DIFFERENTIAL evolution (DE) [3] has become a popular

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1 Self-adative Differential Evolution with Neighborhood Search Zhenyu Yang, Ke Tang and Xin Yao Abstract In this aer we investigate several self-adative mechanisms to imrove our revious work on [], which is a recent DE variant for numerical otimization. The selfadative methods originate from another DE variant, [2], but are remarkably modified and extended to fit our. And thus a Self-adative (Sa) is roosed to imrove s erformance. Three self-adative mechanisms are utilized in Sa: self-adatation for two candidate mutation strategies, self-adatations for controlling scale factor F and crossover rate CR, resectively. Exerimental studies are carried out on a broad range of different benchmark functions, and the roosed Sa has shown significant sueriority over. I. INTRODUCTION DIFFERENTIAL evolution (DE) [3] has become a oular algorithm in global otimization. It has shown suerior erformance in both widely used benchmark functions [4], [5] and real-world alications [6]. DE conventionally has several candidate mutation schemes, and three control arameters, i.e., oulation size NP, scale factor F and crossover rate CR. Aart from the arameter NP (which is common for all oulation-based algorithms), mutation strategy selection, arameters F and CR adatations are the three most imortant issues of DE research. Many work has been done along these lines. The relationshi between the control arameters and oulation diversity has been analyzed in [7]. Exerimental arameter studies and emirical arameter settings of DE have been carried out in [8]. Selfadative strategy has also been investigated to adat these control arameters [9], as well as different mutation strategies [2]. In [], we roosed a DE variant, namely Differential Evolution with Neighborhood Search (), to adat the scale factor F. Insired by the neighborhood search (NS) strategy in evolutionary rogramming (EP) [], intends to mix search biases of different NS oerators through the factor F. It is well-known that NS is a main strategy underinning EP []. Although DE might be similar to the evolutionary rocess of EP, it lacks relevant concet of neighborhood search. Instead of redefining the factor F as a constant, generates F from Gaussian and Cauchy distributed random numbers, which are beneficial to roducing small and large search ste sizes, resectively []. A robability is introduced to control when to use Gaussian or Cauchy oerator. In the revious work was simly set to a The authors are with the Nature Insired Comutation and Alications Laboratory, the Deartment of Comuter Science and Technology, University of Science and Technology of China, Hefei, Anhui 2327, China. Xin Yao is also with CERCIA, the School of Comuter Science, University of Birmingham, Edgbaston, Birmingham B5 2TT, U.K. ( s: zhyuyang@mail.ustc.edu.cn, ketang@ustc.edu.cn, x.yao@cs.bham.ac.uk). Corresonding author: Ke Tang (Phone: ). constant number. Obviously, it would be more desirable if could be self-adated during the evolution rocess. By these means, the algorithm can automatically adjust between Gaussian and Cauchy oerators, and thereby the erformance can be imroved. Self-adative Differential Evolution () by Qin et at. [2], is a different DE variant that mainly focuses on adatation for arameter CR and mutation strategies of DE. The motivation is to solve the dilemma that CR values and mutation strategies involved in DE are often highly roblem deendent. adots two DE mutation strategies and introduces a robability to control which one to use. The robability is gradually self-adated according to learning exerience. Additionally, crossover rate CR is self-adated by recording CR values that make offsring successfully enter the next generation. Both of the two self-adative mechanisms have achieved significant imrovement over the classical DE with emirical arameter configuration. It can be concluded that and have quite different emhases on imroving DE s erformance. ays secial attention to the crossover rate CR s adatation and the self-adatation between different DE mutation strategies, while intends to mix search biases of different NS oerators through the arameter F, and no self-adatation is adoted. The difference motivates us to introduce s self-adative mechanisms into, study their behaviors, and then roose a self-adative (Sa). The outline and features of the roosed Sa are summarized as follows: ) It inherits the self-adated mutation schemes selection mechanism of ; 2) It adots a selfadative strategy to adjust the arameter