Distributed Probabilistic Model-Building Genetic Algorithm

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1 ,,,, GA PMBGA PCA PCA UNDX MGG Boundary Extension by Mirroring BEM Distributed Probabilistic Model-Building Genetic Algorithm Masaki SANO, Tomoyuki HIROYASU, Mitsunori MIKI, Hisashi SHIMOSAKA, and Shigeyoshi TSUTSUI Graduate School of Engineering, Doshisha University Department of Engineering, Doshisha University Department of Management Information, Hannan University In this paper, a new model of Probabilistic Model-Building Genetic Algorithms(s), Distributed PMBGA (), is proposed. In the, the correlation among the design variables is considered by Principal Component Analysis(PCA) when the offsprings are generated. The island model is also applied in the for maintaining the population diversity. Through the standard test functions, the effectiveness of the is examined. In this paper, some models of are examined. The where PCA is executed in the half of the islands and not executed in the other islands can find the good solutions in the problems whether or not the problems have the correlation among the design variables. From these results, it is clarified that the has higher searching ability than the UNDX with MGG. It is also discussed the treatment of the boundary condition of the design field using the Boundary Extension by Mirroring (BEM). 1 Genetic Algorithm : GA 1) populationselection crossover mutation genetic operator GA 2) Probabilistic Model-Building Genetic Algorithm : PMBGA 3) GA Real-Coded GA 1

2 Unimodal Normal Distribution Crossover : UNDX 4) UNDX 2 UNDX UNDX 5) Eshelman Blend Crossover : BLX- BLX- GA 6) Schwefel correlated mutation PM- BGA Distributed PMBGA : Principal Component Analysis : PCA 5) Population Select promissing individuals Generate new individuals, and Replace old individuals with new ones Estimation of distribution Construct probabilistic model Probabilistic Model 1: Probabilistic Model-Building GA 3 GA GA 7) GA 1 Tanese Distributed GA : DGA 8) DGA (migration) DGA 8) Island 2 3 PMBGA 4 5 Migration Individual 2 PMBGA GA PMBGA GA 2: Distributed Genetic Algorithm 4 PMBGA Distributed Probabilistic Model-Building Genetic Algorithm : DGA PM- BGA PCA 2

3 4.1 PMBGA 1 t 3 v2 x2 v1 y2 4.3 PCA S(t) PCA PCA S(t) PCA T (t) ( 4) T (t) T (t) PCA migration x1 reduction of corelation between design variables y1 generation 0 Island Archive for PCA Select better individuals Psub(0) T(0) generation 1 Island Replace Generate new individuals y2 y1 generation 2 Psub(1) Psub(2) T(1) T(2) 3: 4: PCA with the archive of the best individuals PCA P sub (t) R s S(t) S(t) S(t) T (t) D T (t) T nt (t) D S(t) D T (t) X ns(t) D T S D D S S = 1 ns(t) 1 T T T (1) S λ 1,λ 2,...,λ D v 1, v 2,...,v D D S(t) X V = [v 1, v 2,...,v D ] X V Y Y ns(t) D Y 3

4 selected individual x v2 distribution of archives for PCA x v1 reduce correlation y y 5: Reduction operation of correlation between design variables with PCA 4.4 np (t) Y n n Y Amp Y Y offs np (t) D 4.5 Y offs V X offs = Y offs V 1 (2) X offs P (t) P (t +1) 4.6 R mu 4.7 ne(t) E(t) E(t) P (t +1) 4.8 GA PCA Rastrigin Schwefel, Rosenbrock Ridge Griewank 5 0 Schwefel Rastrigin Schwefel Rosenbrock Ridge Griewank F Rastrigin =10n + F Schwefel = F Rosenbrock = n i=1 n ( x 2 i 10 cos(2πx i ) ) (3) i=1 ( 5.12 x i < 5.12) ) x i sin( xi C (4) (C : optimum.) ( 512 x i < 512) n ( 100(x1 x 2 i ) 2 +(1 x i ) 2) (5) i=2 F Ridge = F Griewank =1+ ( x i < 2.048) n ( i ) 2 x j (6) i=1 j=1 ( 64 x i < 64) n x 2 n i 4000 ( cos ( x i ) ) (7) i=1 i ( 512 x i < 512) i=1 5.2 PCA 4

