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Engineering Optimization
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A robust evolutionary algorithm for
global optimization
Jinn-Moon Yang a , Chin-Jen Lin b & Cheng-Yan Kao b a
Department of Biological Science and Technology and Institute of Bioinformatics , National Chiao Tung University , Hsinchu, 30050, Taiwan
b
Department of Computer Science and Information Engineering , National Taiwan University , Taipei, 106, Taiwan
Published online: 17 Sep 2010.
To cite this article: Jinn-Moon Yang , Chin-Jen Lin & Cheng-Yan Kao (2002) A robust
evolutionary algorithm for global optimization, Engineering Optimization, 34:5, 405-425, DOI:
10.1080/03052150214019
To link to this article: http://dx.doi.org/10.1080/03052150214019
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A ROBUST EVOLUTIONARY ALGORITHM FOR
GLOBAL OPTIMIZATION
JINN-MOON YANGa,*, CHIN-JEN LINband CHENG-YAN KAOb
a
Department of Biological Science and Technology and Institute of Bioinformatics, National Chiao Tung University, Hsinchu, 30050, Taiwan;bDepartment of Computer Science and Information
Engineering, National Taiwan University, Taipei 106, Taiwan (Received 20 July 2001; In final form 25 February 2002)
This paper studies an evolutionary algorithm for global optimization. Based on family competition and adaptive rules, the proposed approach consists of global and local strategies by integrating decreasing-based mutations and self-adaptive mutations. The proposed approach is experimentally analyzed by showing that its components can integrate with one another and possess good local and global properties. Following the description of implementation details, the approach is then applied to several widely used test sets, including problems from international contests on evolutionary optimization. Numerical results indicate that the new approach performs very robustly and is competitive with other well-known evolutionary algorithms.
Keywords: Evolutionary algorithms; Family competition; Multiple mutation operators; Adaptive rules; Global optimization
1 INTRODUCTION
Many practical applications from engineering, natural sciences, economics, and business can be formulated as global optimization problems, whose objective functions often possess many local minima in the region of interest. Global optimization is the task of finding the absolutely best minimum among all local minima of optimizing an objective function. In general, it is difficult to obtain an exact solution for such problems.
A general form of the global optimization problem is stated as follows [25]: Given a function f : M Rn! R, M 6¼ ;, for x2 M, the value f f ðxÞ> 1 is called a
global minimum if and only if
f ðxÞ f ðxÞ; 8x 2 M ð1Þ
Then xis a global minimum point, f is called the objective function, and the set M is called
the feasible region. Note that finding a global maximum is equivalent to finding a minimum of f ðxÞ. Therefore, this work only considers the problem of minimization. In addition, this paper only considers the unconstrained optimization problem, that is, M ¼ Rn.
* Corresponding author. E-mail: moon@cc.nctu.edu.tw
ISSN 0305-215X print; ISSN 1029-0273 online # 2002 Taylor & Francis Ltd DOI: 10.1080=0305215021000033726
Many traditional approaches solve global optimization by developing an analogue that closely resembles the original function and is solvable by traditional mathematical stra-tegies such as linear and nonlinear programming. Such techniques often require simplifica-tion of the original problem which may drive the computed solusimplifica-tion far away from the global optimum [25]. To obtain a global solution, several probabilistic methods have been proposed. Among them, evolutionary algorithms are very promising.
Currently, there are three main independently developed but strongly related implementa-tions of evolutionary algorithms: genetic algorithms [14], evolution strategies [23], and evo-lutionary programming [11]. These perform differently when applied to global optimization. For genetic algorithms, both practice and theory ½9; 15; 19 entail the disadvantages of apply-ing a binary-represented implementation to global optimization. The codapply-ing function of genetic algorithms may introduce an additional multimodality, making the combined objective function more complex than the original function. To achieve better performance, real-coded genetic algorithms ½9; 17; 4 have been introduced. In contrast, evolution strate-gies and evolutionary programming mainly use real-valued representations and focus on self-adaptive Gaussian mutations. This type of mutation has succeeded in continuous optimi-zation and has been widely regarded as a good operator for local searches. Unfortunately, experiments ½30; 33 show that self-adaptive Gaussian mutation leaves individuals trapped near local optima for rugged functions.
Because none of these three types of original evolutionary algorithms are very efficient tools, many modifications have been proposed to improve solution quality and to speed up convergence. In particular, a recent trend ½13; 34 is to incorporate local search techniques into evolutionary algorithms. Such a hybrid approach can combine the merits of an evolu-tionary algorithm with those of a local search technique. Generally speaking, a hybrid approach usually can make a better tradeoff between computational cost and the global optimality of the solution. However, for existing methods, local search techniques and genetic operators often work separately during the search process.
Another technique is to use multiple genetic operators ½22; 28. This approach works by assigning a list of parameters to determine the probability of using each operator. Then, an adaptive mechanism is applied to change these probabilities to reflect the performance of the operators. The main disadvantage of this method is that the mechanism for adapting the probabilities may mislead evolutionary algorithms toward local optima.
To further improve the above approaches, in this paper a new method called family competition evolutionary algorithm (FCEA) is proposed for solving global optimization problems. FCEA is a multi-operator approach which combines three mutation operators: decreasing-based Gaussian mutation, self-adaptive Gaussian mutation, and self-adaptive Cauchy mutation. It incorporates family competition and adaptive rules ½29; 35 for con-trolling step sizes to construct the relationship among these three operators. In order to balance exploration and exploitation, each of these operators is designed to compensate for the disadvantages of the other. In addition, FCEA markedly differs from previous approaches because these mutation operators are sequentially applied with equal probability 1.
