Multimodal Optimization Using Crowding Differential Evolution with Spatially Neighbors Best Search

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

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 S i 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

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). guarantee converging to the nearby optima thus For low CF values, a common problem named maintaining of found optima. The DE variant known as replacement error may emerge which is caused by the DE/best/1 adopt the best fit vector of the current offspring replaces another individual that may not be population to generate donor vectors. The scheme similar to the offspring [12][16]. enhances exploitation ability since all the vectors are To reduce the replacement error problem caused by attracted by the single global vector. At the same time, canonical crowding with low CF values, a number of this scheme weakens the exploration abilities, and may works have been done to overcome this problem. Such as get trapped in some local optima. Deterministic crowding proposed by [17], Probabilistic In this study, we propose a new algorithm named crowding [18], or just set CF equal to the number of spatially neighbors best search crowding differential individuals in the population as crowding DE (CDE) [12] evolution (SnbDE). Different from the ring topology of did. The algorithm of CDE can be depicted as Algorithm neighborhood, which use indices based neighborhood. 2. Let's consider a DE population PXX= [,12 ,,L XNP ],

where each X i (1,2,,iNP= L ) is a D-dimensional Algorithm 2 The algorithm of crowding DE vector in the feasible solution space. The vector indices 1: t ⇐ 0 are randomly generated, say as their generating order at 2: Initialize population P with NP randomly generate population initialization step, and are kept constant individual throughout the run. A vector Xi 's 3: Evaluate Pi,1,2,, NP i = L rrNP (0< <− ( 1) / 2) neighborhoods are 4: while ( not termination condition) do 5: Use the conventional DE to generate offspring S Xir− ,,LLXX i ,, ir+ . Note that topological 6: for all Individual i in S do neighborhoods may belong to a different peak in the 7: Calculate the Euclidean distance values of the fitness landscape because they were chosen randomly (in individual S to the individuals in population P . terms of fitness landscape). In order to overcome this i problem, the SnbDE algorithm using a spatial 8: Find the nearest individual Pj in population P neighborhood based on Euclidean distance around the

to individual Si . considered individual. 9: if f ()SfP> () The SnbDE is based on the CDE but differently in the ij generating of offspring. In SnbDE, we use the /DE/best/1/ 10: Replace individual Pj with individual Si . scheme, where the “best” represent the best fit individual 11: end if around the current optima. The crowding with nearest 12: end for neighbors replacement proposed by Li et al. [11] was 13: tt⇐+1 similar to the crowding scheme with spatially neighbors 14: end while best search strategy proposed in this study. In [11], the lowest fitness individual in the selected q -nearest subset based on the Euclidean distance will be replaced by the offspring if it less fit. In our proposed algorithm, in order III. SNBDE to enhance the exploitative ability, in the meanwhile, to keep the explorative ability of crowding DE, similar As previously mentioned, DE, similar to other EAs, process as proposed in [11] is adopted. For each has shown highly efficient for global optimization individual x ,(iNP= 1,2, , ) in population P , the problems. However, DE is incapable to locate multiple i L global optima or detect other desirable suboptima individuals in population P are sorted by ascending simultaneously. The CDE which extend DE with the order based on the Euclidean crowding scheme [12] shows good performance in distance dx(,ij x )(, ij= 1,2,,L NPi ; ≠ j). Thus we get

剟 the reordered set denoted as PxxxsNP={,,, 12L − 1 }. The where 01x . Deb's 1st function has 5 equally spaced global optima as showed in Fig. 1. All five global values local best individual x is searched in previously lbest are uniformly distributed at x =0.1, 0.3, 0.5, 0.7, 0.9 rr() NP individuals in Ps . Use xlbest as the best with a value 1.0. individual in /DE/best/1 scheme to generate offspring, 1 then the canonical crowding DE is conducted to select 0.9 individuals thus produce the new population. This 0.8 procedure is described as Algorithm 3. 0.7 0.6 Algorithm 3 The illustration of spatially near best search 0.5 crowding DE 0.4 1: Initialize population P with NP random individuals 0.3 2: Evaluate individuals in P 0.2 3: gen ⇐ 1 0.1 0 4: while (not termination condition) do 0 0.2 0.4 0.6 0.8 1 5: for i = 1 to NP Figure 1. Deb's 1st function 6: for j = 1 to NP 2) Deb's decreasing function 7: Calculate Euclidean distance dx(,ij x ), i≠ j Deb's decreasing function can be described as: 8: end for −−2ln(2)((x 0.1)/0.9)2 6 F2 ()xx= 2 sin(5)π (9) 9: Sort Pi 's neighbors set in ascending order, Deb's decreasing function is similar to Deb's 1st function, Ps={,,, x12 xL xNP− 1 } with the values corresponding to the optima are 1.0, 10: Choose the previously r individuals, 0.934, 0.760, 0.540 and 0.334, respectively (see Fig. 2). 1 Pr={,,, x x x } 12L r 0.9

