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Page 1: Global optimization by an improved differential evolutionary algorithm

Applied Mathematics and Computation 188 (2007) 669–680

www.elsevier.com/locate/amc

Global optimization by an improved differentialevolutionary algorithm

Yong-Jun Wang *, Jiang-She Zhang

School of Science, Xi’an Jiaotong University, Xi’an 710049, PR China

Abstract

A hybrid differential evolutionary (DE) algorithm for global optimization is proposed. In the new algorithm, the sto-chastic properties of chaotic systems are used to spread the individuals in search spaces as much as possible, the patternsearch method is employed to speed up the local exploiting and the DE operators are used to jump to a better point. Theglobal convergence is proved. Three typical chaotic systems are investigated in detail. Numerical experiments on bench-mark examples including 13 high dimensional functions demonstrate that the new method achieved an improved successrate and final solution with less computational effort.� 2006 Elsevier Inc. All rights reserved.

Keywords: Differential evolutionary algorithm; Global optimization; Chaotic systems; Pattern search method

1. Introduction

Global optimization (GO) has emerged as one of the most exciting new areas of mathematical program-ming. It has received a wide attraction from many fields such as computational chemistry and biology,biomedicine, computer science, economics, and engineering design and control. However, the global optimi-zation of nonlinear, non-convex and non-differential problems is still an open challenge for researchers.

In the past few years, many direct or heuristics based techniques have been proposed to meet this challenge[1,2]. For example, the well known direct search methods or stochastic methods such as Nelder and Simplexmethod [3], Hook and Jeekes Pattern search [4,5], genetic algorithm (GA) [6,8], differential evolutionary algo-rithm (DE) [7], simulated annealing method (SA) [9,10], particle swarm optimization (PSO) [11,12] and Antcolony optimization [13]. Especially, some promising algorithms for high dimensional functions such as clas-sical evolution programming (CEP) and fast evolution programming (FEP) [14–16] are proposed recently.Among them, most standard direct search methods use greedy criterion to make decision to search for theglobal minimum of the objective functions. Although the greedy decisions process converge fair fast, it runs

0096-3003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.10.021

* Corresponding author.E-mail address: [email protected] (Y.-J. Wang).

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670 Y.-J. Wang, J.-S. Zhang / Applied Mathematics and Computation 188 (2007) 669–680

the risk of getting trapped in local minima. By contrast, the stochastic methods relax the greedy criterion byoccasionally permitting an uphill move of solution to a global one. Unfortunately, these methods generallyconverge slow.

DE algorithm is a recently proposed population-based method for GOP. It is a very simple and straightforward strategy for GOPs, which borrows the idea from Nelder Mead’s method and use the randomly gen-erated initial population, differential mutation, probability crossover and greedy criterion based selection tofind the minimum of objective functions [7]. By examining the traditional DE, we find that there is no mech-anism in the algorithm to extract and use global information about search space and the evolutionary schemein it is unavoidable to repeat computation on some previously detected points [17–19]. In fact, as a global opti-mization technique, DE algorithm is not as effective as Hook Jeeves pattern search method when exploiting alocal minimum, since pattern search method is basically an effective local minimization method [4,5]. There-fore, DE is in its infancy and can most probably be improved.

Chaos is a kind of characteristic of nonlinear systems, which has been extensively studied and applied inmany fields, such as engineering scientific and optimization [20–22], especially as a component of effective opti-mization algorithms relying on its university, randomcity and sensitivity dependence on the initial conditionsof chaotic mapping. For example, CSA [23] combines chaotic systems and SA for global optimization, CPSO[24] utilizes the property of chaotic systems to speed up the convergence of Particle swarm optimizationmethod.

In order to improve the global performance of DE, in this paper, the chaotic mappings are used to initializeand reinitialize (when necessary) population so that search space information can be extracted and used, andin each generation, pattern search method is employed to speed up the local exploiting (the new method isdenoted by CPDE), and the differential operators used in DE help to jump to a better point. Simulationson test including 13 high dimensional functions demonstrate that the new method performs well in termsof the number of function evaluations, the quality of final solutions and success rate.

This paper is structured as follows. In the next section, three chaotic systems, Hook Jeeves pattern searchmethod and DE algorithm are briefly introduced. In Section 3, the mathematic model of the GOP is definedand the new proposed method (CPDE) is described. And the global convergence is proved. The experimentalsimulation results are reported in Section 4. And conclusions are drawn in the last section.

