An Improved Fireworks Algorithm with Landscape Information for Balancing Exploration and Exploitation

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1 An Improved Fireworks Algorithm with Landscape Information for Balancing Eploration and Eploitation Junfeng Chen,QiwenYang, Jianjun Ni, Yingjuan Xie,ShiCheng College of IOT Engineering, Hohai Universit, Changzhou China Jiangsu Ke Laborator of Power Transmission and Distribution Equipment Technolog, Changzhou China Division of Computer Science, Universit of Nottingham Ningbo, China Abstract Fireworks algorithm is a newl risen and developing swarm intelligence algorithm, the performance of which is determined b the tradeoff between eploration and eploitation. How to develop a satisfactor weight for eploration and eploitation is an interesting and challenging work. In this paper, the landscapes of optimization problem are firstl analzed, and then a new sparks eplosion strateg is designed to represent and mine the landscape information. Moreover, the eploration and eploitation coeist in the improved fireworks algorithm, which can automaticall adjust the search strategies according to landscape structure. Finall, numerical eperiments are performed for the algorithms investigations, performance analsis, and comparisons. The simulation results indicate that the proposed algorithm has a significant performance on all the test functions and can achieve the global minimum for most test functions. Inde Terms Fireworks algorithm, landscape information, eploration, eploitation I. INTRODUCTION Fireworks Algorithm (FWA) [], [] is a newl risen and developing swarm intelligence algorithm, which is inspired b the eplosion and illumination of fireworks in the night sk. It is originall attributed to Tan and Zhu, and was modified for solving practical problems, such as nonnegative matri factorization [], digital filters [], spam detection [5], and oil crop fertilization []. Similar to most swarm algorithms, FWA makes few or no assumptions about the problem being optimized and takes an advantage of swarm searching with a population (called a swarm) of candidate solutions (called sparks) to seek high qualit solutions of complicated optimization problems efficientl and effectivel. Man researchers believe that the behavior of a swarm algorithm is determined b two forces: eploration and eploitation [7]. Ecessive eploitation will be high-efficienc, but eas to induce premature convergence. Ecessive eploration will increase accurac, but consume too much computation time. Thus, how to get a tradeoff between the eploration and eploitation is reall a ke to the global performance of swarm intelligence algorithms. Man strategies for balancing eploration and eploitation were proposed in the past decade. For instance, Blum and Roli pointed out that the mechanisms of metaheuristics to efficientl eplore a search space are all based on intensification and diversification, although the are inspired b different philosophies [7]. Liu et al. introduced an entrop-driven eploration and eploitation approach for evolutionar algorithms [8]. Olorunda and Engelbrecht developed a diversit rate for a particle swarm optimization to quantif the rate of change from eploration to eploitation [9]. Zhao et al. designed fitness function, crossover operator and mutation operator for the genetic algorithms to improve the eplorative capabilities and the eploitation effects []. Epitropakis et al. proposed a hbrid differential mutation method, which was a linear combination of the eplorative and eploitive operators []. Lin and Gen introduced fuzz logic control into genetic algorithm for balancing between eploration and eploitation []. Li et al. presented a well-designed densit estimator for a dnamic neighborhood evolutionar algorithm, which achieves auto-tuning between proimit and diversit []. Pei et al. emploed elite individuals sampling, the elite neighborhood sampling and random sampling strategies into the FWA for approimating fitness landscape and enhancing search capabilit []. Liu et al. provided a new parameter for FWA to control the eploration and eploitation dnamicall [5]. Crepinsek et al. discussed what components of evolutionar algorithms contribute to eploration and eploitation, when eploration and eploitation are controlled and how balance between eploration and eploitation is achieved []. In view of the above-mentioned facts, FWA is not sophisticated enough and can be further improved b developing a satisfactor weight for eploration and eploitation. Moreover, the same search strategies ma result in different optimization effects, when dealing with simple and smooth landscapes, or dealing with bump and rough landscapes. In our view, the search efficienc of FWA is not onl influenced b search strategies, but also related to the fitness landscape of optimization problems to be solved. As a result, the main objective of this paper is to reveal the relationship between the landscape information of optimization problems and the interaction mechanisms of eploration and eploitation of FWA. The remaining paper is organized as follows. In Section /5/$. c 5 IEEE 7

