for population 2 Best2 = best member for population 2 fmin2 = minimum value of the objective function for population 2 V2 = mutated vector for population 2 g2 = constraints for population 2 Steps during the program 1. Initialize all the parameters required to run the code i.e., n, Lb, Ub, itermax, d, etc. 2. After initialization has been done a random population is generated between the upper and lower bounds. 3. The constraints are evaluated and the objective function obtained for the entire population 4. The main part of the program consists of the mutation, crossover and selection which is part of the code and it runs to give the best member of the iteration and the best global member also. 5. During the main program two random vectors are added together with a scaling factor F and this is added to the best member vector calculated and a bound check is imposed. 6. After the mutation process takes place the crossover of the vectors to produce a trial vector starts. 7. The trial vector is obtained and is compared with the present best member and if found to be better it is selected in to the new population for next and further iterations. 8. Finally the results of the best member and the function value is obtained and tabulated.

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Results The results shown below are for 250 iterations and a population size of 2000:

Hence it is seen that not much change is there in the values of fmin for both objective functions even though the value of CR is changing.

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Figure 3: The figures shows the results for F = 0.5 and CR = 0.8 for both the objective functions

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Matlab code from Literature clear all; close all; clc %Function to be minimized D=2; objf=inline('4*x1^2-2.1*x1^4+(x1^6)/3+x1*x2-4*x2^2+4*x2^4','x1','x2'); objf=vectorize(objf); %Initialization of DE parameters N=20; %population size (total function evaluations will be itmax*N, must be>=5) itmax=30; F=0.8; CR=0.5; %mutation and crossover ratio %Problem bounds a(1:N,1)=-1.9; b(1:N,1)=1.9; %bounds on variable x1 a(1:N,2)=-1.1; b(1:N,2)=1.1; %bounds on variable x2 d=(b-a); basemat=repmat(int16(linspace(1,N,N)),N,1); %used later basej=repmat(int16(linspace(1,D,D)),N,1); %used later %Random initialization of positions x=a+d.*rand(N,D); %Evaluate objective for all particles fx=objf(x(:,1),x(:,2)); %Find best [fxbest,ixbest]=min(fx); xbest=x(ixbest,1:D); %Iterate for it=1:itmax; permat=bsxfun(@(x,y) x(randperm(y(1))),basemat',N(ones(N,1)))'; %Generate donors by mutation v(1:N,1:D)=repmat(xbest,N,1)+F*(x(permat(1:N,1),1:D)-x(permat(1:N,2),1:D)); %Perform recombination r=repmat(randi([1 D],N,1),1,D); muv = ((rand(N,D)