2. Chaotic systems, pattern search method and DE algorithm

2.1. Chaotic systems

Chaos is a kind of a characteristic of nonlinear dynamic system which exhibits bounded dynamic unstable,pseudo random, ergodic, non-period behavior depended on initial value and control parameters [25].

Here three chaotic systems are presented, namely logistic map [26], mapping drawn from chaotic neuron[27] and Tent mapping [28], which are defined in (1)–(3), respectively.

chkþ1 ¼ lchkð1� chkÞ; chk 2 ½0; 1�; k ¼ 0; 1; 2; . . . ;K; ð1Þchkþ1 ¼ g chk � 2 tanhðcchkÞ exp½�3ðchkÞ2�; chk 2 ½0; 1�; k ¼ 0; 1; 2; . . . ;K; ð2Þ

chkþ1 ¼chk=a; chk 2 ½0; a�;ð1� aÞð1� chkÞ; chk 2 ða; 1�;

�k ¼ 0; 1; 2; . . . ;K; ð3Þ

where k is the iteration counter, K is the preset maximum number of chaotic iterations, l in (1) and c in(2) are the control parameters, g 2 [0,1] in (2) is a damping factor of nerve membrane, and chk is amapping or self feedback. When we set l = 4 and ch0 62 {0, 025,0.5, 0,75,1} in (1), and if the generatingsequences in (2) can avoid static points ch0 62 {0,025,0.5,0,75,1} and little period chk = chk�i,i 2 {1,2,3,4}, then the mapped variables in (1), (2) or (3) can distribute in search space with ergodicity,randomness and irregularity.

The referenced three chaotic systems show good chaotic properties. If a designed algorithm needs the pointsto distribute in search space as much as possible, the three chaotic systems can meet this need. It has beendemonstrated that these chaotic systems display better randomness than other systems [23,24].

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Y.-J. Wang, J.-S. Zhang / Applied Mathematics and Computation 188 (2007) 669–680 671

2.2. Hook Jeeves pattern search method

Pattern search technique is a classical powerful local direct search method that does not use numerical oranalytic gradients of the objective functions, which is firstly designed by Hook and Jeeves [4]. This methodfulfills local searches by alternations of axis direction moving and pattern moving in iterations, where axisdirection move is used to explore the descent direction and pattern move is used to accelerate the moving speedin the descent direction. Repeat the axis move and pattern move until convergence.

More specially, firstly, one direction move can be executed as follows: Start from one point, which is gen-erally called the consult point, and explore the descent direction with a predefined step length along axis direc-tions ej (j = 1,2, . . . ,n) in order. Take the point obtained by axis move as the one from which the next axismove starts. Thus the descent point is often obtained after axis moves. Secondly, Assume that xk is the consultpoint and xk+1 is the point obtained by axis moves. One pattern move can be described as follows: Startingfrom xk+1 and moving along the direction d = xk+1 � xk, we obtain the point y = xk+1 + ad, where a is theaccelerating parameter. If the moving fails, we contrast the step length and start from xk+1 again. Repeatthe direct move and pattern move until termination. The main steps of the pattern search and more detailscan be found from [4,5].

Finally, it should be pointed out that pattern search method is basically an effective local minimizationmethod for non-differential functions. Incorporating pattern technique into one stochastic global optimizationalgorithm such as DE to execute effective local exploiting is worth investigating.

2.3. The traditional DE algorithm

The DE algorithm is a stochastic population-based method for GOPs over continuous spaces [7]. It initial-izes the population randomly by uniform distribution over search space and maintains a population with NP

individuals in each generation. A new vector is generated by adding the weighted difference between two ran-dom vectors to a third vector, and this operation is called the mutation. The mutated vector are then mixedwith the components of anther predetermined vector, this operation is called crossover. If the generated vectorby crossover (the offspring) has a lower objective function value than the predetermined vector, it replaces thevector, and this operation is called selection. The above evolution process is repeated until termination con-ditions are met, e.g. the predefined solution precision or the maximum number of function evaluations.

To our best knowledge, there are several versions of DE reported [18,7,19,29,30]. However, most of themare applied to functions with dimension not higher than 10. Here, we only give the main steps of the classicalDE algorithm (Algorithm 1), using the notation DE/rand/1/bin in the original paper [7]. This strategy is themost often used in practice [17,18,7,29]. More details can be referred to [7,29].