2 II, the basic optimization process of FWA is reviewed. In Section III, an improved FWA for auto-tuning eploration and eploitation is proposed and the balance mechanism is analzed, followed b eperimental simulation and result discussion on benchmark functions in Section IV. Finall, our concluding remarks are contained in Section V. II. CANONICAL FIREWORKS ALGORITHM For simplicit, we focus on an unconstrained minimization problem in bounded space with finite dimensions in the form of S: minf( ) () where designates a position or candidate solution in the search-space, S is a finite dimensional search space, and f( ) is the fitness or cost function to be minimized. The canonical FWA works on a new search mechanism which describes the evolution of a population of sparks in the search space for problem solving. A set of sparks X =[,,, N ] composed b N sparks with encoding length M, can be written in the following vector and matri forms: T M g X =. = M.... = g. () N N N NM g M where i =[ i, i,, im ] Sis the i-th sparks in X, and ij is the j-th solution component of the i-th sparks, i =,,,N, j =,,,M. For the sake of convenience, the j-th vector which consists of the j-th components of all the sparks in X is called the j-th set component, written as g j =[ j, j,, Nj ] T. Generall speaking, FWA has three main operators: calculating, creating and mapping. More specificall, calculating operator consists of computing eplosion number and amplitude, creating operator refers to generating eplosion sparks and Gaussian sparks, and mapping operator means bounding sparks back to search space. A principle FWA is described as Algorithm. The number of sparks generated b a firework i is defined as follow. f(ˇ) f( i ) H i = K N j= (f(ˇ) f( () j)) where K is a parameter controlling the number of eplosion sparks. i is the i-th sparks, ˇ is the worst fireworks with the maimum fitness value, i.e., f(ˇ) =ma(f( i )), i =,,,N. To avoid overwhelming effects of splendid fireworks, bounds are defined for H i. H min if H i <H min H i = H ma if H i >H ma () H i otherwise where H min and H ma are the lower and upper bounds of sparks numbers. Amplitude of eplosion for each firework is defined as follows f( i ) f(ˆ) A i = R N i= (f( (5) i) f(ˆ)) where R denotes the maimum eplosion amplitude and ˆ is the best fireworks with the minimum fitness value, i.e., f(ˆ) = min(f( i )), i =,,,N. The basic FWA contains a rough thinking of the tradeoff between eploration and eploitation. The eploitation implies the fireworks with better fitness generate a larger population of eplosion sparks within a smaller range. While eploration means the fireworks with lower fitness can onl generate a smaller population with higher eplosion amplitude. III. IMPROVED FWA WITH LANDSCAPE INFORMATION A. Landscape Information of Optimization Problems No free lunch theorem [7] indicates that no search strateg can keep optimal when optimizing arbitrar problems. In other words, a general-purpose and universal optimization algorithm is impossible. The onl wa that one strateg can outperform another is if it is specialized to the structure of the specific problem under consideration. However, we know nothing about the landscape structure when dealing with a black-bo problem. As a result, How to get and sustain the domain knowledge of specific optimization problems is a fundamental issue in the swarm intelligence field. In this paper, we tr to design a new sparks eplosions strateg which is characterized of representing and mining the landscape information of optimization problem. Specificall, the landscape of optimization problem is approimatel described as the distribution structure of candidate solutions during the searching process of FWA. Here, we take a simple mathematical function f(, ) = sin() sin() to illustrate the basic idea of the proposed method. When, [, π], the landscape structure is shown as Fig.. From Fig. (a), the three-dimensional surface image has two maima and two minima, and its corresponding contours describe the curves with the same function values. The color is deeper and cooler and the value is smaller. Fig. (b) is the contour chart with the accumulated candidate solutions of FWA iterating. The black dots represent the candidate