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一种求解连续域优化问题的混合蚁群算法,肖菁,李亮平,近期在群计算中对连续域优化的研究备受研究者的关注。本文提出了一种基于群体的增量学习方法与差分进化算法相结合的混合蚁群算法
国武技论文在线 http:/www.paper.edu.cn 90 Table 1 Notations in ACOPbILc yIluo Lion X=x1,x2…,x a set of n continuous variables in a certain problem h of the solu number of ants used in each iteration spccd of convergence q locality of the search process dimension of the problem arning rate N(.a) aussian function with mcan u and standard deviation o S solution in the solution archive coefficient in differential evolution Algorithm 1: HACO Algorithm ct iteration t=0: initialization of k solutions ( while t < max Numlterations do t=til. for i=l to m/2 for each dirnension i de sampling the valuc of dimension i by the gaussian kernel pdf by ranking for j-m/2+1 to m do for each dimension i de sampling the value of dimension i by the Gaussian pdf by Pbilc ch ant de chive) generate Gaussian functions() For an n-dimension problem, an ant constructs a solution in n steps at each step i. an ant samples a value for variable X. HAco keeps k solutions in the archive for pheromone update 95 calculation. The ith variable of th solution is denoted by Sij intialization _of k solutions( initially generate k solutions from the scratch by uniform sampling update solution archived means that the pheromone update is accomplished by adding the set of newly generated better solutions according to the ranking or pbilc method, then the same number of worst solutions in the archive is removed accordingly. In haco generate Gaussian functionso performs two kinds of 100 Gaussian generation. One of the ant groups applies the rank-based selection mechanism as in ACOR and the other applies the pbilc method which will be introduced later on. Then the two ant groups use the corresponding Gaussian functions to sample their values for each dimension For each dimension i, HACO generates a probability density function called Gaussian kernel 105 pdf. First, each solution constructs an individual gaussian pdf i(u; ai for each dimension. Then the gaussian kernel pdf is making up by the k separate Gaussian functions with different weights These weights are associated with the fitness value of the objective function. Better solution will have higher weight. The weight w of the solution Si is calculated by the following equation. 2a2k2 which defines the wcight to bc a valuc of the Gaussian function with argument 1, mcan 1.0, and standard deviation gk, where q is a parameter to balance bctwcen diversification and intensification. When q is small, the best-ranked solutions are strongly preferred, and when it is large, the probability becomes uniform Sampling the Gaussian kernel pdf is done in two phases. First, choose one of the individual 1 15 Gaussian function that compose thc gaussian kernel with probability pl given by cquation(2) 3 国武技论文在线 http:/www.paper.edu.cn Then sample the chosen Gaussian function We consider the chosen Gaussian function by(2) with ui a standard solution for other ant solutions to explore. To establish the value of the standard deviation i at step i, we calculate the 120 average distance from u; to all solutions in the archive, and multiply it by the parameter 5 k where the parameter 5>0 has a similar effect to the pheromone evaporation rate in traditional ACO. The higher the value of s, the lower the convergence speed of the algorithm Different from ACo which uses rank-based selection scheme only, HAcO increases the search diversity to alleviate the trap of local optima by incorporating the differential evolution de) operator in population-based incremental learning. In [14], empirical comparison of various DE shows that the best/1/bin (using the best solution to find scarch directions and binomial recombination is the most competitive approach regardless the characteristics of the problems 130 Accordingly we use the de of best1/bin in HACO. Al each iteration, u; is calculated