An Ensemble of Differential Evolution Algorithms for Constrained Function Optimization M. Fatih Tasgetiren, P. Nagaratnam Suganthan, Quan-Ke Pan, Rammohan Mallipeddi, Sedat Sarman
Abstract—This paper presents an ensemble of differential evolution algorithms employing the variable parameter search and two distinct mutation strategies in the ensemble to solve real-parameter constrained optimization problems. It is well known that the performance of DE is sensitive to the choice of mutation strategies and associated control parameters. For these reasons, the ensemble is achieved in such a way that each individual is assigned to one of the two distinct mutation strategies or a variable parameter search (VPS). The algorithm was tested using benchmark instances in Congress on Evolutionary Computation 2010. For these benchmark problems, the problem definition file, codes and evaluation criteria are available in http://www.ntu.edu.sg/home/EPNSugan. Since the optimal or best known solutions are not available in the literature, the detailed computational results required in line with the special session format are provided for the competition.
he general constrained optimization problem (P) is r concerned with finding x so as to:
r r min f (x ) x = (x1 , x 2 ,.., x D )∈ ℜ D
(1) r where x ∈ F ⊆ S . The objective function f (x ) is defined on the search space S ⊆ ℜ D and the set F ⊆ S defines the feasible region. In general, S is defined as an D-dimensional space in ℜ D where the domains of the variables are defined by their lower and upper bounds as: L(i ) ≤ x i ≤ U (i ),
1≤ i ≤ D
(2)
The feasible region F is defined by a set of m additional constraints (m ≥ 0 ) as: This research is partially supported by National Science Foundation of China under Grants 60874075, 70871065, 60905039, and Science Research and Development of Provincial Department of Public Edu cation of Shandong under Grant J09LG29. M. Fatih Tasgetiren is with the Department of Industrial Engineering at Yasar University, Izmir, Turkey; Phone: +90 232 411 5301; e-mail: (
[email protected]). P. N. Suganthan is with the School of Electrical and Electronics Engineering at Nanyang Technological University 639798, Singapore (e-mail:
[email protected]). Q-K Pan is with the School of Computer Science, Liaocheng University, China. (e-mail:
[email protected]). R. Mallipeddi is with the School of Electrical and Electronics Engineering at Nanyang Technological University 639798, Singapore (e-mail:
[email protected]) Sedat Sarman is with the Department of Industrial Engineering at Yasar University, Izmir, Turkey; Phone: +90 232 411 5275; (e-mail:
[email protected]).
978-1-4244-8126-2/10/$26.00 ©2010 IEEE
(3) r (4) h j (x ) = 0 , for j = p + 1,.., m . r r A point x ⊆ F is called a feasible solution if x satisfies both r equality and inequality constraints, otherwise x is an infeasible solution. The problem P is said to be feasible if there exists at least one feasible point, infeasible otherwise. If r r r x is feasible and g i (x ) = 0 , the constraint g i (x ) ≤ 0 is said r to be active at x . Equality constraints are also active at all feasible points. The equality constraints can be transformed into inequality form and can be combined with other inequality constraints as Gi (x ) = max{g i ( X ),0}
H i (x ) = max{| hi ( X ) | −δ ,0} p
υ (x ) =
I. INTRODUCTION
T
r g i (x ) ≤ 0 , for i = 1,.., p and
i = 1,... p i = p + 1,..., m
(5)
m
∑ Gi (x ) + ∑ H i ( x ) i =1
i = p +1
m
where υ (x ) is the amount of violation and δ is a tolerance parameter for the equality constraints, which is predefined as δ = 0.0001 in this competition. Differential evolution (DE) is one of the latest evolutionary optimization methods proposed by Storn and Price [2,3]. Like other evolutionary-type algorithms, DE is a population-based, stochastic global optimizer. Since DE was first introduced to solve the Chebychev polynomial fitting problem by Storn and Price [2,3], it has been successfully applied in a variety of applications that can be found in Corne et al. [4], Lampinen [5], Babu and Onwubolu [6], and Price et al. [7]. The applications of DE to the constrained optimization problems can also be found in [8, 9, 10, 11, 12, 13, 14, 15, and 16]. The state of the art survey on constrained optimization is given in [17]. Based on the success of local search refinement in the vpsDE algorithm in [18] as well as the ensemble concept in [20, 27, 28], in this paper, we present an ensemble of differential evolution algorithms where each trial individual is generated by one of the three different strategies taking advantage of different DEs in an ensemble. The DE with the ensemble of three different strategies will be denoted as eDE from now on throughout the paper. Therefore, this paper aims at employing the eDE algorithm to optimize the benchmark suite presented in the special session of Real Parameter Constrained Optimization in CEC2010. This paper is organized as follows. Section II gives the methodology of the proposed eDE algorithm, computational results of test problems are shown in Section III. Finally,
Section IV summarizes the concluding remarks. II. DIFFERENTIAL EVOLUTION ALGORITHMS