11: Find the best individual xlbest ∈ Pr 0.8

0.7 12: Using xlbest as the local best individual in 0.6 /DE/best/1 scheme to generate offspring Si 0.5

13: end for 0.4

14: Using crowding procedure described in 0.3 algorithm 2 to generate new population 0.2 15: gen⇐+ gen 1 0.1 16: end while 0 0 0.2 0.4 0.6 0.8 1 Figure 2. Deb's decreasing function 3) Branin RCOS IV. EXPERIMENT RESULTS 5.122 5 This section presents results obtained by applying Fxy3 (, )= ( y−+−+ x x 6) 4π 2 π SnbDE to search multiple optima on a set of multimodal problems. Then, the performance of the proposed 1 10(1− )cos(x )+ 10 (10) algorithm and standard crowding DE are tested on these 8π functions. The effect of different control parameter values for SnbDE is also investigated. 400

A. Test Functions 300

The benchmark functions used in this study are widely 200 used for comparing the performance of multimodal optimization problems. The number of optima (global or 100 local) of the functions is ranging from 2 to 6, 0 15 dimensionality is from 1 to 2, and the high dimensionality 10 10 problem is our future research. These functions are 5 5 described as follows. 0 0 −5 1) Deb's 1st function Figure 3. Branin RCOS Deb's 1st function is defined as:

6 The Branin RCOS function is a two-dimensional F ()xx= sin(5)π (8) 1 function with 3 global optima, which optimum value is

0.397887 in the region −510剟x and 015剟y (see Fig. z Success rate (%): the percentage of successful 3). runs in which all optima (global and local) are 4) Ursem's F4 successfully found. z Peak ratio: measure the ability of algorithms to (2−+x22y ) locate the multiple optima, the sum of the optima Fxy(, )=+ 3sin(0.5ππ x 0.5) (11) identified by the technique divided by the sum of 4 4 the actual optima in the search space. In this study, where −2,剟x y 2. Ursem's F4 is a two-dimensional an optimum is considered to be detected if it is function with one global optimum and four local peaks. within a Euclidean distance of 0.01 for F1,F 2 and All the four local optima are distributed on the edge of 0.05 for F3,FF 4, 5 from the real optimum. the search space and the global one in the center (see Fig. z NFEs: The number of fitness function evaluations 4). is recorded when the pre-assigned threshold is 1.5 reached. The average and standard deviation of the number of fitness evaluations are used in this work. 1 C. Results and Analysis 0.5 Table I presents descriptive statistics related to the five 0 benchmarks, i.e. the values for the mean of the average success rate, the mean and standard deviation of the −0.5 2 fitness evaluation and the mean peak ratio. The specific 1 2 parameter setting for each of the algorithms are described 0 1 0 as follows. Population size NP was set to 30 for F1,F 2 , −1 −1 −2 −2 and 50 for F3,F 4 and 100 for F5 . The probability of Figure 4. Ursem's F4 crossover was fixed to Cr = 0.9 , and the scaling factor was F = 0.5 . The parameter r was set to 2, and an 5) Six-Hump Camel Back investigation of the effect of different r to the 42 performance of SnbDE will be provided. The crowding xy4(− 1) Fxy(, )=− ((42.1 − x22 + ) x + xy + ) (12) factor for both SnbDE and CDE were set to equal to the 5 2 3 y population size ( CF= NP ). where −1.9剟x 1.9 and −1.1剟y 1.1 . The six-hump TABLE I. camel back function has six optima (2 global and 4 local, EXPERIMENTAL RESULTS(AVERAGED OVER 30 RUNS) see Fig. 5). This function is used by many multimodal Success Peak Func. NFEs optimization algorithms to inspect the effectiveness to Rate (%) Ratio locate all optima. Because among the four local optima, CDE 100 590 ± 163 1.0 two of them are stable, the other two are unstable, and all F1 SnbDE 100 410 ± 145 1.0 optima are near from each other. The specific CDE 100 680 ± 230 1.0 characteristic makes it hard to identify all the optima by F2 SnbDE 100 570 ± 215 1.0 most multimodal optimization algorithm. 100 1.0 F CDE 7955 ± 560 B. Performance Measures 3 SnbDE 100 5850 ± 515 1.0 100 1.0 Several performance measures for multimodal F CDE 1850 ± 125 optimization algorithms have been used in literature. The 4 SnbDE 100 1350 ± 115 1.0 68 0.67 widely used performance measures are described briefly F CDE 8250 ± 1520 as follows: 5 SnbDE 100 5210 ± 1210 1.0