To the best of the authors’ knowledge, FCEA is the first successful approach to integrate self-adaptive mutations and decreasing-based mutations using family competition principles. The rest of this paper is organized as follows. Section 2 introduces the evolutionary nature of FCEA. Next, Section 3 presents ten widely used testing functions, then investigates the main characteristics of FCEA. It demonstrates experimentally how FCEA balances the trade-off between exploitation and exploration of the search. Section 4 compares FCEA with other methods. In Section 5 FCEA is tested on functions from two international contests on evolutionary optimization. Conclusions are drawn in Section 6.
2 THE FAMILY COMPETITION EVOLUTIONARY ALGORITHM
This section presents details of the family competition evolutionary algorithm (FCEA) for global optimization. The basic structure of the FCEA is as follows (Fig. 1): N solutions are randomly generated as the initial population. Then FCEA enters the main evolutionary loop, consisting of three stages in every iteration: decreasing-based Gaussian mutation, self-adaptive Cauchy mutation, and self-adaptive Gaussian mutation. Each stage is realized by generating a new quasi-population (with N solutions) as the parent of the next stage. As shown in Figure 1, these stages differ only in the mutations used and in some parameters. Hence a general procedure ‘‘FC_adaptive’’ is used to represent the work done by these stages. The FC_adaptive procedure employs four parameters, namely, the parent population (P, with N solutions), mutation operator (M), selection method (S), and family competition length (L), to generate a new quasi-population. The main work of FC_adaptive is to produce offspring and then conduct the family competition. Each individual in the population sequen-tially becomes the ‘‘family father’’. With a probability pc, this family father and another
solu-tion randomly chosen from the rest of the parent populasolu-tion are used as parents to do a recombination operation. Then the new offspring or the family father (if the recombination is not conducted) is operated on by a mutation. For each family father, such a procedure is repeated L times. Finally L children are produced but only the one with the best objective value survives. Since this creates L children from one ‘‘family father’’ and performs a selec-tion, this is a family competition strategy. This was thought to be a good way to avoid premature convergence and also to keep the spirit of local searches, and results agree.
After the family competition, there are N parents and N children left. Based on different stages (or the parameter S of the FC_adaptive procedure), various ways are used to obtain a new quasi-population with N individuals. For both Gaussian and Cauchy self-adaptive
FIGURE 1 Overview of the algorithm: (a) FCEA (b) FC_adaptive procedure.
mutations, in each pair of father and child the individual with the better objective value sur-vives. This procedure is called ‘‘family selection’’. On the other hand, ‘‘population selection’’ chooses the best N individuals from all N parents and N children. With a probability Pps,
FCEA applies population selection to speed up the convergence when the decreasing-based Gaussian mutation is used. For the probability ð1 PpsÞ, family selection is still
con-sidered. In order to reduce the ill effects of greediness on this selection, the initial Ppsis set to
0.05, but it is changed to 0.5 when the mean step size of self-adaptive Gaussian mutation is larger than that of decreasing-based Gaussian mutation. Note that the population selection is similar to ðm þ mÞ-ES in the traditional evolution strategies. Hence, through the process of selection, the FC_adaptive procedure forces each solution of the starting population to have one final offspring. Note that we create LN offspring in the procedure FC_adaptive but the size of the new quasi-population remains the same, N.
For all three mutation operators, different parameters are assigned to control their perfor-mance. Such parameters must be adjusted through the evolutionary process. They are modified first when mutations are applied. Then, after the family competition is complete, parameters are adapted again to better reflect the performance of the whole FC_adaptive procedure.
Regarding chromosome representation, each solution of a population is presented as a set of four n-dimensional vectors ðxi;si; vi;ci
Þ, where n is the number of variables and i ¼ 1; . . . ; N . The vector x is the main variable to be optimized; and s, v, and c are the step-size vectors of decreasing-based mutations, adaptive Gaussian mutation, and self-adaptive Cauchy mutation, respectively. In other words, each solution x is associated with some parameters for step-size control. In this paper, the initial x is randomly chosen from the feasible search space while the initial step sizes s, v, and c vary for different problems. For easy description of operators, a ¼ ðxa;sa; va;caÞis used to represent the ‘‘family father’’
and b ¼ ðxb;sb; vb;cbÞ as another parent (only for the recombination operator). The
off-spring of each operation is represented as c ¼ ðxc;sc; vc;ccÞ. Also, the symbol xd j is used
to denote the jth component of an individual d, 8j 2 f1; . . . ; ng. The rest of this section explains each important component of the FC_adaptive procedure: recombination operators, mutation operations, and rules for adapting step sizes (s, v, and c).
2.1 Recombination Operators
FCEA implements two simple recombination operators to generate offspring: modified discrete recombination and blend crossover (BLX-0.5) [9]. The intermediate recombination [1], a special case of BLX-0.5, is also applied. With probabilities 0.5, 0.25, and 0.25 at each stage only one of the three operators is chosen. Probabilities are set according to experimental experi-ence. Here, it is noted again that recombination operators are activated with only a probability pc. The main variable x and a step size are recombined in a recombination operator.
