932 JOURNAL OF SOFTWARE, VOL. 8, NO. 4, APRIL 2013 Multimodal Optimization using Crowding Differential Evolution with Spatially Neighbors Best Search Dingcai Shen State Key Laboratory of Software Engineering, Computer School, Wuhan University, Wuhan, China School of Computer and Information Science, Hubei Engineering University, Xiaogan, China Email: [email protected] Yuanxiang Li State Key Laboratory of Software Engineering, Computer School, Wuhan University, Wuhan, China Abstract—Many real practical applications are often needed options. For example, in the field of mechanical design to find more than one optimum solution. Existing we may choose the less fit solutions as our final choice Evolutionary Algorithm (EAs) are originally designed to due to physical or spatial restrictions that the optimal search the unique global value of the objective function. The solutions may be hard to fabricate, or for the easiness of present work proposed an improved niching based scheme maintenance, or the reliability and so on. named spatially neighbors best search technique combine with crowding-based differential evolution (SnbDE) for Multimodal problems are usually thought as a difficult multimodal optimization problems. Differential evolution work to solve by canonical EAs because of the existence (DE) is known for its simple implementation and efficient of multiple global or local optima. Numerous techniques for global optimization. Numerous DE-variants have been have been applied to EAs to enable them to find and exploited to resolve diverse optimization problems. The maintain multiple optima among the whole search space. proposed method adopts DE with DE/best/1/bin scheme. These techniques can be classified into two major The best individual in the adopted scheme is searched categories [3][4]: (i) iterative methods [5][6], which around the considered individual to control the balance of adopting the same optimization algorithm repeatedly to exploitation and exploration. The results of the empirical acquire multiple optima of a multimodal optimization comparison provide distinct evidence that SnbDE outperform the canonical crowding-based differential problems; (ii) parallel subpopulation models (explicit or evolution. SnbDE has been shown to be efficient and implicit), which achieving multiple solutions for a effective in locating and maintaining multiple optima of multimodal optimization problems by dividing a selected benchmark functions for multimodal optimization population into several groups, species or niches that problems. evolve in parallel, such as Multinational GA [7], Species conservation technique [3][8][9], Crowding [10][11][12], Index Terms—Differential evolution, Multimodal optimiza- AFMDE [13], fitness sharing [14], and so on. The tion, Niching, Crowding canonical crowding and fitness sharing are widely used techniques but at the risk of losing niches during the evolution. I. INTRODUCTION In this study, a new multimodal optimization method Most existing evolutionary algorithms (EAs) are employs a spatially near best search strategy using aiming to find a single global optimum of an optimization crowding-based differential evolution (SnbDE) to find problem. A large number of works have been done to multiple optimal solutions simultaneously. The advantage improve the capability of the EAs to effectively and of SnbDE is weakening the effect of the parameter named efficiently find the global value of the optimization crowding factor (CF), which inappropriate setting may problems. Among the commonly used evolutionary cause the replacement error. Numerical experiments on a algorithms, differential evolution (DE) [1] is one of the set of widely used multimodal benchmark functions show most popular evolutionary algorithm for its simple and that the SnbDE is able to find effective solutions to all efficient. As with other EAs, DE is also a population- tested functions. based stochastic optimization technique for solving The rest of this paper is outlined as follows. The next continues optimization problems, especially for real- section presents relevant work, including crowding parameter optimization [2]. And the main purpose, as scheme used for multimodal optimization problems, and other techniques, is to find the single global value. a brief review of differential evolution and its commonly However, in real-world applications, there often exist employed variants. The proposed SnbDE is described in multiple optima (global or local), and it is often requested detail in section 3. In section 4, the proposed algorithm is to locate either all or most of these solutions for more compared with canonical crowding based DE [12] on © 2013 ACADEMY PUBLISHER doi:10.4304/jsw.8.4.932-938 JOURNAL OF SOFTWARE, VOL. 8, NO. 4, APRIL 2013 933 commonly used multimodal optimization benchmark t individual vij, . The process of the crossover is depicted as functions. Finally, section 5 presents the conclusions and some future work. follows: t ⎪⎧vifrandCrjjij, ,(0,1)or j„ = rnd II. BACKGROUND AND RELATED