2 We compare the results obtained using SnbDE with

0 canonical CDE. Each algorithm is tested on each function for 30 runs. From the Table I we can see SnbDE is able to −2 identify and maintain multiple optima in the search space.

−4 Our approach is comparatively successful to the canonical CDE in all tested functions. Especially for F5 , −6 2 in all 30 runs, SnbDE successfully locate and maintain 1 2 0 1 the six optima, while CDE can only find the 4 global 0 −1 −1 optima in most runs. −2 −2 As Fig. 6 indicates, because of the global selection of Figure 5. Six-hump camel back CDE, some found peaks may be attracted by the higher fitter individuals result in the missing of optima. From Fig. 6(a), we can see the initial population is uniformly distributed among the search space, after the algorithm

evolved 150 generation; most individuals gathered 1.5 around the four global optima, the local individuals are 1 attracted by the global optima. When the algorithm attained its 800 generations, all individuals are gathered 0.5 around the four global optima. This status retained to the end of the evolution. With the same distribution of 0 population, however, the SnbDE successfully identified −0.5 and maintained all six optima. 1.5 −1

1 −1.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 (e) iteration 800 0.5

Figure 6. Individuals distribution of CDE on the F5 ( with a 0 population size of 100), asterisk represents the real optima(global or local) −0.5

−1 Fig. 7 shows the distribution of individuals when the algorithm achieved the pre-specified threshold with the −1.5 same initialization population. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 (a) Initial population 1.5 1.5 1

1 0.5

0.5 0

0 −0.5

−0.5 −1

−1 −1.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−1.5 Figure 7. The distribution of SnbDE on F5 after 44 iterations −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 (b) iteration 150 (with a population size of 100), asterisk represents the real 1.5 optima(global or local)

1 Fig. 8 illustrate the effect of population size on peak ratio (with r = 2 for SnbDE) when the algorithms 0.5 applied to F5 . From Fig. 8 we can see, with the increasing 0 of population size, the peak ratio attained of both algorithms is increased. When the population size was set −0.5 too small (e.g. less than or equal to twenty), the peak ratio

−1 achieved by CDE is higher than that of SnbDE, with the increasing of population size, the peak ratio achieved by

−1.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 SnbDE growing more rapidly than that of CDE. (c) iteration 300 1.2 1.5

1 1

0.8 0.5

0.6 0 peak ratio

0.4 −0.5

0.2 −1 SnbDE CDE −1.5 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 10 20 30 40 50 60 70 80 90 100 (d) iteration 500 population size

Figure 8. The effect of population size on peak ratio of the F5

When the population size reached 80, the peak ratio come to 1.0 for SnbDE, and about 0.67 for CDE, respectively. When the population size reached 100, the

peak ratio reached 1.0 for SnbDE, and CDE keep brief review,” in IEEE Congress on Evolutionary constant at about 0.67. In our experiments, we also notice Computation, 2010, pp. 1-8. that once a new optimum found, the SnbDE can maintain [4] J. Yao, N. Kharma, and P. Grogono, “Bi-Objective it throughout the run, while CDE often lose the found Multipopulation Genetic Algorithm for Multimodal Function Optimization,” IEEE Transactions on local optimum, and in most cases can only maintained Evolutionary Computation, vol. 14, no. 1, pp. 80-102, Feb. four optima. 2010. The effect of r to the peak ratio of algorithm SnbDE [5] J. E. Vitela and O. Castaños, “A sequential niching on F5 is depicted as Fig. 9. As can be seen, there was a memetic algorithm for continuous multimodal function significant effect for r value on peak ratio. With a lower optimization,” Applied Mathematics and Computation, Feb. r (e.g. r„ 2 ), the SnbDE can locate and maintain all 2012, doi:10.1016/j.amc.2011.05.051 [6] D. Beasley, D. R. Bull, and R. R. Martin, “A Sequential optima. With the increasing of r , the selection of best Niche Technique for Multimodal Function Optimization,” individual became the global selection result in the Evolutionary Computation, vol. 1, no. 2, pp. 101-125, Jun. gathering around the global optima. 1993. 1.2 [7] R. Ursem, “Multinational GAs: Multimodal optimization techniques in dynamic environments,” in Proc. of the 1 Genetic and Evolutionary Computation Conference, 2000, pp. 19-26. 0.8 [8] C. Stoean, M. Preuss, R. Stoean, and D. Dumitrescu, “Multimodal Optimization by Means of a Topological 0.6 Species Conservation Algorithm,” IEEE Transactions on Evolutionary Computation, vol. 14, no. 6, pp. 842-864, Peak Ratio 0.4 Dec. 2010. [9] J.-P. Li, M. E. Balazs, G. T. Parks, and P. J. Clarkson, “A