A. Traditional DE As known, there exist several mutation variations in traditional DEs. We follow the DE / rand / 1 / bin scheme of Storn and Price [2,3] for a general description. The traditional DE algorithm starts with initializing the initial target population denoted by NP . Each individual in NP has an D -dimensional vector with parameter values determined randomly and uniformly between predefined search ranges denoted by xijmin and xijmax , respectively:
(
)
x ijt = x ijmin + x ijmax − x ijmin × r
(6)
where xijt is the i th target individual with respect to j th dimension at generation t ; and r is a uniform random number between 0 and 1. Mutant individuals are generated in such a way that DE perturbs vectors from the target population by adding the weighted difference between two randomly selected target population members to a third member in the target population as follows:
(
vijt = x ajt −1 + F × xbjt −1 − x cjt −1
)
(7)
where a , b , and c are three randomly chosen individuals from the target population such that (a ≠ b ≠ c ≠ i ∈ (1,.., NP )) and ( j = 1,2,.., D ) . F > 0 is a mutation scale factor which affects the differential variation between two individuals. Mutation phase is followed by the recombination of mutant individual with its corresponding target individual. To do so, a crossover operator is applied to obtain the trial individual such that: ⎧⎪vijt
uijt = ⎨ t −1 ⎪⎩ xij
if
rijt
≤ CR or
j = Dj
To decide whether or not the trial individual u it should be a member of the target population for the next generation, it is compared to its counterpart target individual xit −1 at the previous generation. The selection is based on the survival of the fittest among the trial and target individuals such that:
( ) ( )
⎧⎪u it if f u it ≤ f xit −1 xit = ⎨ t −1 ⎪⎩ xi otherwise
B. DE with Ensemble As mentioned before, the ensemble is achieved in such a way that each individual is assigned to one of the three distinct differential evolution strategies. In particular, the VPS strategy is especially employed in the ensemble to enhance its local exploitation capability. In fact, VPS is inspired from variable neighborhood search (VNS) algorithm of Mladenovic and Hansen [19]. We apply the VPS procedure when generating the trial individual at each generation. The traditional DE mutation and crossover operators as well as DE mutation strategies can be applied with variable parameters depending on the value of the neighboring counter k . However, since VPS procedure makes a “blind search” with a mutation strategy and a crossover operator for further improvement, it may waste considerable amount of function evaluations when applied to each individual in the population. In order to overcome this difficulty, one of the three different strategies S max = 3 given below is assigned to each individual, which is inspired from the ensemble of mutation strategies in [20]: S1 : VPS
i =1, 2,.., NP
S 2 : DE / best / 1 / bin :
CR = U (0,1) F = U (0,1)
(
F = U (0,1) K = U (0,1)
(
counterpart in the previous generation u it −1 . CR is a
uniform random number between 0 and 1. In other words, the trial individual is made up with some parameters of mutant individual, or at least one of the parameters randomly selected, and some other parameters of the target individual. The Eq. (8) is denoted by u it = CRk vit , xit −1 to ease the
(
i:=1, 2 ,.., NP
)
understanding of the pseudo code. Parameter values violating the search range during the evolution are restricted either to:
(
)
j = 1,2,.., D
(9)
)
CR = U (0,1)
parameter of each trial individual u it differs from its user-defined crossover constant in the range [0,1] , and r is a
i:=1, 2 ,.., NP
S 3 : DE / current − to − rand / 1 / bin :
( j = 1,2,.., D ) , which is used to ensure that at least one
t ij
(
)
t −1 vit = xbest + F x at −1 − xbt −1 , u it = CRk vit , xit −1 i =1,.., NP
where the index D j refers to a randomly chosen dimension
u ijt = xijmin + xijmax − xijmin × r1
( )
uit = VPS xit −1
(8)
otherwise
(10)
) (
u it = x it −1 + K x at −1 − x it −1 + F x bt −1 − x ct −1 i =1,.., NP
)
The above scheme provides an ensemble of VPS procedure and two different mutation strategies in the eDE algorithm. Given that δ and Φ are the temporary individuals, the pseudo codes of the eDE algorithm and VPS procedure are given in Figs. 2 and 3, respectively. Pr ocedure eDE Set NP , MaxFES , k max ,
nFES := 0
(
0 Π = x10 , π 20 ,.., x NP
)
Si
and
N k (v )
%Initialize parameters
% Set target population
S i = rand ()% S max
%Assign a strategy to each
i:=1, 2,.., NP
( )
f x i0 i:=1, 2,.., NP , nFES + +
xi
% Evaluate population via SF
{ ( )}
0 x best = arg min f x i0
%Find best solution via SF
i =1, 2,.., NP
t := 1 while
%Start generation
(nFES < MaxFES )
( ) f (u ) < f (x )
do {
u it = S i xit −1
%switch to each strategy
i =1, 2,.., NP
t t −1 i i i =1, 2,.., NP ; nFES + +
if
then {
x it = u it } i =1, 2,.., NP
%Evaluate trial individual via EC %And accept via SF
x it = x it −1 }
{ ( ) ( )}
t t −1 xbest = arg min f xit , f xbest
%Find best solution via SF
i:=1, 2,.., NP
t := t + 1 }endwhile
%Next generation
t return xbest
%Return best solution
endprocedure Fig. 2. The eDE Algorithm.