of ; 3) It enhances the original CR self-adatation of by adding a weighting strategy. The efficacy of Sa is evaluated on two sets of widely used benchmark functions. The rest of this aer is organized as follows: Section II gives the reliminaries; Section III describes the roosed Sa algorithm; Section IV resents the exerimental studies; Finally, Section V concludes this aer and briefly discusses several other self-adative DE schemes. II. PRELIMINARIES A. Differential Evolution (DE) Individuals in DE are reresented by D-dimensional vectors x i, i,, NP}, whered is the number of objective arameters and NP is the oulation size. According to [3], the classical DE can be summarized as follows: ) Mutation: v i = x i + F (x i2 x i3 ) /8/$25. c 28 IEEE

2 where i,i 2,i 3 [, NP] are random and mutually different integers, and they are also different with the vector index i. Scale factor F > is a real constant factor and is often set to. 2) Crossover: vi (j), if U u i (j) = j (, ) CR or j = j rand x i (j), otherwise. with U j (, ) stands for the uniform random number between and, and j rand is a randomly chosen index to ensure that the trial vector u i does not dulicate x i. CR (, ) is the crossover rate, which is often set to.9. 3) Selection: x i = ui, if f(u i ) f(x i ) x i, otherwise. where x i is the offsring of x i for the next generation (Without loss of generality, we consider only minimization roblem in this aer). There are several schemes of DE based on different mutation strategies [3]: v i = x i + F (x i2 x i3 ) () v i = x best + F (x i x i2 ) (2) v i = x i + F (x best x i )+F (x i x i2 ) (3) v i = x best + F (x i x i2 )+F (x i3 x i4 ) (4) v i = x i + F (x i2 x i3 )+F (x i4 x i5 ) (5) Schemes () and (3), with notations as DE/rand/ and DE/current to best/2, are the most often used in ractice due to their good erformance [2], [3]. B. Differential Evolution with Neighborhood Search () [] is a recent DE variant that utilizes the neighborhood search (NS) strategy in evolutionary rogramming (EP). NS is a main strategy underinning EP, and the characteristics of several NS oerators have been investigated in EP literature []. Although DE might be similar to the evolutionary rocess in EP, it lacks relevant concet of neighborhood search. is the same with the classical DE described in Section II.A excet the scale factor F is relaced by the following equation: Ni (, ), if U F i = i (, ) < (6) δ i, otherwise. where i is the index of current trial vector, U i (, ) stands for the uniform random number between and, N i (, ) denotes a Gaussian random number with mean and standard deviation, and δ i denotes a Cauchy random variable with scale arameter t =. The arameter was set to a constant number in. The advantages of NS strategy in DE have been studied in []. Exerimental results have shown that has significant advantages over classical DE on a broad range of different benchmark functions. It has been found that is effective in escaing from local otima when searching in environments without rior knowledge about what kind of search ste size will be referred. C. Self-adative Differential Evolution () by Qin et al. [2], gives the first attemt to adot two different mutation strategies in single DE variant. The motivation of is to solve the dilemma that mutation strategies involved in DE are often highly deendent on the roblems under consideration. It introduces a robability to control which mutation strategy to use, and is gradually self-adated according to the learning exerience. Additionally, utilizes two methods to adat and self-adat DE s arameters F and CR. Detailed contributions of are summarized as follows: ) Mutation strategies self-adatation: selects mutation strategies Eq. () and Eq. (3) as candidates, and roduces the trial vector based on: Eq. (), if Ui (, ) < v i = (7) Eq. (3), otherwise. Here is set to initially. After evaluation of all offsring, the number of offsring successfully entering the next generation while generated by Eq. () and Eq. (3) are recorded as ns and ns 2, resectively, and the numbers of offsring discarded while generated by Eq. () and Eq. (3) are recorded as nf and nf 2. Those two airs of numbers are accumulated within a secified number of generations (5 in ), called the learning eriod. Then, the robability is udated as: ns (ns 2 + nf 2 ) = (8) ns 2 (ns + nf )+ns (ns 2 + nf 2 ) Here ns, ns 2, nf and nf 2 will be reset once is udated after each learning eriod. 2) Scale factor F setting: In, F is set to F i = N i (,.3) where N i (,.3) denotes a Gaussian random number with mean and standard deviation.3. 3) Crossover rate CR self-adatation: allocates a CR i for each individuals according to: CR i = N i (,.) (9) is set to initially. These CR values for all individuals remain the same for several generations (5 in ) and then a new set of CR values is generated using the same equation. During every generation, the CR values associated with offsring successfully entering the next generation are recorded in an array CR rec. After a secified number of generations (25 in ), will be udated: = CR rec CR rec (k) () CR rec k= 28 IEEE Congress onevolutionarycomutation(cec 28)