0.2 species conserving genetic algorithm for multimodal function optimization,” Evolutionary computation, vol. 10,

0 no. 3, pp. 207-234, Jan. 2002. 2 4 6 8 10 12 14 16 r value [10] M. Li, D. Lin, and J. Kou, “A hybrid niching PSO enhanced with recombination-replacement crowding Figure 9. The effect of $r$ on peak ratio of SnbDE on F5 strategy for multimodal function optimization,” Applied Soft Computing, vol. 12, no. 3, pp. 975-987, Mar. 2012. [11] M. Li and J. Kou, “Crowding with nearest neighbors V. CONCLUSION replacement for multiple species niching and building blocks preservation in binary multimodal functions In this paper we present the SnbDE algorithm based on optimization,” Journal of Heuristics, vol. 14, no. 3, pp. the idea of crowding DE with spatially nearest best search 243-270, Sep. 2008. strategy. The performance of the SnbDE is compared [12] R. Thomsen, “Multimodal optimization using crowding- against the canonical CDE on several commonly used based differential evolution,” in Proceedings of the 2004 Congress on Evolutionary Computation, 2004, vol. 2, pp. multimodal benchmark functions. All experiments have 1382-1389. demonstrated that the SnbDE is a competitive candidate [13] D. Shen, Y. Li, B. Wei, and X. Xia, “Adaptive Forking for multimodal evolutionary algorithm. Further research Multipopulation Differential Evolution Algorithm for will investigate whether SnbDE is effective in handling Multimodal Optimization,” Journal of Convergence complex real-world multimodal optimization problems, Information Technology, vol. 7, no. 5, pp. 57-65, Mar. including ones with higher dimensionality. The ability to 2012. track optima in a dynamic environment will also be an [14] A. Della Cioppa, C. De Stefano, and A. Marcelli, “Where interesting topic for future research. Are the Niches? Dynamic Fitness Sharing,” IEEE Transactions on Evolutionary Computation, vol. 11, no. 4, pp. 453-465, Aug. 2007. ACKNOWLEDGMENT [15] K. D. Jong, “Analysis of the behavior of a class of genetic This work is supported by the National Nature Science adaptive systems,” University of Michigan, 1975. Foundation of China (Grant No.61070009, 61165004). [16] S. Das, S. Maity, B.-Y. Qu, and P. N. Suganthan, “Real- parameter evolutionary multimodal optimization — A survey of the state-of-the-art,” Swarm and Evolutionary REFERENCES Computation, vol. 1, no. 2, pp. 71-88, Jun. 2011. [1] R. Storn and K. V. Price, “Differential evolution-a simple [17] S. Mahfoud, “Crowding and preselection revisited,” in and efficient adaptive scheme for global optimization over Parallel Problem Solving From Nature, 1992, pp. 27-36. continuous spaces,” International computer science [18] O. J. Mengshoel and D. E. Goldberg, “Probabilistic institute, vol. 46, no. 2. Institute of Company Secretaries of Crowding: Deterministic Crowding with Probabilistic India, 24-Jun-1995. Replacement,” in Proceedings of the Genetic and [2] U. K. Chakraborty, Advances in differential evolution. Evolutionary Computation Conference, 1999, pp. 409-416. Springer, 2008. [19] S. Das, A. Abraham, U. K. Chakraborty, and A. Konar, [3] J.-P. Li, X.-dong Li, and A. Wood, “Species based “Differential Evolution Using a Neighborhood-Based evolutionary algorithms for multimodal optimization: A Mutation Operator,” IEEE Transactions on Evolutionary Computation, vol. 13, no. 3, pp. 526-553, Jun. 2009.