Assume the individual i in generation k is denoted by xki ; i ¼ 1; 2; . . . ;NP , and NP is population size.

Algorithm 1 (The traditional DE algorithm)

Step 0. Preset mutation parameter F, crossover parameter CR and the population scale NP. Give the max-imum number of iterations, SSmax and set iteration counter k = 1.

Step 1. Randomly generate points x01; x

02; . . . ; x0

NP from solution space S.Step 2. while (maximum number of iterations, SSmax not reached)

Generate NP new points (individuals) as follows:For each vector xk

i ¼ ðxi;1; xi;2; . . . ; xi;nÞ; i ¼ 1; 2; . . . ;NP :

Step 2.1. (first selection) Chose three vectors xk

r1; xkr2; x

kr3 from the current population, where

r1, r2, r3 2 {1,2, . . . ,NP}.Step 2.2. (mutation) Generate vector v = (vi,1,vi,2, . . . ,vi,n) by v ¼ xk

r1 þ F ðX kr2 � xk

r3Þ.Step 2.3. (crossover) Generate new vector u = (ui,1,ui,2, . . . ,ui,n) according to the preset vector

xki ¼ ðxi;1; xi;2; . . . ; xi;nÞ and v = (vi,1,vi,2, . . . ,vi,n) in Step 2.2 as follows:

ui;j ¼vi;j; if ðrandð0; 1Þ 6 CR or j ¼ rnbrðiÞÞ;xi;j; if ðrandð0; 1Þ > CR or j 6¼ rnbrðiÞÞ;

j ¼ 1; 2; . . . ; n;�

Page 4: Global optimization by an improved differential evolutionary algorithm

672 Y.-J. Wang, J.-S. Zhang / Applied Mathematics and Computation 188 (2007) 669–680

where rand(0, 1) is a uniform random number in range (0,1), and rnbr(i) is a randomly chosen oneindex from set {1,2, . . . ,n}, which insures the new vector to get at least one parameter from thenew generated vector v.Step 2.4. (second selection)

If f ðuÞ < f ðxki Þ, set xkþ1

i ¼ u, otherwise, set xkþ1i ¼ xk

i . Set k = k + 1.End for

End whileStep 4. Output the best results.

As is argued in [7], there are several advantages for this algorithm to outperform some other EAs [11], e.g.DE is a very simple and straightforward strategy, and it is easy to use yet a very powerful algorithm. However,spreading the individuals in the searching space as much as possible and speeding up the local convergence ofDE may further improve a plain DE algorithm.

3. The new proposed algorithm (CPDE)

3.1. Problem formulation

The problem of finding the global minimum of a real-valued function f(x) in n � dimensional search spaceS is denoted by

min f ðxÞ

s:t: x 2 S � Rn;ð4Þ

where f(x) may be nonlinear, non-convex and non-differential. A vector x* 2 S satisfying f(x*) 6 f(x) for allx 2 S is called a global minimizer of f(x) over S and the corresponding value f(x*) is called a global minimum.This paper will focus on box constrained and non-differential functions.

3.2. The new algorithm (CPDE)

As have been talked in Section 2 for DE, the most important operation of it is its generating offspringscheme, which makes DE have the capacity of fulfilling global optimization tasks. Although it has the abilityto maintain the diversity to do local search in a sense, its idea is borrowed from Nelder Simplex method andits local searches are based on randomly generated points to approximate local solution and much time maybe cost on repeated work. Thus using Pattern search to execute the local search in DE algorithm may be agood idea. In addition, the initial population is randomly generated in the original DE. If the initial popula-tion can be spread as much as possible over the objective function surface, that can guide the population inDE towards the more promising areas [7]. Therefore, incorporating chaos technique into DE is also worthinvestigating.

In view of the above, a new algorithm is proposed. Firstly, one chaotic system (1), (2) or (3) presented inSection 2.1 is employed to initialize the population of DE, which ensures the individuals in population to bespread in the search spaces as much as possible. Second, execute the DE algorithm in Section 2.3 for somegenerations (iterations) which is often smaller than that in the traditional DE [7] to find an approximate min-imum of the objective function. Next, the best individual (point) found in the last generation is used as thestarting point to execute the pattern search in Section 2.2 to find a local minimum x*. If the solution with pre-defined precision is found, terminate the computation. Otherwise, one chaotic system (1), (2) or (3) is used toreinitialize some points with half number of the population scale and then replace the worst half part of thepopulation, while the best half part keeps steady. Next, replace the worst individual (point) in the current pop-ulation by the local solution x* found by the pattern search in the third Step. Then a new population is formed.Turn to Step 1 and repeat above steps until convergence.