3 CR rec will be reset once is udated. This selfadatation scheme for CR is denoted as SaCR. For detailed rinciles and exlanations behind s selfadatation strategies, arameter settings, or even simulated results, lease refer to [2]. III. SELF-ADAPTIVE DIFFERENTIAL EVOLUTION WITH NEIGHBORHOOD SEARCH A. Sa: The Incororated Algorithms It can be concluded that and have quite different emhases: The former ays secial attention to selfadatation between different mutation strategies, as well as the self-adatation on crossover rate CR, while the latter intends to mix search biases of different NS oerators through the arameter F, and no self-adatation is adoted. The difference motivates us to introduce s self-adative mechanisms into, study their behaviors, and then roose a self-adative (Sa). Based on the motivations above, we address crucial issues of the roosed Sa as follows: ) Mutation strategies self-adatation: Sa utilizes the same method as in this art. For details, lease refer to Eq. (7) and Eq. (8). 2) Scale factor F self-adatation: Sa inherits the method of controlling the arameter F from, but extending it to: Ni (,.3), if U F i = i (, ) < δ i, otherwise. where will be self-adated as is done in according to Eq. (8), excet here we have to record corresonding F values that make offsring enter the next generation successfully. 3) Weighted crossover rate CR self-adatation: We use a similar strategy to what does with SaCR strategy. But whenever we record a successful CR value in array CR rec, we will also record the corresonding imrovement on fitness value in array Δf rec, with Δf rec (k) =f(k) f new (k). And then, Eq. () is changed to: = CR rec k= w k =Δf rec (k)/ w k CR rec (k) () Δf rec k= Δf rec (k) (2) Note: here CR rec Δf rec. The weighted selfadatation scheme for CR is denoted as SaCRW, and we will exlain why we add the weight mechanism to the original SaCR in Section III.B with details. Due to the significant successes of and, Sa, which incororates enhancements ), 2) and 3), is romising. B. Weighted CR Self-adatation The arameter CR of DE determines how many comonents of mutated vector will be introduced into current candidate for the next generation, so the robability of generating imroved offsring from the same arent with a small CR is higher than that with a large CR. It can referred that the of SaCR has an imlicit bias towards small values during self-adatation rocess. The bias might become harmful when otimizing nonsearable functions, in which interactions exist between variables. Because large CR value is required to change the nonsearable variables together. To illustrate this roblem, we conducted an exeriment with Sa+SaCR on the well-known Generalized Rosenbrock s function []. The evolution curves for S runs and F runs of 25 indeendent runs are given in Fig.. Here S runs means Sa has found the region of otimum, while F runs means Sa failed to do that. For S runs, it can be found that was successfully adated to a large value, and after that the fitness values are imroved quickly. For F runs, was adated to a small value, and traed there from that time on. The algorithm failed to make significant imrovement on fitness values thereafter..2 2 S runs F runs 2 S runs F runs 4 fitness value Fig.. Evolution curves of and fitness value on the Generalized Rosenbrock function. S runs denotes results of successful runs, while F runs denotes results of failed runs of 25 indeendent runs. The vertical axes show the value (u figure) and fitness value (down figure), while the horizontal axes show the number of generations. On the other hand, it is assumed that large successful CR values will achieve larger imrovement on fitness values than small successful CR values for nonsearable functions, because it will be good to change nonsearable variables together []. So we can balance the bias of SaCR with a weight based on the size of fitness value imrovement. This is the basic motivation of SaCRW in Section III.A. To validate the effectiveness, another exeriment with Sa+SaCRW was conducted on the same function. The results are summarized in Table I. It can be found that SaCRW was successfully adats to required large values in all runs. The advantage is also shown by differences of fitness values IEEE Congress onevolutionarycomutation(cec 28)