Modified Discrete Recombination: The original discrete recombination [1] generates a child that inherits genes from two parents with equal probability. Here the two parents of the recombination operator are the ‘‘family father’’ and another solution randomly selected. Experience indicates that FCEA can be more robust if the child inherits genes from the ‘‘family father’’ with a higher probability. Therefore, the operator was modified to be as follows: xcj ¼ x a j with probability 0:8 xbj with probability 0:2 ( ð2Þ
BLX-0.5 and Intermediate Recombination: BLX-0.5 [9] is successfully used in real-coded genetic algorithms. It is defined as:
xcj ¼xaj þbðxb
j xajÞand ð3Þ
wcj ¼waj þbðwbj wajÞ ð4Þ
where b is chosen uniformly in the range [0:5, 1.5]; and w is s, v or c based on the tion operator applied in the family competition. For example, if self-adaptive Gaussian muta-tion is used in this FC_adaptive procedure, x in Eqs. (3) and (4) is v. When b is fixed to 0.5, BLX-0.5 is termed intermediate recombination. The work of the evolution strategies commu-nity [1] was followed to employ only intermediate recombination on step-size vectors, that is, s, v, and c. To be more precise, x is also recombined when the intermediate recombination is chosen.
2.2 Mutation Operators
Mutations are main operators of FCEA. After the recombination, a mutation operator is applied to each ‘‘family father’’ or the new offspring generated by recombination. In FCEA, the mutation is performed independently on each vector element of the ‘‘family father’’ by adding a random value with expectation zero:
xi0¼xiþwDðÞ ð5Þ
where xiis the ith component of x, x0iis the ith component of x
0 mutated from x, DðÞ is a
random variable, and w is the step size. In this paper, DðÞ is evaluated as N(0, 1) or C(1) if the mutations are, respectively, Gaussian or Cauchy.
Self-Adaptive Gaussian Mutation: We adopted Schwefel’s proposal [23] to use self-adaptive Gaussian mutation in global optimization. The mutation is accomplished by first mutating the step size vj and then the variable xj:
vcj ¼vajexp½t0N ð0; 1Þ þ tNjð0; 1Þ ð6Þ
xcj ¼xaj þvcjNjð0; 1Þ ð7Þ
where N(0, 1) is the standard normal distribution. Nj(0, 1) is a new value with distribution
N(0, 1) that must be regenerated for each index j. For FCEA, Ref. [1] was followed in setting t and t0 as ðpffiffiffiffiffi2nÞ1
and ðpffiffiffiffiffiffiffiffiffi2pffiffiffinÞ1, respectively.
Self-Adaptive Cauchy Mutation: A random variable is said to have the Cauchy distribution ðCðtÞÞ if it has the following density function:
f ðx; tÞ ¼ t=p
t2þx2; 1< x < 1 ð8Þ
Self-adaptive Cauchy mutation is defined as follows:
ccj ¼caj exp½t0N ð0; 1Þ þ tNjð0; 1Þ ð9Þ
xcj ¼xaj þccjCjðtÞ ð10Þ
In the present work, t is 1. Note that self-adaptive Cauchy mutation is similar to self-adaptive Gaussian mutation except that Eq. (7) is replaced by Eq. (10). That is, they implement the same step-size control.
Figure 2 compares density functions of Gaussian distribution (N(0, 1)) and Cauchy distri-butions (C(1)). Clearly Cauchy mutation is able to make a larger perturbation than Gaussian mutation. This implies that Cauchy mutation has a higher probability of escaping from local optima than Gaussian mutation. However, the order of local convergence is identical for Gaussian and spherical Cauchy distributions, while nonspherical Cauchy mutations lead to slower local convergence [20].
Decreasing-Based Gaussian Mutations: Decreasing-based Gaussian mutation uses the step-size vector s with a fixed decrease rate g ¼ 0:95 as follows:
sc¼gsa ð11Þ
xcj ¼xaj þscN
jð0; 1Þ ð12Þ
Previous results [30] demonstrated that self-adaptive mutations converge faster than decreasing-based mutations but, for rugged functions, self-adaptive mutations more easily become trapped in local optima than decreasing-based mutations.
It can be seen that step sizes are the same for all components of xain the decreasing-based
mutation, but are different in the self-adaptive mutations. Decreasing-based mutation can be visualised as a search for a better child in a hypersphere centered on the parent. However, for self-adaptive mutation, the search space becomes a hyperellipse. Figure 3 illustrates this difference by two-dimensional contour plots. Therefore, children are searched for in two different types of regions, according to the types of mutations.
2.3 Adaptive Rules
The performance of Gaussian and Cauchy mutations is largely influenced by the step sizes. FCEA adjusts the step sizes while mutations are applied (e.g. Eqs. (6), (9), and (11)).
FIGURE 2 Density functions of Gaussian and Cauchy distributions.
However, such updates insufficiently consider the performance of the whole family. Therefore, after family competition, some additional rules are implemented:
1. A-decrease-rule: Immediately after self-adaptive mutations, if objective values of all offspring are greater than or equal to that of the ‘‘family parent’’, the step-size vectors v (Gaussian) or c (Cauchy) of the parent are decreased:
waj ¼0:95waj ð13Þ
where wa is the step-size vector of the parent. In other words, when there is no
improvement after self-adaptive mutations, a more conservative, that is, smaller, step size tends to make better improvement in the next iteration. This is inspired by the 1/5-success rule of ð1 þ lÞ-ES [1].