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Y.-J. Wang, J.-S. Zhang / Applied Mathematics and Computation 188 (2007) 669–680 673

In fact, using the chaotic systems to initialize and reinitialize population, pattern search method to searchfor local minima effectively and the DE operators to help the found local minimum to jump to a better point,is the main idea of this paper.

Assume chjk is the jth chaotic component in the kth generation, j = 1,2, . . ., n and k 6 K, where n is the

dimension of vector x and K is the maximum number of chaotic iteration. And xmin,j and xmax,j denote thelower and upper boundaries of jth variable of a vector x, respectively. Let even integer NP denotes the pop-ulation scale and xT

i ¼ ðxTi;1; x

Ti;2; . . . ; xT

i;nÞ denotes the ith individual in the Tth generation, i = 1,2, . . ., NP.The main steps of the new method (CPDE) see Algorithm 3 and the method of chaotically initializing the

population see Algorithm 2.

Algorithm 2 (Chaotically initializing population)

Step 0. Set the maximum number of chaotic iteration K P 300, the population scale NP, and the individualcounter i = 0.

While (i 6 NP) do

Step 1. Randomly initialize variables chj0 2 ð0; 1Þ; chj

0 62 f0:25; 0:5; 0:75g; j = 1,2, . . . ,n and set iterationcounter k = 0.

Step 2. While (k < K) do

Generate different chaotic variables chjk; j ¼ 1; 2; . . . ; n according to the formula (1), (2) or (3). Set

k = k + 1.End while.

Step 3. Mapping the chaotic variables chjk to feasible region according to equation xð0Þij ¼ xmin;jþ

chjkðxmax;j � xmin;jÞ, j = 1,2, . . . ,n.

Step 4. Set i = i + 1

End while

Note that in Algorithm 2, a point (individual) in feasible region is generated through one chaotic systemthrough K cycles of Step 1 to Step 3. And a population with NP individuals is formed after NP cycles of Step 1 to 4.

Algorithm 3 (The new algorithm (CPDE))

Step 0. Preset the population size, NP, the maximum number of iteration, ktoal within DE algorithm, andthe maximum iteration TTmax. Set T = 0.

Step 1. Chaotically initialize the population (see Algorithm 2) and evaluate it.Step 2. While (TTmax or solution with preset precision not reached) do

Step 2.1. Execute first selection, mutation, crossover, second selection of DE search in Section 2.3 forktoal iterations.

Step 2.2. Execute the pattern search in Section 2.2 with the best individual in the population as theinitial point. Assume x* is obtained.

Step 2.3. Use the chaotically initializing population method (Algorithm 2) to generate a subpopula-tion Ssub with scale NP/2, Use Ssub to replace the worst half part of the original population,while the best half part keeps steady in the population.

Step 2.4. Use x* obtained in Step 2.2 to replace the worst individual in the current population andform a new population.

Step 2.5. Set T = T + 1 and turn to Step 2.1

End while

Step 3. Output the best results.

Note: In Algorithm 3, ktoal is preset smaller than the maximum number of iterations in literature. If the ktoal

in Step 2.1 is set to the same as that in the original DE, the Step 2.1 is the traditional DE method with cha-otically initializing population.

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674 Y.-J. Wang, J.-S. Zhang / Applied Mathematics and Computation 188 (2007) 669–680

3.3. The global convergence of the new algorithm

Now, we prove the asymptotic global convergence of the new method. Suppose that

1. The feasible region S of the problem (4) is a bounded closed set.2. The local minimization sequence (the best solution obtained in the last of each cycle of Step 2) is strictly

descent and the sequence in each cycle of Step 2 can converge to a local minimizer.3. Let f �Le be the least local minimum found by the algorithm, which is larger than the global minimum f * and

the Lebesgue measure of the set mðS�LmÞ > 0, where

S�Lm ¼ fx : f ðxÞ < f �Le; x 2 Sg: ð5Þ

Definition 1. Let nk be a random vector sequence in probability space. If there exits a vector sequence suchthat

p f � 2 limk!þ1

nk

� �¼ 1 ð6Þ

or

p\þ1m¼1

[þ1k¼m

ðjnk � f �j > eÞ( )

¼ 0; 8e > 0; ð7Þ

where f* is a global minimum of the objective function f(x), we say that the sequence nk converges with prob-ability 1 or converges at almost everywhere to a global one of the objective function f(x).