5 PCA PCA : Parameters Population size 512 Number of elites 1 Number of islands 32 Migration rate Migration interval 5 Archive size for PCA 100 Sampling rate 0.25 Amp. of Variance 2 Mutation rate 0.1/ (Dim. of function) model 1 : PCA model 2 : PCA model 3 : PCA PCA model 1 4 model 2 PCA model 3 PCA GA Distributed Environment GA : DEGA DGA 9) DEGA GA DGA DEGA 2: Number of times that the threshold is reached model 1 model 2 model 3 Rastrigin Schwefel Rosenbrock Ridge Griewank Number of Evaluation 2.5x x x x x10 5 model 1 model 2 model Rastrigin Schwefel Rosenbrock Ridge Griewank 6: Average number of evaluations to reach the threshold Schwefel Rastrigin model 1 PCA Rosenbrock PCA model 2 Ridge model 2 PCA PCA PCA 5

6 model 3 Griewank model 1 model 2 model 3 model Rastrigin PCA PCA PCA Rastrigin PCA model 1 PCA Rastrigin Rosenbrock 7 20 Rastrigin Rosenbrock 8 20 erase/ Rastrigin Num. of Updated Individuals x x x x10 6 Rastrigin Num. of Updated Individuals x x x x10 6 Rosenbrock 7: History of number of updated individuals in archive of the best individuals erase/ x x10 5 Rastrigin erase/ x x10 5 Rosenbrock 8: History of averave of evaluation values in the model in which archive is erased each 10 generation Rosenbrock Rastrigin Rastrigin PCA UNDX MGG GA MGG Unimodal Normal Distribution Crossover : UNDX GA 4) UNDX UNDX Minimal Generation Gap MGG 10) 6

7 MGG 1 MGG α =0.5 β = model 3 Rastrigin Schwefel 5.5 GA Tsutsui Boundary Extension by Mirroring (BEM) 11) BEM extension rate r e (0.0 <r e < 1.0) GA BEM 5.2 model 3 3 3: Domain of objective functions Rosenbrock Ridge Function Optimal solution Domain Rastrigin 0.0 [0, 5.12] Schwefel [-512, 421] Rosenbrock 1.0 [-2.048, 1] Ridge 0.0 [0, 64] Griewank 9: History of average of evaluation values BEM 7

8 BEM BEM 1.0x x x x x10 6 Rastrigin (modified) BEM 1.0x x x x x10 6 Rosenbrock (modified) BEM 1.0x x10 5.0x10 Schwefel (modified) BEM 1.0x x10 5.0x10 Ridge (modified) 10: History of average of evaluation values on functions which have optimum at the edge of search space 6 GA DPM- BGA PCA PCA PCA DPMGA PCA GA UNDX MGG BEM BEM 1) D.E.Goldberg. Genetic Algorithms in Search Optimization and Machine Learnig. Addison- Wesley, ) Annie S. Wu, Robert K. Lindsay, and Rick L. Riolo. Emprical observation on the roles of crossover and mutation. Proc. 7th International Conference on Genetic Algorithms, pp , ) Martin Pelikan, David E. Goldberg, and Fernando Lobo. A Survey of Optimization by Building and Using Probabilistic Models. No , Sep ),,. UNDX GA., Vol. 14, No. 6, pp , ),.. 13, pp , ) Thomas Bäck, Frank Hoffmeister, and Hans-Paul Schwefel. A Survey of Evolution Strategies. Proc. 4th International Conference on Genetic Algorithms, pp. 2 9, ) Erick Cantú-Paz. A survey of parallel genetic algorithms. Calculateurs Paralleles, Vol. 10, No. 2, ) Reiko Tanese. Distributed Genetic Algorithms. Proc. 3rd International Conference on Genetic Algorithms, pp , ) M.Miki, T.Hiroyasu, M.Kaneko, and K.Hatanaka. A Parallel Genetic Algorithm with Distributed Environment Scheme. IEEE Proceedings of Systems, Man and Cybernetics Conference SMC 99, ),,.., Vol. 12, No. 5, pp , ) Shigeyoshi Tsutsui. Multi-parent Recombination in Genetic Algorithms with Search Space Boundary Extension by Mirroring. Proc. the 5th International Conference on Parallel Problem Solving from Nature (PPSN V), pp , Sep,

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