from a linear, combination of the best solution and two other randomly chosen solutions in the solution archive random 1 random 2 where the F parameter scales the influence of the two randomly chosen solutions selected to calculate the mutation value. F is randomly valued from [0.3, 0.9]. Afer the u; is calculaled, the 135 standard deviation value oi is generated by formula(3)in the previous subsection In order to search the continuous domain more dynamically, we introduce population-based incremental learning to update both u; and di i=1, 2,...,nm 1+1 (1-a) =(1-a)o;+a;(6 140 Similar to thc rank-bascd pheromone update, the newly gencratcd bcttcr solutions by PBILc are added to the solution archive and the old worst solutions are removed accordingly It is noted that haco does not deal with the variable correlation and no local search heuristics is applied In this section, the performance of HACO in both low dimensions and high dimensions will be tested. All the simulations performed in this work were made on an Intel Pentium E2160 1 &GHz, IGB of RAM running Windows XP Professional SP2 and the algorithm is implemented in VC6.0. For low dimensions we compare with API, CAco and COACS. For high dimensions we compare with ACoR, CACS and MACACOLOJ l50 We use the 17 minimum problems in 8. The functions are listed in Table 2 in which fl, f2 f3, f5 and f6 are unimodal functions, f4 and f7 to f15 are multimodal functions with local optima f16 and fI7 are rotated multimodal functions. For more details of f10 to f17, please refer to 4 国武技论文在线 http:/www.paper.edu.cn Appendix B in [8]. The dimension for all test functions is n 4. The basic parameters for all the 155 functions showed in Table 3 are as the same as those used in [8. MAXEVALS is the maximum function evaluations in each trial. The value of error is used to measure whether a trial is successful. If the error between the result and the known optimum is smaller than or equal to ERROR, we regard the trial as successful The basic parameters for HACO are: the number of ants m-2, speed of convergence 5-0. 85 160 the learning rate a 0.1, k from (10, 20,50, 100 which can achieve the best results and the locality of the search process g for fl to f6 is o., for f7 to f1705 Each function is tested for 100 independent trials in this experiment. In Table 4, the mean and standard deviation obtained by each algorithm are listed with the best results outlined in bold. In Tablc 5, spccd comparisons arc madc. Thc column" cvals"stands for the avcragc function 165 evaluations in the trials for obtaining the final result, "succ evals " stands for the average function evaluations when the algorithm obtains results within predefined errors listed in Table 3,"succ stands for the number of successful trials among the 100 trials. The results of APl, CACO, COaC are obtained from [8] Compared with the other three algorithms, in most of the test functions, HACO can obtain the 170 best results. In the step function f5, all algorithms can obtain the optimum. In other unimodal functions f1.e2. 33. 6 and multimodal functions 19 l17. CoAC and haco can both obtain the optima. In f10 and fll CACO, COAC and haco achieve the best results. In multimodal functions f7 f12 with the accurate minimum values unknown CoAC and haco can reach the near optima, while in hacO the standard deviation is smaller In f8 and f16, APl achieves the best 175 mean values than other algorithms, and haco is the second best. In the Shekel functions f13 to f15. coac is the best while haco is a little worse. as summarized above. haco can solve both unimodal functions and multimodal functions effectively. For the functions with variable correlation f4, f16 and f17, HACo can also achieve the optima or near optima From Table 5, we can see that the average number of function evaluations in hacO is 180 smaller than other algorithms except for [2 and 116. For n2, HACO only spends a little more function