Lemma 1. With an initial point x0 2 S�Lm, defined in (5), the minimization sequence generated by minimizing f(x)

on S at each generation (each cycle of Step 2 in Algorithm 3) of the new algorithm converges to a global minimizerof f(x) on S.

Proof. with an initial point x0 2 S�Lm, the minimization sequence in each cycle of Step 2 will converge to a localminimizer (denoted by x�1) of f(x) on S according to Assumption 3. Since f ðx0Þ < f ðx�LmÞ and the minimizationsequence in the last of each cycle of Step 2 is strictly descent (Assumption 2), it follows that f ðx�1Þ < f ðx�LmÞ. ByAssumption 3 that f ðx�LeÞ is the least local minimal value of the problem (4) which is larger than f* (f* is aglobal minimum), we have f ðx�1Þ ¼ f �, i.e. the minimization sequence converges to a global minimizer off(x) on S. h

Lemma 2. Let Ak, k = 1,2, . . . is a random event sequence in probability space, and pk = p(Ak). If

Xþ1k¼1

pk < þ1

then

8e > 0; p\þ1m¼1

[þ1k¼m

Ak

( )¼ 0:

And if

Xþ1k¼1

pk ¼ þ1;

and Ak are independent events, then we have

8e > 0; p\þ1m¼1

[þ1k¼m

Ak

( )¼ 1:

Let xk be the best point in the kth iteration (obtained in Step 2.4 of each cycle of Step 2 in Algorithm 3) of the new

algorithm, and yk be the corresponding local minimum.

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Y.-J. Wang, J.-S. Zhang / Applied Mathematics and Computation 188 (2007) 669–680 675

Theorem 1. Let TTmax be the maximum number of the iterations of local searches by combining pattern search

and DE search (Algorithm 3). If TTmax!+1, then yTT maxconverges to the global minimum of f(x) of the prob-

lem (4) on S with probability 1, Namely

p limTT max!þ1

yTT max¼ f �

� �¼ 1:

Proof. Let m = TTmax. According to the Definition 1, to prove the Theorem 1 is equivalent to prove that

p\þ1m¼1

[þ1k¼m

ðjyk � f �j > eÞ( )

¼ 0; 8e > 0: ð8Þ

According to the descent property of the proposed algorithm, namely the new algorithm is an elitist preservedstrategy, seeing Algorithm 3 and [7,19], it is obvious that f* 6 yk�1 6 yk, i.e. fykg

1k¼1 is monotonically

decreasing. h

Let q ¼ 1� mðS�LmÞmðSÞ , where mðS�LmÞ and m(S) are the Lebesgue measures of S�Lm defined in (5) and the feasible

region S in problem (4). By Lemma 1 and the monotonic property of fykg1k¼1, We have

pfjyk � f �jP eg ¼ p\ki¼1

jyi � f �jP e

( )6 p

\ki¼1

ðxi 62 S�LmÞ( )

¼Yk

i¼1

pfxi 62 S�Lmg ¼Yk

i¼1

1�mðS�LmÞmðSÞ

� �¼ qk:

Thus we have

X1k¼1

qk ¼ q1� q

< þ1

for 0 < q < 1.According to Lemma 2, the formula (8) holds.In fact, "e > 0

p\þ1m¼1

[þ1k¼m

ðjyk � f �j > eÞ( )

6 limm!þ1

p[þ1k¼m

ðjyk � f �j > eÞ( )

6 limm!þ1

Xþ1k¼m

pðjyk � f �j > eÞ

6 limm!þ1

X1k¼m

qk ¼ limm!þ1

qm

1� q:

Since m(SLm) > 0, we have 0 < q < 1. Then if m!+1, qm

1�q! 0. Formula (8) is obtained. Hence the Theorem1 holds.

4. Simulations

In this section, the performance of the new proposed algorithm (CPDE) was evaluated on 6 benchmarkproblems in [23,24] and 13 high dimensional functions in [14–16], where several methods have been investi-gated on all or some of these examples. Since the solution quality (regardless of the initial system parameters),the convergence speed and the frequency of finding optimal solutions are main quality measures of an algo-rithm, 50 times of independent runs are performed to investigate the average number on these indexes in orderto make a fair comparison.