4 TABLE I SIMULATED RESULTS OF SACR AND SACRW ON THE GENERALIZED ROSENBROCK FUNCTION.THE RESULTS OF 25 INDEPENDENT RUNS ARE SORTED FROM ST TO 25TH BASED ON FITNESS VALUES # of Sa+SaCR Sa+SaCRW runs Fitness Final Fitness Final st 7.42e-3 4.e+ 88 5th 3.99e+.5.e+ 34 9th 2.33e+.64.e+ 53 3th 2.34e+.57.e+ 73 7th 2.37e e th 2.39e e th 2.43e+.55.9e Mean.82e+ 4.3e-3 Std 9.76e+ 6.28e-3 SaCRW made the algorithms success in all 25 runs, while SaCR made it achieve only 4 successful runs (st 4th). IV. EXPERIMENTAL STUDIES A. Exerimental Setu Exerimental validations for the roosed Sa are conducted on both a set of classical test functions [], and a new set of benchmark functions rovided by CEC 25 secial session [2]. The algorithms used for comarison are, and Sa (with SaCRW). The oulation size NP is set to for all algorithms, and no other arameters is adjusted during evolution. B. Results on Classical Benchmark Functions The classical test set includes 23 functions, in which f f 3 are high-dimensional (3-D) and f 4 f 23 are low-dimensional functions. Functions f f 5 are unimodal, functions f 8 f 3 are multimodal functions with many local otima, and functions f 4 f 23 are multimodal functions with only a few local otima. Details of these functions can be found in the aendix of []. The number of evolution generations of all algorithms is set to 5 for f f 4, 5 for f 5, 5 for f 6 f 3, 2 for f 4, 5 for f 5 and 2 for f 6 f 23. The average results of 25 indeendent runs are summarized in Tables II IV. TABLE II EXPERIMENTAL COMPARISON ON f f 7 (OVER 25 RUNS). Func Mean Mean Mean t-test t-test f 3.2e e e f e- 6.22e- 4.5e f e-22.2e-8.6e f 4.59e e e f 5 4.3e-3 2.e+.24e f 6.e+.e+.e+.. f 7 7.2e e-3.2e For unimodal functions f f 7, Sa achieved much better results than and, excet on the simle ste function f 6, where all three algorithms erformed exactly the same. The great difference can be seen from results on the Generalized Rosenbrock s Function, f 5. The evolution curves of arameter adatation and fitness value for this function are given in Fig. 2 and 3. As we mentioned before, in Sa has been able to self-adat to roer values. TABLE III EXPERIMENTAL COMPARISON ON f 8 f 3 (OVER 25 RUNS). Func Mean Mean Mean t-test t-test f f 9.84e-5 4.e e f 2.36e-2 9.6e- 6.72e f.e+ 8.88e e f e-23.2e e f 3 3.2e-22.75e e For multimodal functions f 8 f 3, Sa is the clear winner again, excet that it was outerformed by on function f 9. With further observation of curves on Figs. 2 and 3, Sa converged slower than, but still made good imrovement all the way. This might have haened because the self-adated arameters in Sa need more time to find the roer values on this function. TABLE IV EXPERIMENTAL COMPARISON ON f 4 f 23 (OVER 25 RUNS). Func Mean Mean Mean t-test t-test f f 5 3.7e-4 3.7e-4 3.7e-4.. f f f 8 3.e+ 3.e+ 3.e+.. f f f f f Table IV shows the results for low-dimensional functions f 4 f 23. The three comared algorithms showed only very minor differences on f 2 and f 2 f 23 (which cannot be seen from the mean values). That is because all of the algorithms have suerior erformance on this low-dimensional functions. C. Results on CEC 25 Benchmark Functions To evaluate Sa further, a new set of benchmark functions rovided by CEC 25 secial session was used. It includes 25 functions with different comlexity [2]. Functions f cec f cec5 are unimodal while the remaining 2 functions are multimodal. Since functions f cec5 f cec25 are hybrid comosition functions, which are very time consuming for fitness evaluation, we only used the first 4 functions of the set in our exeriments. All of these functions are scalable, and we set their dimensions to 3 in our exeriments. Detailed descrition of these functions can be found in [2]. The number of evolution generations is set to 3 for all 28 IEEE Congress onevolutionarycomutation(cec 28) 3

5 5 Sa.2.7 f f Sa f f Sa f cec5 f cec Sa f cec9 2 f cec9 Fig. 2. The self-adatation curves of, and for f 5, f 9, f cec5, and f cec9. On the vertical axes are shown their values (between and ), while on the horizontal axes are shown the number of generations. Fig. 3. The evolution curves for f 5, f 9, f cec5 and f cec9. The vertical axes show the distance to the otimum and the horizontal axes show the number of generations IEEE Congress onevolutionarycomutation(cec 28)