solutions, plotted b the variable on the horizontal ais and the variable on the vertical ais. In general, a swarm intelligence algorithm onl makes full use of evaluation function rather than other additional information. In other words, swarm algorithms onl consider about whether a candidate solution is superior to another or not, regardless of distribution differences of the candidate solutions. However, there is much useful information in the function landscape. For eample, Fig. (c) represents the distribution structure of candidate solutions with small evaluation values. It can be seen that these candidate solutions outline the local landscapes of minima, which are located on the lower-right and the upper-left parts. In practice, there are lots of difficult optimization problems, which landscapes are relativel complicated and diverse. Some 7

3 Algorithm : Pseudocode of Fireworks Algorithm Initialize N sparks randoml and evaluate their qualit (i.e., fitness); while have not found good enough solution or not reached the pre-determined maimum number of iterations do for each firework i do Calculate the number of sparks H i ; 5 Calculate the amplitude of eplosion A i ; for j =to H i do 7 Randoml select P components from i, P = round(m rand(, )); 8 for for each selected components k =to P do 9 Update the displacement of components; s j i with updated ik ; if s jk <LBor s jk >UBthen Mapping s jk to the search space s jk = LB + rand(, ) (UB LB); for h =to G i do Randoml select Q components from i ; 5 for each selected components l =to Q do Update the components il = rand(, ) ( il il); 7 g h i with updated il ; 8 if g hl <LBor g hl >UBthen 9 Mapping g hl to the search space ghl = LB + rand(, ) (UB LB); Evaluate the fitness values of the sparks s j and g h, j [,H i ],h [,G i ]; Keep the best sparks to net generation; Select the other N sparks from the generated sparks based on the selection probabilit; are multi-dimensional functions with a great number of peaks and valles or with strong variables linkage. However, a multi-dimensional rugged landscape ma be regarded as a combination of several one-dimensional models. Fig. lists the common formats of one-dimensional curves. These landscapes ma be a local part of the one-dimensional landscape of sophisticated problems. Fig.. (a) (c) (e) (b) (d) (f) The tpical one-dimensional landscapes. It can be seen that Fig. (a) is a simple valle, which is often regarded as an eas landscape for almost all the swarm algorithms. Adding some noise to the function in Fig. (a) will form the rugged landscape in Fig. (b). A multi-funnel landscape is shown in Fig. (c). If the global optimal solution is far awa from the local optima to some etent, some swarm algorithms ma be trapped in those local optima. According to the curve in Fig. (d), the introducing fluctuations into the landscape contribute to increasing the optimization hardness. Fig. (e) and Fig. (f) is an opposite pulse function without or with noise signal. In contrast to the curve in Fig. (e), the local disturbance helps lessen the optimization hardness, because there is some useful information in the landscape in Fig. (f). B. Creating sparks with Landscape Information In this paper, an improved fireworks algorithm is developed with eploration and eploitation coeisting. The search strategies can be switched according to landscape structure adaptivel. The useful information in the landscape is represented b the dispersing etent and dispersion degree of the candidate solutions. Algorithm describes the proposed operator of creating sparks with landscapes information. As shown in Algorithm, the landscape structure is represented b the distribution information of the selected sparks, denoted as a set of vectors { u, v, w}, where u is the kernel, v is the coverage and w is the dispersion of the selected sparks. And then this landscape information is used to create 7

4 5 f (, ).5.5 (a) The landscape stereogram 5 (b) The contour chart with candidate solutions. f (, ).5 (c) The landscape information represented b selected candidate solutions. Fig.. The landscape structure of function f(, ) =sin() sin() Algorithm : Pseudocode of the creating sparks operator Select L sparks from the generated sparks and the fitness values of selected sparks meet the predicate conditions; Determine the best spark as the kernel in the selected sparks u =[u,u,,u M ]=ˆ; Calculate the coverage of the selected sparks v =[v,v,,v M ]= L L i= ( i u) ; Mapping the fitness values f( ) to the confidence interval (, ]; 5 for each couple of sparks and their fitness values ( i,f( i )) do Calculate o i = ( i u) lnf( ; i) 7 Calculate the mean of o i as o, i.e., o = L L i= o i; 8 Calculate the dispersion of the selected sparks w =[w,w,,w M ]= L L i= ( o i o); 9 Create the N new sparks based on the landscape information { u, v, w}; for j =to N do Generate a normall distributed random vector p with the kernel v and the coverage w, i.e., p = NormRand( v, w); Generate a normall distributed random spark i with the kernel u and the coverage p, i.e., i = NormRand( u, p); 75