2. D-increase-rule: It is difficult to decide the rate g for decreasing-based mutations. Unlike self-adaptive mutations which adjust step sizes automatically, its step size goes to zero as the number of iterations increases. Therefore, it is essential to employ a rule which can enlarge the step size in some situations. The step size of the decreasing-based mutation should not be too small, when compared to step sizes of self-adaptive mutations. In this work s is increased if one of the two self-adaptive mutations generates better offspring. To be more precise, after a self-adaptive mutation, if the best child with step size v is better than its ‘‘family father’’, the step size of the decreasing-based mutation is updated as follows:
sc¼maxðsc;bvcmeanÞ ð14Þ
where vc
meanis the mean value of the vector v; and b is chosen to be 0.2. Note that this rule is
applied in the stages of self-adaptive mutations but not in those of decreasing-based mutations.
3 ANALYSIS OF FCEA
This section discusses several characteristics of FCEA by numerical experiments and mathematical explanations. The core idea of FCEA is that the mutation operators are able to cooperate with each other in order to achieve good performance by applying the adap-tive rules and family competition.
Ten well-known functions, summarized in Table I, were used for the tests. The test functions reflect a different degree of complexity. Functions f7 to f9 are unimodal but FIGURE 3 The different search spaces for self-adaptive (a) and decreasing-based (b) mutations. Using two-dimensional contour plots, the search spaces from parents are ellipses and circles.
T ABLE I The Fun ction T est Bed. Func tion Limit Na me fmin f1 ðx Þ ¼ P m i¼1 ci ½e k xi Ai k 2= pcos ðp k xi Ai k 2Þ , w her e m ¼ 15 xi 2½ 0 ; 10 Modifie d Lang erman 1.5 f2 ðx Þ¼ P n i¼ 1 sin ðyi Þ sin 2 mðð iy 2Þ=i p Þ; where yi ¼ xi cos p =6 xiþ 1 sin p =6i f i mod 2 ¼ 1 xi 1 sin p = 6 þ xi cos p =6i f i mod 2 ¼ 0 and i 6¼ n xi if i ¼ n 8 > > > < > > > : xi 2½ 0 ; p Epistatic Michale w icz 9.66 f3 ðx Þ¼ 20 exp ð 0 :2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 = n P n i¼ 1 x 2 i p Þ exp ð1 =n P n i¼ 1 cos ð2 p xi ÞÞ þ 20 þ ex i 2½ 30 ; 30 Ackle y 0 f4 ðx Þ¼ P n i¼ 1 ½x 2i 10 cos ð2 p xi Þþ 10 xi 2½ 5 :12 ; 5 :12 Rastrigin 0 f5 ðx Þ¼ P n i¼ 1 xi sin ð ffiffiffiffiffiffi ffi jxi j p Þ xi 2½ 500 ; 500 Schw efel 420.9687 n f6 ðx Þ¼ P n i¼ 1 x 2=i 4000 Qn i¼ 1 cos ðxi = ffiffi i p Þþ 1 xi 2½ 600 ; 600 Grie w ank 0 f7 ðx Þ¼ P n 1 i¼ 1 ð1 xi Þ 2þ 100 ðxiþ 1 xi Þ 2 xi 2½ 5 :12 ; 5 :12 Rosenb rock 0 f8 ðx Þ¼ P n i¼ 1 ð P i j¼ 1 xj Þ 2 xi 2½ 65 :536 ; 65 :536 Ridge 0 f9 ðx Þ¼ P n i¼ 1 10 i 1x 2 i xi 2½ 10 ; 10 Ellipsoid 0 f10 ðx Þ¼ð P n i¼ 1 x 2Þi 0 :25 ½sin 2ð50 ð P n i¼ 1 x 2Þi 0 :1Þþ 1 :0 xi 2½ 100 ; 100 V sinw av e 0 *: Ai are constant v ectors. See http: == homepages.ulb .ac.be = gseront = ICEO =Functions = Functions.html
f1 to f6 and f10 are multimodal functions each of whose number of local minima increases
exponentially with the problem dimension [25]. Functions f1and f2are selected from the
sec-ond international contest on evolutionary optimization. Function f7 is considered to be
diffi-cult because the minimum is located in a narrow curved valley. In addition, functions f8and
f9are quadratic problems. Figures 4(a) and 4(b) show respectively the two-dimensional plots
of f5 and f7. It can be seen that they are quite different.
Many well-known test problems are separable functions [26] which can be defined as follows: a function f : Rn! Ris separable if and only if
f ðxÞ ¼X
l i¼1
giðxiÞ ð15Þ
Thus a separable function f (x) can be decomposed into a sum of l functions gi. The optimal
value of a separable function f can be obtained in a sequence of l independent optimization processes for each parameter xi.
From Table I, only f4, f5, and f9 are separable. Function f6, a nonseparable function,
becomes simpler and smoother as the dimensionality is increased; the problem approaches a separable function because the second term may be neglected.
3.1 Parameters of FCEA
Table II indicates the setting of FCEA parameters, such as family competition lengths and recombination probabilities. They are used for most of functions defined in this work. Ld,
FIGURE 4 Two-dimensional plots of Schwefel’s function ðf5Þand Rosenbrock’s function ðf7Þ.
s, and pcD are the parameters for decreasing-based mutation; La, v, c, and pcAare for
self-adaptive mutations. These parameters were decided after experiments on 50 functions from previous studies ½3; 7; 9; 10; 12; 18; 24; 32 with various values. For each problem FCEA is tested by 50 independent runs. Of course, not all combinations of various parameter values were tested. FCEA stops if it exceeds a maximum number of function evaluations or jf ðxbestÞ f ðxÞj e, where xbestis the solution found by FCEA and xis a global optimum.