( )
Pr ocedure VPS xit −1 k max := 2 k := 1
)
%Set temporary individual
( )
δ = f xit −1 i =1,.., NP
do{
%Set temporary individual
a ≠ b ≠ c ≠ i ∈ (1,.., NP ) if (k = 1) then
%Select three individuals
CR = 0.9 F = 0. 9 (k = 2) then
if
CR = U (0,1) F = U (0,1)
( ) = CR (v , δ ) f (Φ ) < f (δ )
v it = N k v it −1
%Mutant individual
Φ ti
%Generate individual
i =1, 2,.., NP
if
then {
%Evaluate via NFT
δ = Φ ti i =1,.., NP
k := 1}
%Back to first neighborhood
else{
δ =δ if
%Next neighborhood k := k + 1} nFES > MaxFES then break
}while(k ≤ k max )
return δ
( ) N (v ) = x
( + F (x
t i
i =1,.., NP
t −1 a i =1,.., NP
t −1 b
− x ct −1
) )
(11) (12)
Regarding the parameters of the eDE algorithm, care must be taken. Traditional DE uses only three control parameters. These are Population Size ( NP ), Mutation Scale Factor ( F ) and Crossover Rate ( CR ). The choice of these parameters is critical for its performance [21]. F is generally related to the convergence speed. It is necessary for F to be of sufficient magnitude to avoid premature convergence. F = 0.9 is suggested as a good compromise between convergence speed and convergence probability [21]. Among F and CR , CR is much more sensitive to problem’s properties and multimodality. It is suggested in [22] that CR = 0.9 is a good choice. For these reasons, we choose CR = 0.9 and F = 0.9 as default parameters for the first neighbourhood (i.e., when k = 1 ) in the eDE algorithm. However, as suggested in [21], there exists no best parameter setting for all types of problems. Considering this suggestion, we randomly and uniformly assign CR value between 0 and 1 while F value is determined randomly and uniformly between 0 and 1for the second neighbourhood (i.e., when k = 2 ). Regarding the remaining three strategies, CR , F and K values are determined randomly and uniformly between 0 and 1 again. In addition to the above, the population size is also a critical choice for the performance of DE [22]. We fixed our population size to NP = 30 .
III. CONSTRAINT HANDLING
t i i:=1, 2,.., NP
t i i =1,.., NP , nFES + +
For the VPS loop, first of all, the traditional DE mutation strategies can be used to generate neighboring solutions. In order to avoid wasting objective function evaluations, we fixed the maximum neigborhood counter at k max = 2 . We did not carry out extensive experiments to determine which neighborhood is better. However, our intial runs indicated that taking the DE / rand / 1 / bin scheme slightly yielded better results. Thus we employed the DE / rand / 1 / bin scheme for both neighborhoods as follows:
2
i =1, 2,.., NP
f(
Fig. 3. The VPS Procedure.
N 1 v it = x at −1 + F x bt −1 − x ct −1
else{
δ = x it −1
endprocedure
%Return best trial individual
Since evolutionary operators may generate infeasible solutions, care must be taken with those violating the constraints. There exist different approaches to handle the constraints [17]. In this paper, we employed three popular approaches to handle the constraints: A. Superiority of Feasible Solutions (SF) In SF [23,24], when two solutions x a and xb are compared, x a is treated as superior to x b under the following conditions for a minimization problem: • x a is feasible and x b is not.
• •
ε (0 ) = υ (xθ )
x a and xb are both feasible and x a has a smaller
objective function value than xb . x a and x b are both infeasible, but x a has a smaller overall constraint violation υ (x) as computed by using Eq. (5).
B. The Adaptive Penalty Function (NFT)
The adaptive penalty approach is presented in [1] where the notion of a “near feasibility threshold” (NFT) corresponds to a “promising region” beyond the feasible region. The NFT is defined as a threshold distance from feasible region such that the search within feasible region and the NFT-neighborhood of the feasible region is encouraged whereas it is discouraged beyond that threshold. Additionally, an adaptive term is added to the penalty function to consider the gap between the best feasible value and best infeasible value found so far. Then the adaptive penalty function is defined as:
(
f p (x ) = f (x ) + f feas − f all
⎛ v i (x ) ⎞ ⎟ ∑ ⎜⎜ NFT ⎟ i =1 ⎝ i ⎠
)
m
αi
(13)
where f all denotes the unpenalized value of the best solution found yet and f feas denotes the value of the best feasible solution yet found. As noted in [17], the adaptive term may lead to zero-or over-penalty. For this reason, only the dynamic part of the above penalty function with NFT threshold is used in this paper as follows: α
⎛ G (x ) ⎞ f p (x ) = f (x ) + ∑ ⎜⎜ i ⎟⎟ + i =1 ⎝ NFTi ⎠ p
⎛ H j (x ) ⎞ ∑ ⎜⎜ NFT ⎟⎟ j = p +1 ⎝ j ⎠ m
α
The general form of the NFT is given by: NFT =
(14) NFT0 1+ λ *t
where NFT0 is an upper bound for the NFT; the lambda is a user-defined positive parameter; and t is the generation counter. The severity parameter α and the positive constant λ of the penalty function are taken as 0.5 and 0.004, respectively. Since the equality constraints are converted to the inequality constraints by subtracting δ from the absolute value of the constraint value and δ is predefined, the NFT0 is chosen as 1e-4. C. ε -Constraint (EC) The ε -constraint handling method was proposed in [25] in which the relaxation of the constraints is controlled by using the ε parameter. As solving a constrained optimization problem becomes tedious when active constraints are present, proper control of the ε parameter is essential [25] to obtain high quality solutions for problems with equality constraints. The ε level is updated until the generation counter k reaches the control generation t C . After the generation counter exceeds t C , the ε level is set to zero to obtain solutions with no constraint violation. The basic idea behind the EC method is that solutions with total violation less that ε (t ) are treated as feasible solutions when making selection for the next generation. The general structure is as follows:
(15) cp
⎧ ⎛ t ⎞ ⎪ε (0)⎜⎜1 − ⎟⎟ ε (t ) = ⎨ t C ⎠ , 0 < t < tC . ⎝ ⎪ t ≥ tC ⎩0,
(16)
where xθ is the top θ -th individual and the following parameters are employed in this paper: θ = 0.25 × NP , t C = 0.8 * MaxGen and cp = 2 . It is very important to note that we employed these three constraint handling methods in the eDE algorithm as follows: SF is only used when the best so far solution is updated. NFT is used in generating the trial individuals through VPS. However, ε -constraint is used during the selection process of eDE together with SF. IV.