evaluations than the smallest but with smallest"succ evals. For f16, CAco spends the Icast function evaluations but has only 25 succcssful trials, API and HACO can obtain thc 100 successful trials but HACO needs much less average evaluations than API does. The average function evaluations within the errors reflect the speed of the algorithm to find the right search 185 direction. HACO can find the right search direction faster in most of the test functions. For some multimodal functions such as f7, f8 and f]6. haco gets almost the same "evals"and" succ evals" Bccausc thesc functions havc many local optima, to cscapc from bcing trapped in a local optimum, the algorithm takes more evaluations to find new search direction, and when the right direction is found, the algorithm can converge to the global optimum faster. As analyzed above, HACO has a 190 main advantage in converge speed We also test the performance of HACO in high dimensions. The 6 test functions are listed in Table 6 and all functions are with dimension n=30 The parameters of the ACOR and HACO are both set as suggested in [12]: the number of 195 ants m-2, speed of convergence 5=0.85, locality of the search process q=10-4 and archive size k-50. The learning rate in HACO is a=0.9. While in CaCS and MACACO, the number of ants is 1000. The max function evaluations 6X106 in rosenbrock. and the other test functions are 106 国武技论文在线 http:/www.paper.edu.cn Table 7 shows the experimental results for 30 independent trials of HACO. The results of ACOR, CACS and MACACo are obtained from [10] 200 From Table 7, we can see that overall the covariance matrix based MaCaCO performs the best. Compared with ACOR, haco obtains better results in all of the six functions. HACO also outperforms Cacs in five of the six functions, just slightly worse on the Salomon function HACO outperforms MACACO in function of Rosenbrock. ACOR, CACS and HACO all perform worse in Schwefel since it is composed of a great number of peaks and valleys and the function 205 has a second best minimum far from the global minimum where search algorithms are easy to trap Since haco does not take the correlation between variables into account and does not employ any local search, we consider it is promising compared to the algorithms which do not utilize the covariance analysis 210 Table 2 Test function lists i Test functions Domain Optimum -100,100 f f2=∑。x|+ =∑ f4=∑100(x1-x,2)2+(x -100,10O 0 f=∑(x1+0.5) [-100,100y 1.28,1.28] 0 x,sin(√x1|) L-500,50-1675.93 [5.12,512]0 f=∑[x2-10c0s(2a)+10 [-32,32 f,=-20exp-021∑,x2)-expl/n∑”cos2x+20+e L-50.50 x/n{10sin2(3m1)+∑(y-1)+sin2(3y+1)+ (n-1)+sin2(2z)}+∑,(x251004) [-50,50] 0 f1=1/10i2(3x)+∑1(x1-1)[+n(3x1) (xn-1)[1+sin2(2xxn)]}+ /1(x,10,100,4) 11 3.075e-4 ∑[a1-x(2+bx)(b2+bx 「O,10 10.1532 [(x-a:)(x-a1)+c1 10.402 ∑【(x-a1)x-a)+cJ s=∑(x-a1)(x-a1)2+c;J 10.5364 「-5.12,5.12 0 fi=1 Iy2-10cos(2yv;)+10], 12=-20s-0V/n∑ expl/n L-32,32」 +20+e,yT=M c approximate value, the accurate minimum values of these functions are unknown 国武技论文在线 http:/www.paper.edu.cn Table 3 Basic Parameter Setting MAXEⅤALS ERROR l7000 .1 FffGfffffff 170000 0.1 170000 0.1 400000 l.0 170000 0.1 170000 0.1 900000 1.0 900000 0.1 900000 0.1 400000 0.1 f 400000 1000000 0.0001 170000 170000 0.1 900000 0.1 900000 0. Table 4 Accuracy Comparisons between APL, CACO, COAC and ACOPBILc API CACO COAC F Optimum ACOPBIl Intan Stddev clean stade Imean stddev Clean stade 0104e-3559c4877c-35409c-34 0 6.35e-31.70e-33.85e-212.36e-20 2B科66⑦ 0 2.34e-31.34e-3666e-231.49c-22 0 4.91e-1 2.54 7.35e-1 1.97 1.98e-15.44e-1 0 5.18c-14540c-142.89c-75 1675.93*14272111963-1675934.34e-12-1675934.32e-12 0 995c-3995c-2.99e-21.30e-1995c-39.95e-2 1.44e24.6le-3695e-16609e-16 f10 0 1.27e4742e-5 f1 1 0 2.67e-41.58e-4 f123.075-4*4.09e-3 67e-34.49e-41.16e-43.09e-48.73e-6 f13-10.1532*-8.736522828-783333.4229 -7.6810 f14-10.4029*934752.1210-944782.4887 -10.20701.1322 f15-10.5364*-941042.194 l0.2491 4180 10.2930 3909 f16 1.08 