The maximum number of function evaluations is 1,200,000 for f7 and f10; 400,000 for all
others. In order to compare with other evolutionary algorithms (e.g. Refs. ½16; 18), the same value of e ¼ 103 was used. Parameters listed in Table II were chosen based on the
following observations:
1. Because the family length is a critical factor in FCEA, Figure 5 tests the performance of different LT values. Figure 5(a) shows the relation between LT and the success rate while
Figure 5(b) shows the relation between LT and the number of function evaluations. It can
be seen that the number of function evaluations increases with increasing family com-petition length. FCEA has the worst success rates when both La and Ld are set to 1.
Except for f10, FCEA is unable to obtain benefits when Laand Ld exceed 3. Therefore,
both La and Ld were set to 6 for f10 which is multimodal and nonseparable function. To
reduce the number of function evaluations, La was set to 4 and Ld to 2 for unimodal and
nonseparable functions, such as f7and f8. Laand Ld are set to 2 for the others. We
recom-mend that FCEA should use enlarged family competition lengths (Ld and La) on
nonse-parable functions in order to achieve better performance.
2. FCEA was implemented on test functions with recombination probabilities between 0.0 to 1.0. Based on experimental results, pcDand pcAwere set to 0.8 and 0.2 respectively. The
performance of FCEA is insensitive to these recombination probabilities when pc> 0:1.
3. The step sizes (v and c) of self-adaptive mutations are set to 0:1jbiaij and to 10 if
0:1jbiaij 10. Decreasing-based mutation with a large initial step size ðs ¼ 4vÞ is a
global search strategy in FCEA. FCEA is less sensitive than evolution strategies on multi-modal problems [5] because FCEA applies both self-adaptive (Eqs. (9) and (6)) and A-decrease-rule (Eq. (13)) mechanisms to adjust viand ciaccording to experimental results.
4. Generally, FCEA should use an increased population size for multimodal functions. The population size should be increased when the problem size (n) becomes larger.
TABLE II Parameters of FCEA and Notation Used in This Paper. Parameter name Value of parameter
Recombination pcD¼0.8 (recombination rate in decreasing-based stage) probability (pc) pcA¼0.2 (recombination rate in two adaptive stages) Family competition Ld¼2 (length in decreasing-based stage)
length La¼2 (length in two adaptive stages) Step sizes vi¼ci¼ 0:1jbiaij; jbiaij 10 10; otherwise si¼4vi Decreasing rate g ¼ 0.95
Other notation n: Number of variables; N: population size;
Mean: the average of the best solutions found; LT: total family competition length (2LaþLd);
SR: percentage of finding a global optimum on 50 independent runs; FE: the average number of function evaluations on 50 independent runs.
3.2 The Effectiveness of Multiple Operators and Family Competition
Using multiple mutations in each iteration is one of the main features of FCEA. For the successful working of a global optimization algorithm, it should consist of both global and local search strategies to facilitate both exploitation and exploration. With family selec-tion and block deleselec-tion, FCEA uses a high selecselec-tion pressure along with a diversity-preserving mechanism. With a high selection pressure it become necessary to use highly disruptive search operators such as the series of three mutation operators used in FCEA. Since it always performs a selection after each mutation procedure, the sequence of three mutation operators is similar to a local search. Using numerical experiments, it can be demonstrated that the three operators work well with one another and possess good local and global properties.
Seven different uses of mutation operators are compared in Tables III and IV. Each use combines some of the three operators applied in FCEA: decreasing-based Gaussian mutation ðMdgÞ, self-adaptive Cauchy mutation ðMcÞ, and self-adaptive Gaussian mutation ðMgÞ. For
example, the Mcapproach uses only self-adaptive Cauchy mutation; the MdgþMcapproach
integrates decreasing-based Gaussian mutation and self-adaptive Cauchy mutation; and FCEA is an approach integrating Mdg, Mc, and Mg. The FCEAncr approach is a special
case of FCEA without adaptive rules, that is, without the A-decrease-rule (Eq. (13)) and
FIGURE 5 The success rates and the average numbers of function evaluations of FCEA on different family competition lengths for five test problems. Each problem is tested over 50 runs for each length.