5 new sparks for the net generation. As a matter of fact, we can control eploration and eploitation b selecting different sparks in the creating sparks operator. Fig. and Fig. show two different selecting and creating processes for function f(, ) =sin(.5) cos(), [, π] and [,π]. As shown in Fig. (a), the function values of selected sparks are lesser than average value and the corresponding landscape vectors are calculated as a set of vectors { u, v, w }, where the kernel u =[.7,.595], the coverage v = [.8,.77], and the dispersion w =[.577,.75]. It can be seen that the elite of the selected sparks is situated on [.7,.595]. The coverage of the selected sparks on the horizontal ais is greater than the coverage on the vertical ais. Meanwhile, the sparks in ais are scattered, while in ais are concentrated. Fig. (b) shows the new generated sparks on account of these landscape information. As seen in Fig. (a), we select the superior 5 sparks and calculate landscape vectors, denoted as { u, v, w }.Thekernel u is equivalent to u, i.e., u = u =[.7,.595]. The coverage v =[.9,.9], which is less than v. The dispersion w = [.,.78], which is less than w. The new creating sparks based on landscape information are shown in Fig. (b). Moreover, we can see that the new generated sparks have distinctive features, b comparing Fig. (b) with Fig. (b). Therefore, the proposed creating operator mingles eploration with eploitation to identif the regions with high qualit solutions in the search space quickl b mean of landscapes information. C. Improved FWA for Auto-tuning Eploration and Eploitation As we know, optimization problems come in a variet of shapes. It is a good choice to select the eplosion operator to generate new sparks for simple unimodal landscapes. And the creating operator with landscapes information contributes to discover the promising region of rugged landscapes in the search space. As for astonishingl comple problem with little useful information in its landscape, random search without preference for an region is an ecellent strateg. In this section, the improved FWA integrates the advantages of three creating operators, which not onl improves the performance greatl, but also reduce the calculation cost. The improved FWA for auto-tuning eploration and eploitation can be described as Algorithm. As shown in Algorithm, the proposed algorithm doesn t simpl run the three creating sparks methods in order or in parallel, but select the proper creating strateg for each component of sparks b using conditional rules. The reason is that the landscapes in each dimension are diverse and variational, especiall for multi-dimension and complicated optimum problem. The sole creating algorithm will depress diversit in some components of sparks and then induce premature convergence. IV. NUMERICAL EXPERIMENTS AND PERFORMANCE ANALYSIS For better understanding the search behavior of the proposed FWA, a set of numerical eperiments are performed for the algorithm investigations, performance analsis, and comparisons. First, we define several benchmark functions which have been widel used in the swarm intelligence communit. Then, the eperimental results of three versions of FWAs are provided and the empirical results are discussed in this section. A. Test Functions Eight well-known benchmark functions are selected as a test suite. Table I illustrates the names, mathematical epressions, dimensions, search space and optimal solution. Based on their properties, these functions can be divided into two groups. Functions f f are high dimensional unimodal functions, and functions f 5 f 8 are high dimensional multimodal functions with man local minima. Some of them possess the highl non-linear characteristics. In addition, shift values are added to functions f 5 and f to form new functions f 7 and f 8, which means the global optimum can be converted to somewhere else. To be specific, the optimal positions of the two functions are shifted and their minimal fitness values remain the same. In addition, we transform all the optimal fitness values to zero for ease of comparison of eperiment results. B. Simulation Results and Performance Analsis Three versions of FWAs are selected to perform the simulation eperiments. The are conventional fireworks algorithm (CFWA) [], enhanced fireworks algorithm [] (EFWA), and improved fireworks algorithm (IFWA) proposed in this paper, respectivel. In all the following eperiments, the population size is set to 5, namel N =5. As FWAs are nondeterministic algorithms, we set maimum iteration and a small positive value. for each test to keep the algorithms from falling into an infinite loop. The parameters of both the CFWA and the EFWA are set in accord with Ref. [], which is applied in all the comparison eperiments. For IFWA, R =, K =, Q =5, H min =5and H ma =. All the FWAs are implemented in Matlab 8. and the simulations are run on the same PC with Intel Core i 5M.GHz, GB memor capacit and the Windows 7 operating sstem. Table II lists the comparison results of benchmark functions obtained b CFWA, EFWA and IFWA. Mean and Std indicate the average evaluation values and standard deviation for each benchmark function, respectivel. Fig. 5 shows the stereograms of Sumcan and Sumcan with logarithmic transformation. From Table II, it can be seen that IFWA outperforms CFWA and EFWA for all the test functions. And it shows significant performance on the shifted functions, which is increasingl difficult to be optimized. Meanwhile, IFWA achieves the global minima of most test functions, ecept the Sumcan function. This is because there is much useful landscape information in these functions, from which IFWA can benefit 7

6 (a) Calculating landscape vectors { u, v, w } based on selected sparks (b) Creating sparks based on landscape information { u, v, w } Fig.. Creating processes with landscape information { u, v, w } (a) Calculating landscape vectors { u, v, w } based on selected sparks (b) Creating sparks based on landscape information { u, v, w } Fig.. Creating processes with landscape information { u, v, w } Algorithm : Improved FWA for Auto-tuning Eploration and Eploitation Initialize N sparks randoml and evaluate their qualit (i.e., fitness); while a termination criterion is unmet do Calculate the coverage of all the sparks; for each set component g j, j =,,...,M do 5 if coverage > half of search space of the component then Generate a certain number of new components of sparks with a random position; 7 else if one-fiftieth of search space < coverage < half of search space then 8 Call Algorithm to create the new components; 9 else if coverage < one-fiftieth of search space then Call Algorithm to generate eplosion components of sparks; 77

7 TABLE I BENCHMARK FUNCTIONS. Function Name Mathematical Epressions Dim. Search Space Optimal fitness (D) (S) (f min) Sphere Rosenbrock Sumcan Sumcan with log Ackle f ( ) = D i= i (, ) D f ( ) = D [ i= (i+ i ) +( i ) ] (, ) D [ f ( ) = 5 + ] D i= i +. 5 (.,.) D =, i = i + i [ f ( ) = log 5 + ] D i= i +5 (.,.) D =, i = i + [ i ( D ) ] f 5( ) = ep. D i= i (, ) D [ ( D ) ] ep. i= cos (πi) ++e D Rastrigin Shifted Ackle Shifted Rastrigin f ( ) = D [ i= i cos (π ] i)+ f 7( ) =f 5( o new + o old ) ( 5., 5.) D o old =[,,, ] (, ) D o new =[.,.,,.] f 8( ) =f ( o new + o old ) o old =[,,, ] ( 5., 5.) D o new =[.5,.5,,.5] (a) Visualization of two dimensional Sumcan function. (b) Visualization of two dimensional Sumcan function with logarithmic transformation. Fig. 5. The stereograms of the Sumcan and Sumcan with logarithmic transformation. greatl. While the Sumcan function is a difficult optimization problem like fishing for a needle in a hastack. Fig. 5 (a) is the visualization of two dimensional Sumcan function, which is hard to locate the global minimum for FWAs. But it gets much easier when we invert the fitness function b use of logarithmic transformation. Fig. 5 (b) shows visualization of two dimensional Sumcan function with logarithmic transformation. We can see that the landscape has eperienced the procedure from graduall development to mature. It is worth pointing out that the FWAs are meta-heuristic algorithms, which performance is not onl related to search strategies, but also to initial distribution of solutions, parameter values and maimum iteration involved in algorithms. For eample, CFWA can get better optimization results for the Rosenbrock function, if we etend the maimum iteration. V. CONCLUSIONS Fireworks algorithm which is inspired b the eplosion of fireworks, aims to seek high qualit solutions of complicated optimization problems efficientl and effectivel. However, how to balance eploration and eploitation is reall a ke to the global performance of FWAs. In this paper, we hold that the search efficienc of FWA is not onl influenced b search strategies, but also related to the fitness landscape of optimization problems to be solved. The landscape information of optimization problem is firstl analzed and described as a set of vectors: the kernel, the coverage and the dispersion of the selected sparks. And then an improved FWA is proposed to tune eploration and eploitation adaptivel. Finall, the 78