6 functions. Error value, i.e. the difference between current fitness value and otimum, is used to comare algorithm s erformance. The average error values of 25 indeendent runs are summarized in Tables V and VI. TABLE V EXPERIMENTAL COMPARISON ON f cec f cec5 (OVER 25 RUNS). Func Error Error Error t-test t-test f cec.e+.e+.e+.. f cec2 5.68e-4.25e-3 4.e f cec3 5.43e+4.77e+5.67e f cec4.22e-4.89e e f cec5 2.45e-.e+3.5e For unimodal functions, Sa erformed better than the other two algorithms on all 5 functions, and significantly better on functions f cec2, f cec4 and f cec5. This is consistent with conclusions drawn on classical unimodal functions. The effectiveness and efficiency on f cec5 can be seen from evolution curves in Fig. 2 and 3. TABLE VI EXPERIMENTAL COMPARISON ON f cec6 f cec4 (OVER 25 RUNS). Func Error Error Error t-test t-test f cec6.59e- 2.99e+ 2.89e f cec7 8.57e-3.65e-2.2e f cec8 2.9e+ 2.9e+ 2.9e+.9.8 f cec9.e+ 2.27e-5.99e f cec 4.2e+ 5.5e+ 4.24e f cec.2e+ 2.7e+.48e f cec2 4.6e e+4.74e f cec3 2.2e+ 2.e+ 5.e f cec4.27e+.26e+.32e For multimodal functions, Sa obtained better results on almost all functions, excet on f cec3 and f cec4,whereit was outerformed by. All algorithms erformed badly on the two functions, and haened to show a minor sueriority. f cec3 and f cec4 are exanded functions, which are comosed of other different functions. This makes their characteristics unclear for further case study. The analysis of algorithms evolutionary behaviors on functions like them is one of the focuses of our future work. Fig. 3 shows the evolution curves of f cec9. It can be seen that Sa found the otimum in less than the maximum number of available generations. The curves in Fig. 2 of this functions showed Sa required different values for, and during different stages, and the self-adatation strategies were able to adjust these arameters as needed (from large to small, then to large again). V. CONCLUSIONS AND DISCUSSIONS In this aer, we roosed a new self-adative DE variant, Sa, which is an imroved version of our revious algorithm. The Sa can be viewed as a hybridization of [2] and []. In Sa: ) We utilized the self-adatation strategy of to adat between candidate mutations; 2) We alied a self-adatation to adjust arameter F ; 3) We illustrated the ill-condition of original CR self-adatation in, and roosed an enhanced version with weighting. The erformance of the roosed Sa algorithm is evaluated and discussed on both a set of 23 classical test functions[], and a new set of 4 benchmark functions rovided by CEC 25 secial session [2]. Sa has shown significant sueriority over both and. Besides the mentioned in this aer, several other self-adative DE variants (s) have also been roosed. Omran et al. roosed a SDE [3], [4] by adating arameters F and CR based on normal distribution. Brest et al. resented the jde, which attaches F and CR values to all individuals of oulation, and evolves these control arameters at individual level [9]. F and CR are udated in each generation according to some heuristic rules. In their later work [5], an imroved version of jde, namely jde-2, has also been roosed by imorting the mutation strategies self-adatation from (Qin et al.). In some resects, Sa is the inheritor of and, while it is different from other self-adative DE algorithms in two major asects: ) By mixing the search biases of both Gaussian and Cauchy oerators, Sa considers a trade-off between small and large search ste sizes; 2) Sa self-adats all its control arameters according to statistical learning exerience during evolution, rather than other heuristic udating rules. We have comared the erformance of Sa with these latest s, but the lack of sace revents showing the results of those exeriments. In general, Sa achieved comarable results to that of the other methods. ACKNOWLEDGMENT This work is artially suorted by the National Natural Science Foundation of China (Grant No ), the Fund for Foreign Scholars in University Research and Teaching Programs (Grant No. B733), and the Graduate Innovation Fund of University of Science and Technology of China (Grant No. KD2744). REFERENCES [] Z. Yang, J. He, and X. Yao, Making a Difference to Differential Evolution, in Advances in Metaheuristics for Hard Otimization, Z. Michalewicz and P. Siarry, Eds. Sringer, 28, [2] A. K. Qin and P. N. Suganthan, Self-adative differential evolution algorithm for numerical otimization, Proceedings of the 25 IEEE Congress on Evolutionary Comutation, vol. 2, , 25. [3] K. Price, R. Storn, and J. Laminen, Differential Evolution: A Practical Aroach to Global Otimization. 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