COMPUTATIONAL RESULTS
The eDE algorithm was coded in C++ and run on an Intel P4 1.33 GHz Laptop PC with 256MB memory. In line with the guidelines described in [26], The maximum number of function evaluations is fixed at 2e4, 1e5 and 2e5 FES for 10 dimensions whereas it is fixed at 6e4,3e5 and 6e5 FES for 30 dimensions. The eDE algorithm was run on the 18 benchmark functions and the performance evaluation of the eDE algorithm is also conducted through the guidelines described for this session. 25 replications are conducted for each benchmark function to record the objective function value f (x ) after each termination criterion. The objective function values achieved at different FES levels are given in details in Tables I and II. Since the optimal or best known solutions are not known, no comparisons are provided. V.
CONCLUSIONS
In this paper, an ensemble of differential evolution algorithms is presented to solve the benchmark problems designed for the special session on real-parameter constrained optimization problems at CEC2010. Since we do not know how the other sophisticated algorithms performed for the benchmarks presented for this session, only the computational results with the necessary statistics are presented to be compared with the competing DE and non-DE algorithms during the conference and afterwards. REFERENCES [1] [2] [3] [4] [5]
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FES 2e4
Proceeding of the World Congress on Evolutionary Computation, Technology, Department of Information Technology, Laboratory of (CEC2009), pp. 1247-1254, 2009, Norway. Information Processing. [19] N. Mladenovic, P. Hansen, Variable neighborhood search, Computers Babu, B. V. and Onwubolu, G. C. (eds.) (2004) New Optimization and Operations Research 24 (1997) 1097-1100. Techniques in Engineering, Springer Verlag. [20] R. Mallipeddi, P. N. Suganthan, Q-K Pan, M. F. Tasgetiren, Price, K., Storn, R., and Lampinen, J. (2005) “Differential Evolution – Differential evolution algorithm with ensemble of parameters and A Practical Approach to Global Optimization”, Springer-Verlag. mutation strategies. Accepted by Applied Soft Computing. 2010. Storn, R. (1999) “System Design by Constraint Adaptation and [21] R. Gämperle, S. D Müller, and P. Koumoutsakos, (2002) “A parameter Differential Evolution,” IEEE Transactions on Evolutionary study for differential evolution,” in Proc.WSEAS Int. Conf. Advances Computation, vol. 3, pp. 22-34. Intell. Syst., Fuzzy Syst., Evol. Comput., 2002, pp. 293–298. Lampinen J. (2002) “A Constraint Handling approach for the [22] J. Rönkkönen, S. Kukkonen and K. Price, “Real-parameter Differential evolution Algorithm” Proc. of the Congress on optimization with differential evolution,” in Proc. 2005 IEEE Congr. Evolutionary Computation (CEC2002), pp. 1468-1473. Evol.Comput., Sep. 2005, pp. 506–513. Koziel S. And Michalewicz Z. (1999) “Evolutionary Algorithms, [23] K. Deb, "An efficient constraint handling method for genetic Homomorphous Mappings, and Constrained Parameter Optimization” algorithms," Computer Methods in Applied Mechanics and Evol. Comput., 7(1) pp. 19-44. Engineering, vol. 186, pp. 311-338, 2000. Lampinen J. (2001) “Multi-Constrained Optimization by the [24] D. Powell and M. Skolnick, "Using genetic algorithms in engineering Differential Evolution,” Proc. of the IASTED International Conference design optimization with non-linear constraints," in