Firstly, the CPDE was compared with the standard DE [7], the chaotic PSO (CPSO) [24], the standard PSO[24] and the standard GA [24] in terms of the average best function values found and their standard deviationsprovided the total number of function evaluations are fixed. Secondly, the robustness of CPDE was tested andcompared with DE, CPSO, PSO, GA, pure random search (PRS), multi-start (MS), simulated annealing(SA1,SA2), tabu search (TS), tree annealing search (TA) and chaotic SA (CSA) [23] in terms of the averagenumber of function evaluations and the success rate of finding a global minimum of an objective function withpredefined precision within the predetermined maximum number of function evaluations. The detailed

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676 Y.-J. Wang, J.-S. Zhang / Applied Mathematics and Computation 188 (2007) 669–680

reference sources on PSO, GA, PRS, MS, SA1, SA2, TS and TA can be found from the references [23,24], andwe do not list them here.

If an algorithm finds the global minima with predefined precision within the maximum number of functionevaluations, we say the algorithm succeeds and terminate computation; else, we say the algorithm fails. FromTables 1–5, the notations’ meanings are as follows: denoted by BF and SD the average best function valuesfound and standard deviation over 50 runs, denoted by SR% the number of successfully finding the globalminimum divided by 50, denoted by AVEN the total number of function evaluations that algorithm succeeds,divided by the number of runs that algorithm succeeds over the 50 runs.

Since the chaotic system can be generated in three different Eqs. (1), (2) or (3), three symbols are used todenote the CPDE with different chaotic system initializing or reinitializing population method. More specif-ically, CPDE1, CPDE2 and CPDE3 are used to denote the chaotic variables in CPDE generated by Eqs. (1),(2) or (3), respectively.

In the last subsection, one version of the new algorithm, CPDE1, which performs best among CPDE1,CPDE2, CPDE3, was selected to compare with CEP and FEP [16] on 13 high dimensional functions to furtherevaluate the new method.

4.1. Preset the parameters in the new algorithm

In the new algorithm, the parameters NP, CR, F in CPDE are fixed to 10 · n (except the Section 4.3) 0.9and 0.5 according the suggestion of the original paper [7], where n is the dimension of vector x. In the chaoticsystems presented in Section 2, the parameters l, g, c and a in Eqs. (1)–(3) are set to 4, 6, 0.9 and 0.5, respec-tively. And the parameters d0, b, a and � in pattern search method in Section 2 are set to 1, 0.25, 1.2 and 0.01,respectively. The maximum number of iterations in generating population in Algorithm 3 by chaotic system isset to K = 300. In Section 4.3, the population was set according to the preset maximum number of iterationssuch that a fair comparison with CEP and FEP [16] can be made.

4.2. Test the average best results of the new algorithm

In this subsection, the performance of the new method was investigated with the maximum number of func-tion evaluations 2000, which is the same as that in [24]. The average best function values and its standard devi-ations of 50 independent runs found by several methods were listed in Table 1. Since the CPSO, PSO and GAalgorithm was executed to test problems in [24], Table 1 includes all of the available results from [24].

As can be seen from Table 1, CPDE1, CPDE2 and CPDE3 outperform CPSO, PSO, GA and DE in mostcases in terms of the average best function values found or standard deviations in 50 independent runs withinthe fixed number of function evaluations. Among the examples, H3, RA and SH are typical ones. The optimafound by the new method are approximate to theoretical global minima. That is to say the new designedmethod can find solution with higher quality than some similar methods.

4.3. Test the robustness of the new algorithm

In this subsection, the robustness of the new method is tested within predetermined number of functionevaluations 2000 and the predefined solution precision is set to 3.5% of the global optimal value [24]. SinceCPSO, PSO, GA, PRS, MS, SA1, SA2, TS and CSA2 are tested on some test problems, Table 2 includesall of the available results from [23,24].

From Table 2, it can be seen that CPDE1, CPDE2 and CPDE3 can find global minima with high frequencyfor every functions with predefined accuracy, compare with other meta-heuristics. Compared with the stan-dard DE algorithm, the chaotic systems provide approximate equal probabilities to spread in search spacesfor trail points and the incorporated pattern search method significantly reduce the blind searches in localoptimization. Therefore, the success rates are improved. Compared with CPSO, the new method outperformsit in success rate for all cases except the H6.

