Fourier Niching Approach for Multi-modal Optimization 2 Yan Pei Hideyuki Takagi 2 Graduate School of Design, Kyushu University 2 2 Faculty of Design,

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1 九州大学学術情報リポジトリ Kyushu University Institutional Repository 多峰性最適化のためのフーリエ ニッチ法 裴, 岩九州大学大学院芸術工学府 高木, 英行九州大学大学院芸術工学研究院 Pei, Yan Graduate School of Design, Kyushu University Takagi, Hideyuki Faculty of Design, Kyushu University 出版情報 : 第 2 回進化計算学会研究会第 8 回進化計算フロンティア研究会合同研究会, pp.89-9, 進化計算学会バージョン :accepted 権利関係 :

2 Fourier Niching Approach for Multi-modal Optimization 2 Yan Pei Hideyuki Takagi 2 Graduate School of Design, Kyushu University 2 2 Faculty of Design, Kyushu University Abstract: We propose a niching method for solving multi-modal optimization problems from their frequency information and evaluate its performance with si multi-modal benchmark functions. We use Fourier transform as the main analysis tool to obtain the frequency information and estimate the potential modal locations of the fitness landscape using the obtained frequency and phase information. We choose the estimated potential modal locations as elite points, directly put them into the population, and find more peaks within local search areas restricted by given search radiuses. Eperimental evaluation results show that our proposed Fourier niching method has the better performance, especially from the computational cost point of view. [] F shared (i) = SH(d ij ) = F original (i) N j=i SH(d ij) { ( dij σ share ) α if d ij > σ share () otherwise F shared (i) F original (i) SH(d ij ) 2 d ij σ share peiyan@ieee.org [2] [] 2 [3] [4] [6] clearing [7] GA [8] 2

3 3 DE/best//bin 6 4 Amplitude Frequency content of Test Function Frequency (Hz) (a) Amplitude Frequency content of Test Function Frequency (Hz) (b) 2 2. DFT [9] global dimension [9] FFT FFT X(k) = N 2 [(n) + ( ) k (n + N 2 n= )]W nk N W N kn N, ( k N ) (n), X(k), n, N DFT 2 F F2 F F Test Function = e 2kπ.9.8 Test Function 2 2: F F2 -D f() = a + n (a i cos(ω i ) + c i sin(ω i ) + b i ) i= a p cos(ω p ) + c p sin(ω p ) a 2 p + c 2 p cos(ω p + b p ) b p = tan (a p /c p ) cos() T i = /F i (T i /2) + b i ( ) k i + b i 2 F i Range i i k i Rangei k i = i N F i ma(k, k 2,..., k n ) N n i= k i (a) (b) : F F2 2.2 F i b i A i [9]

4 2 = Range i m k i (2) m m = F F2 F3 F6 2 Shekel Foholes Rastrigin Schwefe Griewank 3 F F2 F3 F6 2, 2, 49, 9 F F2 F3 F6, 2 σ share =.3 F, F2, F4 26 F3 F6 64 F 8 2 (N P eak ) 2 4 2: DE F F2 ms Func. Method Time N P eak F none 4.9 F Sharing F Fourier 7.83 F Sharing+Fourier 7.46 F2 none 8. F2 Sharing.33 2 F2 Fourier 6.24 F2 Sharing+Fourier 6.2 F3 F F4 F F F2 2. DE F3 F6 3: Shekel Foholes Rastrigin Schwefe Griewank DE/best//bin 3. F F2 2 F3 F6 4. F3 F6