Proceedings of Artificial Intelligence Applications (AIA 2001), pp. 177-184. Fifth International conference on Genetic Algorithms San Mateo, Lampinen J. (2001) “Solving Problems Subject to Multiple Nonlinear Calfornia: Morgan Kaufmann, 1993, pp. 424-431. Constraints by the Differential Evolution”, Proc. of the MENDEL2001, 7th International Conference on Soft Computing, pp. 50-57. [25] T. Takahama and S. Sakai, "Constrained Optimization by the Constrained Differential Evolution with Gradient-Based Mutation and Lin Y-C, Hwang K-S, and Wang F-S, (2002) “Hybrid Differential Feasible Elites," in IEEE Congress on Evolutionary Computation Evolution with Multiplier updating method for Nonlinear Constrained Sheraton Vancouver Wall Centre Hotel, Vancouver, BC, Canada, 2006, Optimization” ,Proc. of the Congress on Evolutionary Computation pp. 1-8. (CEC2002), pp. 872-877. [26] R. Mallipeddi, P. N. Suganthan “Problem definitions and evaluation Chiou J-P and Wang F-S. (1999) “Hybrid Method of Evolutionary criteria for the CEC 2010 special session on single objective Algorithms for Static and Dynamic Optimization Problems with constrained real-parameter optimization,” in Technical Report, Applications to a Fed-Batch fermantation Process”, Computers and Nanyang Technological University, Singapore,2009. Chemical Engineering, 23, pp. 1277-1291. [27] M. Fatih Tasgetiren, P. N. Suganthan, Quan-Ke Pan. “An ensemble of Sarimveis H and Nikolakopoulos A. (2005) “A Line Up Evolutionary discrete differential evolution algorithms for solving the generalized Algorithm for Solving Nonlinear Constrained Optimization Problems”, traveling salesman problem”, Applıed Mathematıcs And Computatıon Computers & Operations Research, 32, pp. 1499-1514. Volume: 215 Issue: 9 Pages: 3356-3368, JAN 1 2010 Becerra R. L and Coello Carlos A. Coello. (2005) “Cultural Differential [28] Quan-Ke Pan; P. N Suganthan, M. Fatih Tasgetiren. “A Harmony Evolution for Constrained Optimization”, Comput. Methods Appl. Search Algorithm with Ensemble of Parameter Sets”, Evolutionary Mech. Engrg., (To appear). Computation, 2009. CEC '09. IEEE Congress on 18-21 May 2009 Coello Carlos A. Coello. (2002) “Theoretical and Numerical Page(s):1815 - 1820 Constraint-Handling Techniques Used with Evolutionary Algorithms: A Survey of the State of the Art”, Comput. Methods Appl. Mech. Engrg., 191(11-12), pp. 1245-1287. M. Fatih Tasgetiren, P. N Suganthan, Quan-Ke Pan, Yun-Chia Liang, “A Differential Evolution Algorithm with a variable Parameter Search for Real-Parameter Continuous Function Optimization, In the TABLE I FUNCTION VALUES ACHIEVED FOR 10D WHEN FES=2E4, FES=1E5, FES=2E5 C01 C02 C03 C04 C05 C06 Best
-7.473101E-001(0)
-7.220922E-001(0)
7.319701E-009(0)
5.597308E-001(1)
8.876102E+000(0) 1.996474E+000(1) 5.060443E+001(1)
1.622523E+002(1)
-5.690670E-001(0)
4.871975E+000(1)
5.115722E+001(1) 4.121813E+001(1) 3.011054E+002(1)
5.176603E+002(1)
c
0,0,0
0,0,1
0,0,0
0,0,3
0,2,2
0,1,2
V
0
1.042058E-004
0
5.514209E-003
1.299397E-001
8.733496E-003
Mean
-6.667501E-001
9.367292E-001
1.546557E+001
4.779142E+000
4.059339E+001
1.014672E+002
std
6.288644E-002
1.567518E+000
1.655396E+001
1.002633E+001
1.368660E+002
2.632677E+002
FR
1.00
0.16
0.92
0.00
0.04
0.08
-7.473103E-001(0)
-2.269559E+000(0)
0(0)
8.020929E-004(0) -4.752021E+002(0) -3.908274E+002(0)
Median -6.782508E-001(0)
-1.035763E+000(0)
8.875553E+000(0) 2.844044E-001(1) 1.472616E+001(0)