T A BLE III Compa rison of One-operator A pproac hes of FCEA Based on 50 Runs. Mc Mdg Mg nN LT SR FE Mean SR FE Mean SR FE Mean f1 10 150 6 80% 222, 383 1.31 3 96% 111, 493 1.403 88% 158, 034 1.34 7 f2 10 150 6 54% 337, 131 9.64 4 100% 186, 131 9.659 92% 230, 045 9.65 7 f3 10 10 6 100% 14,4 09 0.00 1 100% 19,3 94 0.001 100% 8893 0.00 1 f4 20 40 6 100% 82,2 03 0.00 1 100% 60,4 53 0.001 98% 54,6 19 0.02 1 f5 10 40 6 100% 37,4 01 0.00 1 92% 43,1 29 9.474 100% 22,7 57 0.00 1 f6 10 40 6 100% 53,6 94 0.00 1 100% 43,9 39 0.001 100% 34,6 40 0.00 1 f7 10 10 10 98% 468, 916 0.08 1 0 % 1,250,04 0 6.100 100% 287, 946 0.00 1 f8 10 20 10 100% 163, 059 0.00 1 66% 1,44,534 0.004 100% 92,9 22 0.00 1 f9 10 10 6 100% 19,7 86 0.00 1 0 % 400, 000 3212 4.2 100% 16,1 21 0.00 1 f10 10 100 18 20% 1,13 7,59 7 0.00 6 100% 1,026,14 0 0.001 50% 923, 740 0.00 3
T A BLE IV Compa rison of Multi-operator A pproache s o f FCEA and FC EA ncr (Which Does not Imp lement Adap ti v e Rules) . M dg þ Mc Mdg þ Mg FCEA FCEA ncr nN LT SR FE Me an SR FE Mean SR FE Me an SR FE Me an f1 10 150 6 86% 156,484 1.339 86% 137, 939 1.34 2 98% 140, 033 1.45 2 92% 194, 541 1.374 f2 10 150 6 94% 180,354 9.657 100% 152, 911 9.65 9 100% 213, 389 9.65 9 76% 313, 373 9.656 f3 10 10 6 100% 17,870 0.001 98% 13,2 90 0.02 4 100% 14,588 0.00 1 98% 19,8 92 0.001 f4 20 40 6 100% 60,886 0.001 98% 58,9 32 0.02 1 100% 59,397 0.00 1 96% 65,3 02 0.061 f5 10 40 6 100% 32,493 0.001 98% 27,1 36 2.36 7 100% 27,638 0.00 1 100% 35,0 96 0.001 f6 10 40 6 100% 45,391 0.001 100% 41,6 48 0.00 1 100% 43,330 0.00 1 100% 49,8 63 0.001 f7 10 10 10 96% 482,837 0.160 98% 301, 272 0.00 1 100% 306, 330 0.00 1 18% 1,12 1,01 8 5.589 f8 10 20 10 100% 142,880 0.001 98% 87,2 73 0.00 1 100% 91,358 0.00 1 0 % 400, 000 10.400 f9 10 10 6 100% 29,456 0.001 100% 23,0 02 0.00 1 100% 29,793 0.00 1 0 % 400, 000 4411 .70 f10 10 100 18 100% 956,378 0.001 98% 926, 266 0.00 1 100% 932, 015 0.00 1 98% 1,,04 2,10 4 0.001
D-increase-rule (Eq. (14)). Except for FCEAncr, the other uses employ adaptive rules. In
order to have a fair comparison, the total length of family competition ðLTÞ of all
seven approaches was set to the same value. For example, if Ld ¼La¼2 in FCEA,
LT ¼6 for one-operator approaches (Mdg, Mc, and Mg) and Ld ¼La¼3 for two-operator
approaches (MdgþMcand MdgþMg).
There are some observations from experimental results:
1. One-operator approaches (Mdg, Mc, and Mg) have widely different performances. Table III
shows that self-adaptive mutations (Mcand Mg) outperform the decreasing-based
muta-tion ðMdgÞon nonseparable unimodal functions (f7and f8). On the other hand, the former
performs worse than the latter on nonseparable multimodal functions (f1, f2, and f10).
2. Self-adaptive Gaussian mutation converges faster than Cauchy mutation. This is consistent with the theoretical results [20].
3. Generally, strategies with a suitable combination of multiple mutations (FCEA and MdgþMg) perform better than one-operator strategies, in terms of the solution quality.
However, the number of function evaluations does not increase much when using multi-operator approaches. Sometimes the number even decreases (e.g. FCEA versus Mdg).
Overall FCEA has the best performance and the number of function evaluations is very competitive. The different approaches have very different performance on f1, f2, f7, and
f10. We claim that mutation operators used in FCEA are able to cooperate with each other.
The decreasing-based mutation may lead self-adaptive mutations into the global basin. 4. Family competition is a useful strategy. Evolution strategies [33] and evolutionary
pro-gramming [32] are often unsatisfactory on multimodal functions, but Table III shows that Mg performs very well on such functions (f4 to f6). The main reason appears to that Mg
applies adaptive rules and family competition.
5. The control of step sizes is important, according to the comparison of FCEAncrand FCEA
in Table IV. Similar observations were made when FCEA was applied in training neural networks [31].
6. The family competition length is crucial for solving complex problems, such as multi-modal and nonseparable functions. For example, the length must be increased for solving f10 to obtain robust performance.
4 COMPARISON WITH OTHER METHODS
Following the detailed discussion of FCEA in the last section, it is now compared with other methods. The same test problems f1 to f10 are used, but since not all previous studies have
released results on all functions, some of the comparisons here are not complete. In addition, the scalability (i.e. the same function with different sizes) of FCEA and other approaches was examined.
Before showing results, some interesting observations about existing methods may be made:
1. The genetic algorithms community ½3; 6–8; 17; 18 has often studied separable functions, but not large ðn > 10Þ nonseparable functions for both unimodal and multimodal functions.
2. In contrast to genetic algorithms, research on evolution strategies and evolutionary pro-gramming ½10; 22 had seldom studied multimodal functions whose numbers of local optima are large, even when those functions are separable.
3. Traditional global optimization algorithms (e.g. ½24; 25) are usually tested by using small problems.
First the scalability of FCEA is examined. To the best of the authors’ knowledge, among existing methods only BGA [17] solved functions f3 to f6 with n > 100. They claimed that
the number of function evaluations scales only as n lnðnÞ. FCEA was applied to these func-tions with n from 10 to 400 by enlarging the population size from 40 to 600. Table V shows that for functions f3 to f6the number of function evaluations increases approximately in the
order of n or n lnðnÞ. FCEA also can solve f7 and f8 with n 50. However, BGA cannot
locate global minima for these two problems with n > 10.
Table VI presents results of FCEA on f10 with n ¼ 2, 5, 10, and 15. FCEA can locate a
local optimum 0.00536 when n > 20. Nevertheless, to the best of our knowledge no other methods have located the global optimum of f10when n > 12. Again, note that it is necessary
to enlarge family competition lengths for multimodal and nonseparable functions.