5 : Range, n, C M, U, N, S No. Name Test function Range n C F Niching f() = sin 6 (π) [,] MN. 2(ln2)( ) F2 Niching2 f() = e 2.8 sin 6 (π(.7.)) [,] MN F3 Shekel s Foholes f() = [ j= j+ 2 ] [ 6.36, 6.36] 2 MS i= ( i a ij ) 6 F4 Rastrigin f() = (n) + n i= (2 i cos(2π i )) [-.2,.2] 2 MS F Schwefel 2.26 f() = n i= ( i sin( i )) [-2,2] 2 MS F6 Griewank f() = + n 2 i i= n 4 i= cos( i i ) [-2,2] 2 MN F F2 F3 F6 [9] FFT DFT f = f s N, f 2 f s N DFT FFT N 2 8, 6, 32, 64, 28, 26 F Schwefel 2.26 Contour with Fourier Niching DE by 8 sampling Schwefel 2.26 Contour with Fourier Niching DE by 6 sampling 4 (F with 8 sampling) (F with 6 sampling) Schwefel 2.26 Contour with Fourier Niching DE by 32 sampling Schwefel 2.26 Contour with Fourier Niching DE by 64 sampling (F with 32 sampling) (F with 64 sampling) 2 3 Schwefel 2.26 Contour with Fourier Niching DE by 28 sampling 4 Schwefel 2.26 Contour with Fourier Niching DE by 26 sampling 4 (F with 28 sampling) (F with 26 sampling) : F

6 Normal DE Sharing Niching Method Niching Method Based on Fourier Analysis Sharing Niching Method Based on Fourier Analysis (F Normal) (F Sharing) (F Fourier) (F Fourier+Sharing) Normal DE Sharing Niching Method Niching Method Based on Fourier Analysis Sharing Niching Method Based on Fourier Analysis (F2 Normal) (F2 Sharing) (F2 Fourier) (F2 Fourier+Sharing) Shekel Foholes Contour with Normal DE Shekel Foholes Contour with Sharing DE Shekel Foholes Contour with Fourier Niching DE Shekel Foholes Contour with Fourier Niching + Sharing DE (F3 Normal) (F3 Sharing) (F3 Fourier) (F3 Fourier+Sharing) Rastrigin Contour with Normal DE Rastrigin Contour with Sharing DE Rastrigin Contour with Fourier Niching + Sharing DE (F4 Normal) (F4 Sharing) (F4 Fourier) (F4 Fourier+Sharing) Schwefel 2.26 Contour with Normal DE Schwefel 2.26 Contour with Sharing DE Schwefel 2.26 Contour with Fourier Niching DE Schwefel 2.26 Contour with Fourier Niching + Sharing DE (F Normal) (F Sharing) (F Fourier) (F Fourier+Sharing) 2 Griewank Contour with Normal DE Griewank Contour with Sharing DE Griewank Contour with Fourier Niching DE Griewank Contour with Fourier Niching + Sharing DE (F6 Normal) (F6 Sharing) (F6 Fourier) (F6 Fourier+Sharing) 4: F F2, F3 F6

7 category [] D. Goldberg and J. Richardson, Genetic algorithms with sharing for multi-modal function optimization, 2nd Int. Conf. on Genetic Algorithms (ICGA987), pages 44 49, Cambridge, MA, USA, July, 987. [2] X. Yin and N. Germay, Investigations on solving load flow problems by genetic algorithms, Electric Power System Research, vol.22, no.3, pages 63, Elsevier, 99. [3] Samir W. Mahfoud, Crowding and preselection revisited, Parallel Problem Solving from Nature (PPSN992), vol. 2, pages 27 37, Brussels, Belgium, Sept., 992. [4] O. Mengsheol and D. Goldberg, Probabilistic crowding: Deterministic crowding with probabilistic replacement, 999 Genetic and Evolutionary Computation Conference (GECCO999), pages 49 46, Orlando, FL, USA, July, 999. [] Kenneth A. De. Jong, An analysis of the behavior of a class of genetic adaptive systems, Technical report, University Microfilms No , Doctoral dissertation, University of Michigan, 97. [6] Georges Harick, Finding multi-modal solutions using restricted tournament selection, 6th Int. Conf. on Genetic Algorithms (ICGA99), pages 24 3, Pittsburgh, PA, USA, July, 997. [7] A. Pétrowski, A clearing procedure as a niching method for genetic algorithms, 3rd IEEE Int. Conf. on Evolutionary Computation (ICEC996), pages Nagoya, Japan, May, 996. [8] J. P. Li, Marton. E. Balazs., T. P. Geoffrey, and P. John Clarkson, A species conserving genetic algorithm for multi-modal function optimization, Evolutionary Computation, vol., no.3, pages , MIT Press, 22.

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