-6.202839E+001(0)
Median -6.782478E-001(0) worst
1e5
Best
worst
6.025697E-002(1) -2.546843E+001(0) -2.081857E+002(0)
-5.690675E-001(0)
1.691681E+000(1)
5.662135E+001(0) 9.369220E-001(1) 2.463553E+002(1)
4.541161E+002(1)
c
0,0,0
0,0,0
0,0,0
0,0,1
0,0,0
0,0,0
V
0
0
0
7.445441E-004
0
0
Mean
-6.667538E-001
-6.214606E-001
9.365370E+000
2.984739E-001
-6.177128E+001
-2.261494E+001
std
6.288966E-002
1.758163E+000
1.038582E+001
3.814705E-001
2.043058E+002
2.657196E+002
FR
1.00
0.96
1.00
0.28
0.84
0.76
2e5
Best
-7.473104E-001(0)
-2.277574E+000(0)
0(0)
Median -6.782508E-001(0)
-1.598583E+000(0)
8.875552E+000(0) 6.824028E-001(0) -7.378337E+001(0) -1.620113E+001(0)
worst
-5.690675E-001
7.157223E-002(0)
5.675149E+001(0) 9.249729E-001(1) 1.640942E+002(0)
-2.760900E+002(1)
c
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
V
0
0
0
0
0
0
Mean
-6.667539E-001
-1.581643E+000
9.725524E+000
1.509818E+000
-9.875375E+001
-1.900159E+002
std
6.288971E-002
6.551232E-001
1.022735E+001
3.791275E+000
1.489190E+002
4.460722E+002
FR
1.00
1.00
1.00
0.76
1.00
0.96
C07
C08
C09
C10
3.925242E-006(0)
4.578169E-011(0)
2.195501E-001(0)
1.181493E-005(0) -4.521675E-002(1)
FES 2e4
Best
Median 1.980709E+000(0)
1.089948E+003
6.414518E+010(1) 5.797001E+012(1) 4.945327E-001(1)
-8.361516E+002(1)
c
0,0,0
0,0,0
0,0,0
0,0,0
1,1,1
0,1,1
V
0
0
0
0
8.901266E+000
3.053936E-001
Mean
1.952579E+000
1.349478E+002
3.979892E+009
2.323815E+011
-4.838970E-001
-1.425700E+002
std
1.944569E+000
2.789521E+002
1.420725E+010
1.159298E+012
1.276571E+000
2.579423E+002
FR
1.00
1.00
0.68
0.80
0.00
0.00
Best
0(0)
0(0)
0(0)
0(0)
-1.496452E-003(0)
-5.683828E+002(0)
2.503812E+001(0) 8.356718E+000(0) 4.310304E+001(0) 2.388128E-002(1)
-1.979111E-001(0)
3.986579E+000
1.087798E+003(0) 2.338911E+012(0) 1.620829E+008(0) -2.260507E-001(1)
-8.870370E+002(1)
c
0,0,0
0,0,0
0,0,0
0,0,0
0,1,1
0,0,0
V
0
0
0
0
6.057732E-002
0
Mean
1.116242E+000
1.434094E+002
9.355662E+010
6.483849E+006
-2.214518E-002
-1.297672E+002
std
1.826880E+000
2.765691E+002
4.677822E+011
3.241647E+007
7.506348E-002
2.367717E+002
FR
1.00
1.00
1.00
1.00
0.20
0.80
Best
0(0)
0(0)
0(0)
0(0)
Median 2.518044E-027(0) worst
-1.522713E-003(0)
-5.700899E+002(0)
2.503796E+001(0) 4.408153E+000(0) 4.172685E+001(0) -1.522689E-003(0)
-1.992426E-001(0)
3.986579E+000(0)
1.087798E+003(0) 5.787857E+002(0) 1.786663E+002(0) -8.734150E-002(1)
1.975195E+002(1)
c
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
V
0
0
0
0
0
0
Mean
9.567790E-001
1.434094E+002
5.523870E+001
4.560774E+001
-9.492869E-003
-5.373528E+001
std
1.737710E+000
2.765691E+002
1.217592E+002
3.273376E+001
3.044710E-002
1.704461E+002
FR
1.00
1.00
1.00
1.00
0.80
0.84
C13
C14
C15
C16
C17
-6.842748E+001(0)
2.305003E-007(0)
3.520024E-007(0)
0(0)
1.045508E-001(0) 1.423341E-004(0)
FES Best
Median -6.350827E+001(0)
C18
3.986579E+000(0) 4.542582E+000(0) 1.019537E+000(0)
1.738354E+000(0) 7.062603E-001(0)
-5.959380E+001(0)
4.266561E+002(0) 1.883410E+003(0) 1.007614E+000(1)
8.867116E+001(1) 2.521258E+001(0)
c
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
V
0
0
0
0
0
0
Mean
-6.369728E+001
5.306097E+001
1.692527E+002
5.168636E-001
1.119411E+001
2.623988E+000
std
2.129535E+000
1.109514E+002
4.715130E+002
4.806400E-001
1.987006E+001
5.470150E+000
FR
1.00
1.00
1.00
0.60
0.52
1.00
-6.842936E+001(0)
0(0)
0(0)
0(0)
1.593491E-003(0) 0(0)
2.025795E-026(0)
4.497468E+000(0) 0(0)
worst
1e5
-5.699294E+002(1)
4.715873E+000
worst
2e4
C12
2.122805E+000(1)