Table VII shows the comparison of FCEA with other well-known genetic algorithms, such as CHC [16] and a real-coded genetic algorithm, BGA [18], on functions f4to f7. FCEA offers
more stable performance than all other approaches, except BGA [18]. Though CHC used
TABLE V The Average Number of Function Evaluations of FCEA on f3to f8with n from 10 to 400 Where n is the Number of Variables.
n f3 f4 f5 f6 f7 f8 10 18,215 34,785 29,459 43,445 301,272 19,451 30 66,410 97,887 115,652 77,512 1,775,917 385,025 50 103,726 344,114 219,053 89,293 4,670,092 1,886,624 100 120,369 653,580 424,247 150,325 N=A* N=A* 200 304,272 1,300,573 872,269 382,344 N=A* N=A* 400 913,528 2,854,456 2,076,059 750,524 N=A* N=A* *Not applicable.
TABLE VI The Performance and Average Number of Function Evaluations of FCEA on f10with n from 2 to 15 by Using Various Family Competition Lengths where n is the Number of Variables.
n N La Ld Mean FE 2 20 2 2 0.001 21,871 5 30 2 2 0.001 83,595 8 50 4 4 0.001 301,810 10 80 9 6 0.001 1,027,294 13 250 12 9 0.001 5,240,723 15 400 12 12 0.0049 7,995,965
TABLE VII Comparison of FCEA with CHC and a Real-coded Genetic Algorithm (BGA) on f4 to f7. FCEA CHC [16, 26] BGA [17] n FE SR FE SR FE SR f4 20 59,115 100% 158,839 100% 9900 100% f5 10 29,459 100% 9803 100% 8699 100% f6 10 43,445 100% 51,015 100% 59,520 92% f7 2 3709 100% 9455 100% 1671 100%
f7 10 301,272 100% N=A* 3% N=A* N=A*
*Not available in the original papers.
T ABLE VIII Comparison o f FCE A with Ev olutio n Strate gies (ES Gau and ES Cau ), Ev olutio nary Pro gramming (EP Gau and EP Cau ), and a Multi-operator Ev olutio nary Pro gramming Method (EP Cau þ Gau ). FCE A ES Gau [33] ES Cau [33] EP Gau [32] EP Cau [32] EP Cau þ Gau [22] n F E Mean FE Mean Mean Mean Mean Mean f3 30 66,4 10 0.00 1 150, 000 9.07 0.01 2 9.2 0.018 0.00 04 f4 30 97,8 87 0.00 1 500, 000 70.8 2 0.16 89.0 0.046 N =A* f5 30 115, 652 12629.01 900, 000 7549 .9 1255 6.4 7917 .1 1255 4.5 N =A* f6 30 77,5 12 0.00 1 200, 000 0.38 0.03 7 0.08 6 0.016 N =A* f7 30 1,775,91 7 0.00 1 2,00 0,000 6.69 33.2 8 6.17 5.06 1.58 *Not av ailabl e in the original papers.
T ABLE IX T est Bed of the 1st International Contes t o n E v olutio nary Optimization. Func tion Limit Na me fmin fB1 ðx Þ¼ P n i¼ 1 ðxi 1 Þ 2 xi 2½ 5 ; 5 Spher e 0 fB2 ðx Þ¼ 1 = 4000 P n i¼ 1 ðxi 100 Þ 2 Q10 i¼ 1 cos ðð xi 100 Þ= ffiffi i p Þþ 1 xi 2½ 600 ; 600 Grie w ank 0 fB3 ðx Þ¼ P m i¼ 1 1 = ðk xi Ai k 2þ ci Þ, w here m ¼ 30 xi 2½ 0 ; 10 Shekel ’s fo xhole s 10.2 078 fB4 ðx Þ¼ P n i¼ 1 sin ðx Þ sin 2 mðð ix 2Þ=i p Þ, w here m ¼ 10 xi 2½ 0 ; p Michale w icz 9.66 fB5 ðx Þ¼ P m i¼ 1 e 1 = p k xi Ai k 2 cos ðp k xi Ai k 2Þ, w here m ¼ 5 xi 2½ 0 ; 10 Lang er man 1.5 Ai are constant v ectors. See http: == homepages.ulb .ac.be = gseront = ICEO = Functions = Functions.html
Gray coding rather than binary coding to achieve better performance, it cannot stably solve the Rosenbrock problem ðf7Þ with n > 10 [26]. In addition, though BGA performs well in
general, it only solves 92% of trials on f6. Salomon’s work (21) is used to compare FCEA
with BGA on coordinate rotation of functions f3to f5, f8, and f9. Rotation means that a linear
and orthogonal transformation matrix T is applied so the new problem is to minimize f ðT ; xÞ. The same method used in Ref. [21] is used here to generate T . FCEA performs better than BGA on these rotation functions and nonseparable function f8. According to Ref. [21],
Gaussian mutations with one global step size are rotationally invariant. Since the decreas-ing-based mutation uses one global step size, FCEA still performs well on rotated functions. Overall, FCEA seems more stable than these modified genetic algorithms for a test bed containing various types of functions.