It can be observed from Table 3 that CPSO, CSA, CPDE and DE are relatively more effective algorithmscompared with other methods in terms of the number of the function evaluations. Compared with PRS, MS,

Page 9: Global optimization by an improved differential evolutionary algorithm

Table 1The average best function values (BF) and their standard deviations (SD) found by CPSO to the standard DE method

CPSO PSO GA CPDE1 CPDE2 CPDE3 DE

GP 3.0000 ± 5.0251e�15 4.6202 ± 11.4554 3.1471 ± 0.9860 3.0000 ± 0.03479 3.0000 ± 0.0354 3.0000 ± 0.0332 3.0000 ± 3.8086e�15BR 0.3979 ± 3.3645e�16 0.4960 ± 0.3703 0.4021 ± 0.0153 0.3979 ± 0.0025 0.3979 ± 0.0029 0.3979 ± 0.0024 0.3988 ± 0.3524H3 �3.8610 ± 0.0033 �3.8572 ± 0.0035 �3.8571 ± 0.0070 �3.8628 ± 0.0387 �3.8628 ± 0.0490 �3.8628 ± 0.0399 �3.8610 ± 0.0000H6 �3.1953 ± 0.1352 �2.8943 ± 0.3995 �3.0212 ± 0.4291 �3.2040 ± 0.0384 �3.300 ± 0.0438 �3.2024 ± 0.0000 �3.2004 ± 0.0352RA �1.9940 ± 0.0248 �1.9702 ± 0.0366 �1.9645 ± 0.0393 �2.0000 ± 0.0226 �2.0000 ± 0.0219 �2.0000 ± 0.0245 �2.0000 ± 0.0000SH �186.7274 ± 0.0218 �180.3265 ± 10.1718 �182.1840 ± 5.3599 �186.7309 ± 2.3771 �186.7309 ± 2.4201 �186.7309 ± 2.1735 �180.7100 ± 10.3638

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Table 2The average success rates (SR%) of method CPSO to DE

CPSO PSO GA CPDE1 CPDE2 CPDE3 DE

GP 100 98 98 100 100 100 100BR 100 94 92 100 100 100 92H3 90 96 16 100 100 100 100H6 96 26 94 90 88 91 60RA 98 100 84 100 100 100 98SH 100 98 98 100 100 98 78

Table 3The average number of function evaluations (AVEN) from PRS to DE method

PRS MS SA1 SA2 TS TA GA PSO CPSO CSA CP-DE1 CP-DE2 CP-DE3 DE[16] [16] [16] [16] [16] [16] [16] [16] [16] [15] [9]

GP 5125 4400 5439 N/A 486 6375 536 1397 192 300 298 305 320 315BR 4850 1600 2700 505 492 4172 1682 743 154 281 279 286 288 385H3 5280 2500 3416 1459 508 1113 112 183 119 379 210 230 246 249H6 18,090 6000 3975 4648 2845 17,262 5727 3796 2551 1865 1966 1800 1875 2880RA 5964 N/A N/A N/A 540 N/A 238 1160 653 441 210 195 371 480SH 6700 N/A 241,215 780 727 N/A 1516 1337 360 289 211 229 308 810

The number in the square brackets means the reference of the corresponding method.

Table 4Comparison of CPDE1, CEP and FEP on function f1–f13 with dimension 30 [16]

Function Number ofgenerations

CPDE1 FEP CEP

Mean best Std Dev Mean best Std Dev Mean best Std Dev

f1 1500 0 0 5.7 · 10�4 1.3 · 10�4 2.2 · 10�4 5.9 · 10�4

f2 2000 0 0 8.1 · 10�3 7.7 · 10�4 2.6 · 10�3 1.7 · 10�4

f3 5000 0 0 1.6 · 10�2 1.4 · 10�2 5.0 · 10�2 6.6 · 10�2

f4 5000 7.5 · 10�2 9.0 · 10�2 0.3 0.5 2.0 1.2f5 20,000 1.5 · 10�6 2.2 · 10�6 5.06 5.87 6.17 13.61f6 1500 0 0 0 0 577.76 1125.76f7 3000 3.4 · 10�3 6.7 · 10�4 7.6 · 10�3 2.6 · 10�3 1.8 · 10�2 6.4 · 10�3

f8 9000 �12,505.5 97 �12,554.5 52.6 �7917.1 634.5f9 5000 4.5 24.5 4.6 · 10�2 1.6 · 10�2 89.0 23. 1f10 1500 5.3 · 10�1 6.6 · 10�2 1.8 · 10�2 2.1 · 10�3 9.2 2.8f11 2000 1.7 · 10�4 2.4 · 10�2 1.6 · 10�2 2.2 · 10�2 8.6 · 10�2 0.12f12 1500 0 0 9.2 · 10�6 3.6 · 10�6 1.76 2.4f13 1500 0 0 1.6 · 10�4 7.3 · 10�5 1.4 3.7

The results are averaged over 50 runs.