Median 2.518044E-027(0)
2e5
C11
1.641748E+001(0) 1.275454E+005(0) 9.538031E+001(0) -2.690157E-001(1)
worst
1e5
-9.999998E-006(0) -5.348183E+002(0) -1.883469E+003(0)
Best
Median -6.557847E+001(0) worst
-6.227604E+001(0)
4.617233E+002(0) 4.669372E+002(0) 9.723811E-001(0)
4.880180E-001(0) 2.890081E-018(0) 1.804004E+001(0) 2.332720E-002(0)
2e5
c
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
V
0
0
0
0
0
0
Mean
6.638453E+001
3.824831E+001
9.206100E+001
8.967750E-002
1.551352E+000
9.774940E-004
std
2.349319E+000
1.103934E+002
1.390490E+002
2.645828E-001
3.562367E+000
4.661470E-003
FR
1.00
1.00
1.00
1.00
1.00
1.0
-6.842936E+001(0)
0(0)
0(0)
0(0)
3.171377E-021(0) 0(0)
7.130356E-026(0)
3.673239E+000(0) 0(0)
1.175613E-001(0) 0(0)
Best
Median -6.842935E+001(0) worst
-6.351742E+001
4.617233E+002(0) 3.439705E+002(0) 4.345939E-002(0)
3.379144E+000(0) 4.111441E-020
c
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
V
0
0
0
0
0
0
Mean
-6.718731E+001
3.958610E+001
3.264273E+001
5.737898E-003
5.062111E-001
1.648278E-021
std
1.941324E+000
1.115921E+002
7.679445E+001
1.256665E-002
8.364819E-001
8.222132E-021
FR
1.00
1.00
1.00
1.00
1.00
1.00
C01
TABLE II FUNCTION VALUES ACHIEVED FOR 10D WHEN FES=6E4, FES=3E5, FES=6E5 C02 C03 C04 C05
-7.488633E-001(0)
-7.272362E-001(0)
2.869798E+001(0) 1.630668E-001(1)
-1.950263E+001(0) -4.623536E+001(0)
Median -5.444592E-001(0)
3.279845E+000(0)
1.055229E+002(1) 5.269406E+000(1)
5.271616E+001(1)
3.361628E+002(1)
-4.253164E-001(0)
3.517532E+000(1)
2.059145E+003(1) 6.572543E+000
1.716351E+002(1)
6.696626E+002(1)
c
0,0,0
0,0,0
0,1,1
2,4,4
0,1,2
0,1,2
V
0
0
2.738495E-002
9.314544E-001
8.681249E-003
1.097306E-002
Mean
-5.679536E-001
2.549633E+000
2.669401E+002
5.473364E+000
5.972786E+001
2.476287E+002
std
8.247382E-002
1.448899E+000
5.115535E+002
1.160492E+001
7.683692E+001
2.208535E+002
FR
1.00
0.84
0.04
0.00
0.08
0.16
-7.488767E-001(0)
-1.979172E+000(0)
2.933926E-010(0)
2.938230E-001(1)
-1.788677E+002(0) -2.202901E+002(0)
Median -4.931375E-001(0)
FES 6e4
Best
worst
3e5
Best
-9.634528E-001(0)
9.288789E+001(1) 1.336509E+000(1)
-4.360851E+001(0) -1.591183E+001(0)
-4.049152E-001
3.257141E+000(0)
1.610083E+002(1) 7.420037E+001(1)
-1.880364E+002(1) 5.595110E+002(1)
c
0,0,0
0,0,0
0,1,1
0,3,4
0,0,0
0,0,0
V
worst
6e5
0
0
1.043481E-002
1.041418E-001
0
0
Mean
-5.317052E-001
-6.232063E-001
8.247347E+001
6.360947E+000
-5.359423E+001
1.950298E+001
std
9.246723E-002
1.311447E+000
5.938258E+001
1.604813E+001
9.580866E+001
2.080141E+002
FR
1.00
1.00
0.28
0.00
0.96
0.88
-7.482488E-001(0)
-2.231465E+000(0)
1.921477E-022(0)
2.983242E-001(1)
-3.077500E+002(0) -3.671389E+002(0)
Median -5.230630E-001(0)
-1.607853E+000(0)
6.360706E+001(1) 9.589583E-001(1)
-1.311831E+001(0) -1.545794E+000(0)
Best
worst
-4.068255E-001
1.073545E+000(0)
1.473725E+002(1) 1.276053E+000(1)
6.689167E+001(0)
8.211721E+002(1)
c
0,0,0
0,0,0
0,0,1
0,2,3
0,0,0
0,0,0
V
0
0
4.617963E-003
1.990307E-002
0
0
Mean
-5.359776E-001
-1.472252E+000
8.484414E+001
2.310892E+000
-6.685173E+001
-1.569566E+001
std
8.116107E-002
9.045195E-001
6.456265E+001
2.416570E+000
1.098696E+002
2.175542E+002
FR
1.00
1.00
0.40
0.00
1.00
0.96
C07
C08
2.197021E-001(0)
2.523581E+001(0) 1.367625E+002(0) 8.354441E+001(0) -1.348577E+000(1) -2.632975E+002(1)
FES 6e4
C06
Best
Median 3.614204E+001(0) worst c
C09
C10
C11
C12
4.120177E+002(0) 2.503337E+007(0) 2.841024E+004(0) -2.027081E-002(1)
-1.503891E+001(1)
1.324067E+002(0)