Table VIII shows the comparison of FCEA with evolution strategies [33] and evolutionary programming [32] approaches which use Gaussian or Cauchy mutations on functions f3to f7
with n ¼ 30. Also include are results of a two-operator approach [22] which in general outperforms one-operator approaches. For other approaches than FCEA, solutions after exceeding a fixed number of function evaluations are reported. FCEA again offers a stable performance. From Tables VII and VIII, it seems that evolution strategies and evolu-tionary programming are better than modified genetic algorithms on the Rosenbrock problem ðf7Þbut worse on the Schwefel problem ðf5Þ. Some previous results [30] also demonstrated
the same observation.
5 TESTING FUNCTIONS FROM THE INTERNATIONAL CONTESTS ON
EVOLUTIONARY OPTIMIZATION
The aim of the first [2] and second international contests on evolutionary optimization is to allow researchers to compare their algorithms on a common test bed. Two indices are defined to measure the performance of the proposed algorithm: the expected number of evaluations per success, FE, and the best value reached, BV. Here BV is the best value reached during the 20 runs by an algorithm.
Functions from these two international contests are shown in Table IX [2] and Table XI. Results of comparing FCEA with the three evolutionary algorithms [2] on problems from the first contest are given in Table X. Sto-Pri, Van-Ke, and Se-Be represent the authors of these algorithms which are the best among comparative approaches, while non-evolutionary approaches are not considered. Table XII presents results of the second contest. It can be observed that FCEA is very competitive with state-of-the-art evolutionary algorithms.
TABLE X Average Results of 50 Runs of FCEA on Problems of the 1st International Contest on Evolutionary Optimization, and Comparison with Three Hybrid Evolutionary Algorithms. Sto-Pri, Van-Ke, and Se-Be Represent the Authors of These Algorithms.
FCEA
Sto-Pri [2] Van-Ke [2] Se-Be [2]
Function n N FE BV* SR FE FE FE fB1 10 2 1658 0.0004 100% 1892 3462 1099 fB2 10 40 43,905 0.0004 100% 13,508 19,125 6446 fB3 10 150 99,169 10.17 96% 744,250 363,685 259,477 fB4 10 50 42,201 9.6597 100% 10,083 41,765 236,348 fB5 10 60 30,982 1.498 100% 44,733 61,729 1,032,627
*The best value reached in 50 runs.
T ABLE X I T est Be d o f 2nd Inte rnational Co ntest on Ev olutio nary Optimiz ation. Func tion Limit Na me fmin fC1 ðx Þ¼ P n 1 i¼ 1 ð1 xi Þ 2 þ 100 ðxiþ 1 xi Þ 2 xi 2½ 5 :12 ; 5 :12 Rosenb rock 0 fC2 ðx Þ¼ e ðk xi Ai k 2 1 Þ= 2 pcos ðp k xi Ai k 2 1 Þð 1 þ c1 ðk xi Ai k 2Þ= ðk xi Ai k 2 1 þ 0 :01 ÞÞ xi 2½ 5 p ; 5 p Odd square 1.023 fC3 ðx Þ¼ P m i¼ 1 ci ½e k xi Ai k 2= pcos ðp k xi Ai k 2Þ , w here m ¼ 15 xi 2½ 0 ; 10 Mod ified Lang erman 1.5 fC4 ðx Þ¼ P m i¼ 1 1 = ðk xi Ai k 2Þ, w here m ¼ 30 xi 2½ 0 ; 10 Sheke l’ s fo xholes 10.2078 fC5 ðx Þ¼ P n i¼1 sin ðyi Þ sin 2 m iy 2 i p ; w here yi ¼ xi cos p = 6 xiþ 1 sin p = 6i f i mod 2 = 1 xi 1 sin p = 6 þ xi cos p = 6i f i mod 2 = 0 and i 6¼ n xi if i ¼ n 8 > < > : xi 2½ 0 ; p Epistatic Michale w icz 9.66 fC6 ðx Þ¼ð P n i¼ 1 cos 4ðx i Þ 2 Qn i¼ 1 cos 2ðx z ÞÞ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P n i¼ 1 ix 2 i p , subje ct to Qn i¼ 1 xi 0 :75 ; P n i¼ 1 xi 0 :75 n xi 2½ 0 ; 10 Bump 0 Ai are constant v ectors. See http: == homepages.ulb .ac.be = gseront = ICEO =Functions = Functions.html
Function fB2is essentially identical to f6, except its solution is shifted. Comparing Tables X
and IV, it is seen that FCEA is little affected by such shifts. Function fB3is a 10-dimensional
Shekel’s foxholes problem, while in DeJong’s test bed [3], only a smaller problem ðn ¼ 2Þ is included. Table X shows that fB3 is difficult for most algorithms. FCEA must change the
recombination rate to be less than 0.05 in order to solve this problem.
It is noted that functions fC3and fC5increase the epistasis between variables of fB5and fB4,
respectively. FCEA needs more function evaluations to reach optimal solutions for fC3and fC5.
6 CONCLUSIONS AND FUTURE RESEARCH
The ‘‘No Free Lunch Theorem’’ [27] shows that there is no efficient algorithm for global optimization, in general. However, instead of seeking for a fast algorithm, this research con-centrate more on stability. This study has demonstrated that FCEA is a robust approach for global optimization. Experience suggests that a global optimization method should consist of both global and local search strategies. For FCEA, decreasing-based mutations with a large initial step size is a global search strategy; self-adaptive mutations with family competition procedure and replacement selection are local search strategies. These mutation operators can closely cooperate with one another.
Experiments on several well-known functions verify that the proposed approach is very competitive with other algorithms, including genetic algorithms, evolution strategies, and evolutionary programming. The authors believe that the flexibility and robustness of FCEA make it a highly effective global optimization tool.
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