8.50e-12.06e-14.27e-14.97e2 2.18e-1 f17 141e240le-3660e-165.00e-16 215 Table 5 Speed Comparisons between API, CACO, COAC and ACOPBILc API CACO COAC ACOpⅡ evals Succ succ evals Succ evals succ evals succ evals evals evals evals succ f1887544202100169609215110044076915100 100 f2 567910016946015071009495396810091685 638565561571001698203705100524731061100 100 142534905788396399148141365883958084608697 95 f5364136411001537 1537100743 743 100 f6925: 156l10016773120310026498 100 100 f741758155476 69202 100264230158730100 6256100 f85073546057010050235 1373958499198 2830499 19428845820610022761626331005852210901100 f1019175659291001092241619100376975418100 100 fl20275936511001209241744100396371850100 100 f125857701054266295707128006444582348187214100 f13100652139397227904686068462097510100 f14934197129368045205353087492527339100 97 f15967411337679366 96457748798100 f1647749646333100 25298129268555806249658387100 f1747124581131002042402767100474099078100 100 国武技论文在线 http:/www.paper.edu.cn Table 6 Test function lists ll Function name roblem Optimum f(x)=∑ -10)0.10 0 Sphere Rosenbrock f(x)=∑ N100(X+1 +(x,-1) [-30,30 Rastrigin f(x)=10n+∑x2-10c0s(2x 5.12,5.12P Griewank f(x)=1+0.0025∑x;-II COS [-500,500] 0 Schwefel r; sinx 「-500,500 125695 Salomon f(x)=1-V2m1x)+0.1/厂2 -100,1001 220 approximate value, the accurate minimum values of these functions are unknown Table 7 Accuracy Comparisons between ACOR, CACS, MACACO and ACOPBILe ACoyddcv CACS MACACO ACO F BILc mcan std mcan stacy mcan stady mcan stddevy ere Rosenbrock 29.93 35.14 1.23 191 48 11.68 Rastrigin 101.6521.0 57.17 14.14 47.39 9.81 griewank 0.09 0.18 0.02 0.02 0.01 0.03 Schwcfcl-8703.26721.53-8934.57633.78 9658.48 salomon 3.05 1.43 0.05 0.16 Table 8 Accuracy Comparisons between ACOPBILc with best/1/bin and ACOPBILc with rand/2/dir aCOpou with best/l bin F ACOpDIL with rand/2/dir Mcan stddev mcan stddev Sphere Rastrigin 47.39 981 43.15 10.80 Schwcfcl -9658.48 735.74 -5767.35 1776.98 Salomon 0.48 0.16 0.36 0.22 225 In [14, the authors conduct empirical comparison of some de variants to solve global optimization problems. Through a set of statistical tests they draw the conclusion that the most competitive approach, regardless the characteristics of the problem to be solved is the best/1/bin 230 which we employed in HACO. And the rand/2/dir which includes fitness function to the mutation and recombination operators provides competitive results for multimodal and nonseparable functions, We also do comparison experiments in some selected functions to see how the two different de variants affect HACO. Table 8 gives the accuracy comparison between HACO with best/1 bin and HACO with rand/2/dir. Fig. I depicts the convergent curves comparing the 235 scalability of hAco with bcst/1/bin and HACO rand /2/dir From Table 8. we can see that the standard deviations of haco with best/l/bin are smaller than those of HAcO with rand/2/dir in all cases. It indicates that the best/1/bin operator is more stable in this algorithm. For unimodal function Sphere and multimodal functions of Rastrigin and Salomon, HACO with rand /2/dir performs the same or slightly better than HACo with best/l/bin 240 docs. But for Schwcfcl function the rand, 2/dir is much worse the best/1/bin fig. 1 shows that for unimodal function of Sphere, HACO with best/1/bin converges faster than HACO with rand/2/dir does. For multimodal functions of Rastrigin and Salomon. the lwo variants perform similarly with slight faster convergence in the late generations of HAco with rand/2/dir. HACO with best/1/bin converges much faster than HACO with rand/2/dir does in function of SchwefeL. Generally 8 国武技论文在线 http:/www.paper.edu.cn 245 speaking, we think the best/1/bin operator wins in HACO Fig. 1(a) Sph Fig. 1(b)Rastrigin 9 国武技论文在线 http:/www.paper.edu.cn 250 Fig. 1(c) Schwefel Fig. 1(d)Salomon Fig. I Convergent curvcs comparing the scalability of two de 10

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