7.537921E+003(0) 6.823706E+011(1) 1.100777E+012(1) 1.969280E+000(1)
3.809540E+001(1)
0,0,0
0,0,0
1,1,1
0,0,0
0,0,0
1,1,1
V
3e5
0
0
0
0
1.775208E+002
1.345599E+000
Mean
4.695272E+001
1.086923E+003
3.991892E+010
3.332912E+011
1.980522E-002
-4.556365E+001
std
3.900452E+001
1.683230E+003
1.402592E+011
1.438814E+012
4.998196E-001
2.909498E+002
FR
1.00
1.00
0.64
0.64
0.00
0.00
7.000473E-021(0)
3.010465E-012(0)
4.129823E-009(0)
3.209700E+001(0) 2.098938E-003(1)
Best
Median 4.236877E-010(0)
1.492257E+002(0) 1.423284E+004(0) 9.465205E+001(0) -3.381378E-002(1)
-1.148510E+001(1)
3.986627E+000
1.365254E+004(0) 4.723380E+008(1) 2.254641E+009(0) -1.988179E-003(1)
1.583186E+002(1)
c
0,0,0
0,0,0
0,0,0
0,0,0
1,1,1
0,1,1
V
0
0
0
0
2.055855E+001
3.227131E-001
Mean
1.594650E+000
8.637338E+002
8.948123E+008
9.049612E+007
2.452089E-002
1.823641E+001
std
1.993312E+000
2.742705E+003
3.989476E+009
4.508662E+008
2.558441E-002
2.968546E+002
FR
1.00
1.00
0.96
1.00
0.00
0.32
5.719750E-025(0)
1.148639E-024(0)
1.068211E-022(0)
1.272140E+001(0) -3.912429E-004(0)
worst
6e5
Best
Median 2.398912E-023(0)
-1.610163E+002(1)
3.986624E+000(0)
3.014445E+003(0) 4.246200E+010(0) 1.554460E+009(0) 1.030492E-001(1)
1.465098E+002(1)
c
0,0,0
0,0,0
0,0,0
0,0,0
1,1,1
0,1,1
V
0
0
0
0
6.629790E+000
3.547296E-001
Mean
1.435185E+000
3.568835E+002
1.705983E+009
6.217998E+007
2.162975E-002
-5.366291E+001
std
1.953039E+000
6.765979E+002
8.490915E+009
3.108916E+008
2.506039E-002
2.756924E+002
FR
1.00
1.00
1.00
1.00
0.16
0.40
FES
C13
C14
-6.436935E+001(0)
5.645312E+000(0) 8.511272E+000(0) 6.189759E-002(0)
3.833016E+000(0) 2.661235E+000(0)
Median -6.113895E+001(0)
6.795598E+001(0) 1.276186E+003(0) 8.290196E-001(0)
1.247754E+001(0) 2.012545E+001(0)
-5.684696E+001(0)
2.992355E+004(0) 7.089523E+004(0) 9.781349E-001(1)
4.735251E+001(0) 7.745144E+001(0)
c
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
V
0
0
0
0
0
0
Mean
-6.139038E+001
3.775279E+003
8.321901E+003
5.455574E-001
1.304866E+001
2.118068E+001
std
1.963421E+000
8.517712E+003
1.662116E+004
3.333478E-001
1.096604E+001
1.532531E+001
FR
1.00
1.00
1.00
0.72
0.64
1.00
-6.561392E+001(0)
6.973603E-021(0) 1.858584E-019(0)
0(0)
1.321554E-001(0) 1.084172E-002(0)
Best
worst
3e5
Best
Median -6.283921E+001(0) worst
6e5
-1.992606E-001(0)
1.305522E+002(0) 1.468337E+003(0) 1.244323E+002(0) 2.995825E-002(1)
worst
6e4
-1.990736E-001(0)
C15
C16
C17
C18
9.079366E-010(0) 2.394905E+001(0) 2.220446E-016(0)
3.086905E+000(0) 2.730369E+000(0)
-5.898163E+001(0)
4.852486E+000(0) 1.204280E+003(0) 1.005010E+000(0)
4.949345E+001(0) 3.431224E+001(0)
c
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
V
0
0
0
0
0
0
Mean
-6.279446E+001
9.914522E-001
1.497994E+002
9.238378E-002
5.030621E+000
4.818549E+000
std
2.137841E+000
1.807831E+000
2.722319E+002
2.535755E-001
9.605814E+000
7.413382E+000
FR
1.00
1.00
1.00
1.00
1.00
1.00
-6.533407E+001(0)
2.733987E-025(0) 1.891143E-024(0)
0(0)
5.830600E-002(0) 9.205394E-004(0)
Best
Median -6.283953E+001(0)
5.913363E-022(0) 2.160359E+001(0) 1.110223E-016(0)
2.198766E+000(0) 6.652159E-001(0)
-5.810822E+001
3.986624E+000
1.359514E+003(0) 3.978648E-001(0)
1.945178E+001(0) 3.492679E+001(0)
c
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
V
0
0
0
0
0
0
Mean
-6.235285E+001
1.275720E+000
9.527798E+001
4.437608E-002
3.298513E+000
2.583090E+000
std
1.948846E+000
1.898013E+000
2.737839E+002
1.025769E-001
5.072083E+000
7.037550E+000
FR
1.00
1.00
1.00
1.00
1.00
1.00
worst