8 TABLE II COMPARISON RESULTS OF CFWA, EFWA AND IFWA. CFWA EFWA IFWA f Mean.e+ 9.7e-.e+ (Std) (.e+) (.e-) (.e+) f Mean.87e+ 7.9e+.e+ (Std) (.e+) (8.7e+) (.e+) f Mean 9.88e+ 9.57e e+ (Std) (.e+) (5.99e+) (.7e+) f Mean.e+.e+.e+ (Std) (.e+) (.e+) (.e+) f 5 Mean.e+.e-.e+ (Std) (.e+) (.e-) (.e+) f Mean.e+.597e+.e+ (Std) (.e+) (.e+) (.e+) f 7 Mean 7.e- 8.75e+.e+ (Std) (.e-) (9.e+) (.e+) f 8 Mean 9.e+.8e+.e+ (Std) (.e+) (.8e+) (.e+) proposed algorithm is applied to numerical optimization problems in comparison with the other two FWAs. The statistical results show the proposed method can identif the regions with high qualit solutions in the search space quickl b using the landscape information. Besides that, we eplain the reasons wh IFWA is hard to locate the global minimum of the Sumcan function. ACKNOWLEDGMENT The research work reported in this paper is supported b the National Natural Science Foundation of China under Grant Number, 77, the Fundamental Research Funds for the Central Universities under Grant Number B8 and B9, and the Ningbo Science & Technolog Bureau with the Science and Technolog Project Number B55. REFERENCES [] Y. Tan and Y. Zhu, Fireworks algorithm for optimization, in Advances in Swarm Intelligence, ser. Lecture Notes in Computer Science, Y. Tan, Y. Shi, and K. C. Tan, Eds. Springer Berlin Heidelberg,, vol. 5, pp. 55. [] S. Zheng, A. Janecek, and Y. Tan, Enhanced fireworks algorithm, in Proceedings of IEEE Congress on Evolutionar Computation, (CEC ). Cancún, Méico: IEEE,, pp [] A. Janecek and Y. Tan, Swarm intelligence for non-negative matri factorization, International Journal of Swarm Intelligence Research (IJSIR), vol., no., pp., October-December. [] H. Gao and M. Diao, Cultural firework algorithm and its application for digital filters design, International Journal of Modelling, Identification and Control, vol., no., pp.,. [5] W. He, G. Mi, and Y. Tan, Parameter optimization of localconcentration model for spam detection b using fireworks algorithm, in Advances in Swarm Intelligence, ser. Lecture Notes in Computer Science, Y. Tan, Y. Shi, and H. Mo, Eds. Springer Berlin Heidelberg,, vol. 798, pp [] Y.-J. Zheng, Q. Song, and S.-Y. Chen, Multiobjective fireworks optimization for variable-rate fertilization in oil crop production, Applied Soft Computing, vol., no., pp. 5,. [7] C. Blum and A. Roli, Metaheuristics in combinatorial optimization: Overview and conceptual comparison, ACM Computing Surves, vol. 5, no., pp. 8 8, September. [8] S.-H. Liu, M. Mernik, and B. R. Brant, Entrop-driven eploration and eploitation in evolutionar algorithms, in Proceedings of the second International Conference on Bioinspired Optimization Methods and their Applications (BIOMA ),, pp. 5. [9] O. Olorunda and A. P. Engelbrecht, Measuring Eploration / Eploitation in Particle Swarms using Swarm Diversit, in Proceedings of the 8 Congress on Evolutionar Computation (CEC 8), 8, pp. 8. [] G. Zhao, W. Luo, H. Nie, and C. Li, A genetic algorithm balancing eploration and eploitation for the travelling salesman problem, in Proceedings of the Fourth International Conference on Natural Computation (ICNC 8), 8, pp [] M. Epitropakis, V. Plagianakos, and M. Vrahatis, Balancing the eploration and eploitation capabilities of the differential evolution algorithm, in Proceedings of the 8 Congress on Evolutionar Computation (CEC 8), 8, pp [] L. Lin and M. Gen, Auto-tuning strateg for evolutionar algorithms: balancing between eploration and eploitation, Soft Computing, vol., no., pp. 57 8, 9. [] K. Li, S. Kwong, J. Cao, M. Li, J. Zheng, and R. Shen, Achieving balance between proimit and diversit in multi-objective evolutionar algorithm, Information Sciences, vol. 8, no., pp., Januar. [] Y. Pei, S. Zheng, Y. Tan, and H. Takagi, An empirical stud on influence of approimation approaches on enhancing fireworks algorithm, in IEEE International Conference on Sstems, Man, and Cbernetics (IEEE SMC ),, pp. 7. [5] J. Liu, S. Zheng, and Y. Tan, The improvement on controlling eploration and eploitation of firework algorithm, in Advances in Swarm Intelligence, ser. Lecture Notes in Computer Science, Y. Tan, Y. Shi, and H. Mo, Eds. Springer Berlin Heidelberg,, vol. 798, pp.. [] M. Črepinšek, S.-H. Liu, and M. Mernik, Eploration and eploitation in evolutionar algorithms: A surve, ACM Computing Surves, vol. 5, no., pp., June. [7] D. H. Wolpert and W. G. Macread, No free lunch theorems for optimization, IEEE Transactions on Evolutionar Computation, vol., no., pp. 7 8, April