WORK t uij, = ⎨ t (6) ⎩⎪xij, , otherwise. This section presents a brief review of differential evolution and its extension to use crowding scheme for where rand (0,1) is the uniform random variable drawn solving multimodal optimization problems. j from the interval [0,1], jrnd is a random integer number A. Differential Evolution in {1,2,..., NP } , and Cr ∈[0,1] is the crossover constant Differential evolution (DE) was first proposed by Storn namely crossover rate or recombination factor. and Price [1] in 1995. The DE gained significant The selection operator is performed at the last step of popularity since its original definition because of its DE to sustain the most promising trial individuals into the simplicity and easiness of implementation and efficiency. next generation. The selection operator can be defined as Like other popular used EAs, DE is a population-based ttt algorithm designed to solve optimization problems over ⎪⎧uiffufx,()() > xt+1 = iii (7) continues search space. i ⎨ t Without loss of generality, we refer to the ⎩⎪xi , otherwse. i maximization problem, both global and local, of an The DE algorithm performs repeated applications of objective function f ()x throughout this paper, where the above process until a specified number of iterations x is a vector of D-dimensional variables in feasible have been reached, or the number of maximum solution space, which can be denoted evaluations has been exceeded. The pseudo-code of D canonical differential evolution algorithm is shown in as Ω= [,LU ]. At the initialization step, a ∏i=1 ii Algorithm 1. 0000 Algorithm 1 The algorithm of canonical Differential population xiiiiD==(,xx,1 ,2 ,,LL x , ),1,2,, i NP, is Evolution (DE/best/1) randomly generated from the feasible solution space, 1: iter ⇐ 0 where NPis the population size. 2: Initialize population P with NP randomly generated Following initialization, the mutation operation is individuals. conducted with the creation of a mutant vector 3: Evaluate individual Pii ,1,2,, = NP. ttt t t L vxxxiii= (,,1 ,2 ,,L iD , ) for each individual xi in current 4: while (termination criteria are not satisfied) do population. The five widely used mutation strategy can be 5: Find the best fit individual Pbest of current described as follows. population 1) DE/rand/1: 6: for i = 1 to NP do tt t t 7: Select rr1, 2∈ { 1, 2, , NP } and vxir=+123 Fxx·( r − r ) (1) L rr12≠ ≠ i 2) DE/best/1: 8: Copy individual Pi to Si tt t t vxibestrr=+ Fxx·(12 − ) (2) 9: nrandD= (1, ) 10: for j =1 to D do 3) DE/current-to-best/1: 11: if rand(0,1) < Cr or jD= tt t t t t vxFxii=+⋅()() besti − x +⋅ Fxx rr12 − (3) 12: Snibestr[]= P [] n+− F *([] P12 n P r []) n 4) DE/best/2: 13: end if 14: nn⇐ (1)mod+ D tt t t t t vxibestrr=+⋅−+⋅− Fxx()()12 Fxx r 34 r (4) 15: end for 16: Evaluate individual Si 5) DE/rand/2: 17: if f ()SfPii< () tt tt tt 18: SPii⇐ vxFxxir=+⋅−12345()() r r +⋅− Fxx r r (5) 19: end if t 20: if fS()> fP ( ) where xbest denote the fittest individuals of the ibest generation t , r1 , r2 , r3 , r4 , r5 are integers randomly 21: PSbest⇐ i drawn from the discrete set {1, 2,L ,NP } and not equal to 22: end if the index of the considered individual, and F > 0 is the 23: end for mutation factor used to scale differential vectors. 24: iter⇐ iter +1 The crossover operation is applied subsequently on 25: end while each component j(1,2,,)jD= L of the mutant © 2013 ACADEMY PUBLISHER 934 JOURNAL OF SOFTWARE, VOL. 8, NO. 4, APRIL 2013 B. Crowding locating peaks and maintaining found peaks throughout De Jong [15] first presented his conception of the run. CDE produce many niches around the found crowding to deal with multimodal optimization problems. optima (global or local). Ideally, all the individuals are Crowding is inspired by the natural phenomenon of the gathered around the global optima when the algorithm competition amongst similar members for limited terminated. However, since the polytropic size of basin of resources. The proposal of crowding technique is aimed attraction of various multimodal optimization problems, to preserve the diversity of the population. In order to some suboptima may be vanished during the process of achieve this goal, a subset of the population is randomly evolution. chosen to be replaced by the most similar individuals of DE, as a population-based stochastic search algorithms, the offspring if it is fitter. The determination of similarity should consider two crucial contradictory aspects, is judged by the distance metric either in genotypic or exploration and exploitation [19], in order to guarantee phenotypic search space. In real-valued coding acceptable success on a given task. For multimodal optimization problems, the Euclidean distance between problems, the ability of exploration impulse the algorithm two points is most widely used. The number of to explore every area of the feasible search space so as to individuals selected to be considered for replacement is locate the global optima, while the ability of exploitation decided by the parameter named crowding factor (CF).