Table 5Comparison of the mean generations (Mean Iter) required to reach the average best function values (Mean Best) obtained by FEP [16] inTable 4 between CPDE1, CEP and FEP on function f1–f13 with dimension 30 [16]

Function f1 f 2 f 3 f4 f5 f6 f7 f8 f 9 f 10 f 11 f 12 f 13

CPDE1 450 440 1550 3000 2030 1500 930 7850 5000 1500 296 464 416FEP 1500 2000 5000 5000 20,000 1500 3000 9000 5000 1500 2000 1500 1500CEP 1500 2000 5000 5000 20,000 1500 3000 9000 5000 1500 2000 1500 1500

The results are averaged over 50 runs.

678 Y.-J. Wang, J.-S. Zhang / Applied Mathematics and Computation 188 (2007) 669–680

SA1, SA2, TS, TA, GA and PSO, the computational effort are greatly reduced. Although CPDE1, CPDE2and CPDE3 have not obvious better behaviors than CPSO in most cases in terms of the average number

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of the function evaluations, they perform better than it in terms of the final solution quality and success rates(see Tables 1 and 2).

4.4. Results for more high dimensional functions [16]

In this subsection, the CPDE1 algorithm, which performs best in CPDE1, CPDE2 and CPDE3, wasselected as the typically new designed algorithm to compare with two typically evolution algorithms FEPand CEP [16] on 13 benchmark functions with dimension 30, which have been tested in [14–16]. A detaileddescription of each test problem can be found in [16].

First, Mean Best (the average best function values found in the last generation) and Std Dev (the standarddeviation) was compared with the same preset maximum number of the generations as that [16]. Second, themean number of the iterations (Mean Iter) required for CPDE1 to reach the mean best function valuesobtained by FEP [16]. All the results are the average value of 50 independent runs. The results see Tables 4and 5.

It can be seen from Table 4 that with the same preset maximum number of iterations, CPDE1 can obtainedbetter Mean Best and Std Dev than FEP and CEP on the 13 functions except the function f8, f9 and f10. Forexample, CPDE1 significantly improves the final solution quality on function f1, f2, f3, f5, f6, f12 and f13.Furthermore, it is also observed that the mean best values obtained by CPDE1 are approximate to the theo-retic global minima in some cases, e.g. f1, f2, f3, f5, f6, f12 and f13. However, CPDE1 performs better thanCEP but worse than FEP on function f8, f9 and f10. Thus it can be said that CPDE1 outperforms CEP andFEP in most test cases.

From Table 5, it can be seen that CPDE1 needs fewer average number of the generations to reach the meanbest function values (Mean Best) obtained by FEP [16] in all cases except f 9 and f 10 with the same values.That means CPDE1 converges to global minima more rapid than FEP and CEP in most cases.

In addition, although CPDE1 does not perform better than FEP on function f8 in terms of Mean Best andStd Dev (see Table 4), it needs fewer average number of the generations (see Table 5). That implies thatalthough CPDE1 does not obtain Mean Best over the average of 50 runs, it can quickly reach the Mean Bestin some runs within the 50 runs, which can be very useful to search an approximate global minimum rapidly.

5. Conclusions and further research

A simple but improved DE algorithm is proposed for global optimization of real-valued functions. In thenew evolution method, the population is generated by chaotic systems and its local searches is executed bypattern search technique, which significantly enhance the performance of the standard DE for GOP. Threeversions of the new algorithm are investigated and comparisons with other existing methods are made in termsof the average success rate, solution quality and the number of the function evaluations. The numerical exper-iments demonstrate that the new algorithm is more effective and efficient.

Inspired by this paper, we can combine the chaotic systems and pattern method with other versions of DE,e.g. another version of DE, using the notation as DE/best/1/bin to design effective methods for GOPs. As anextension of this paper, we may also incorporate the dynamic parameters selection techniques [16] into the newmethod (CPDE) to make it more promising.

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