Reproduction Strategy Based on Self-Organizing Map for Real-coded Genetic Algorithms

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1 Neural Information Processing - Letters and Reviews Vol. 5, No. 2, November 2004 LETTER Reproduction Strategy Based on Self-Organizing Map for Real-coded Genetic Algorithms Ryosuke Kubota Graduate School of Computer Science and Systems Engineering Kyushu Institute of Technology Kawazu, Iizuka-shi, Fukuoka, , Japan kubota-ryosuke@edu.brain.kyutech.ac.jp Takeshi Yamakawa and Keiichi Horio Graduate School of Life Science and Systems Engineering Kyushu Institute of Technology 2-4 Hibikino, Wakamatsu-ku, Kitakyushu-shi, Fukuoka, , Japan yamakawa@brain.kyutech.ac.jp, horio@brain.kyutech.ac.jp (Submitted on August 17, 2004) Abstract - A novel reproduction strategy employing a Self-Organizing Map (SOM) for Realcoded genetic algorithms (RcGA) is proposed to preserve genetic diversity of population. In the proposed reproduction, a set of new chromosomes is generated by the learning of the SOM with modified updating equation. The continuous update facilitates the preservation of genetic diversity. Keywords - Real-coded genetic algorithm, self-organizing map, reproduction, genetic diversity, fitness 1. Introduction A Real-coded genetic algorithm (RcGA) [1][2] is a modified genetic algorithm (GA) [3] employing realvalued vectors for representation of the chromosomes, and is widely-applied to many optimization problems [4]. The RcGA generally employs a reproduction strategy based on roulette wheel selection (RWS). However, this strategy may lose genetic diversity of population in an early stage [5], because it can not generate new chromosomes which are different from present chromosomes. In this paper, we propose a novel reproduction strategy employing an idea of Self-Organizing Map (SOM) [6][7] to cope with this problem. In the proposed reproduction, a set of new chromosomes of next generation is generated by learning of the SOM. The updating equation used in learning of the SOM is modified by adding new coefficients with respect to fitness values of chromosomes. The continuous update facilitates the preservation of genetic diversity and the effective search. The effectiveness of the proposed reproduction is verified by applying it to benchmark optimization problems of DeJong. 2. Real-coded Genetic Algorithm The procedure of the RcGA is summarized as follows. 0: All chromosomes are initialized using random values. 27

2 Reproduction Strategy based on SOM for Real-coded GAs R. Kubota, T. Yamakawa, and K. Horio x 2 x 2 x 2 (a) x 1 (b) x 1 (c) x 1 Figure 1. Reproduction employing RWS and desirable reproduction. (a) Present population. (b) Population after RWS. (c) Population after desirable reproduction. Each axis and dashed circles represent the element of chromosomes and contours of fitness value, respectively. Area in the circle with small radius means that fitness values is high. x i X = ( x 1,,,, x M ) 1 i M Input Layer W 1 1 j -2 j -1 j j +1 j +2 N Competitive Layer Figure 2. Construction of SOM. 1: New chromosomes (offsprings) are produced from the randomly-selected parents using crossover and mutation strategies. 2: Chromosomes of next generation are selected from those of present generation using the RWS. 3: 1 to 2 are repeated until the given conditions are satisfied. In the reproduction strategy based on RWS, genetic diversity of population may be lost in early stage, because the chromosomes of the next generation are generated by copying those of the present generation. In other words, the RWS can not generate new chromosomes which are different from the present chromosomes. Fig.1(a) shows present population before RWS. Fig.1(b) shows population after RWS and decrease of genetic diversity. The chromosomes are searching points in the present generation, and they are also base points of crossover in the next generation. Thus, decrease of genetic diversity leads to ineffectiveness of the search. The mutation with high probability can preserve the decrease of genetic diversity. However, this method is not effective, because the fitness values of new chromosomes which are produced by mutation should not be high. To achieve effective search, The reproduction based on the distribution of high fitness values in Fig.1(c) is desirable. 3. Reproduction Strategy using SOM In this paper, to cope with this problem, we propose the novel reproduction strategy employing the idea of the SOM. Fig.2 shows construction of SOM. The SOM consists of the input and the competitive layers that include M 28

3 Neural Information Processing - Letters and Reviews Vol. 5, No. 2, November 2004 and N units, respectively. The j-th unit in the competitive layer is connected to all units in the input layer by the weight vector w j = [w j1,, w ji,, w jm ]. In the learning, the weight vector is continuously updated toward the input vector x = [x 1,, x i,, x M ] by: x j (t + 1) = w j (t) + α(t)(x w j (t)), (1) where t represents a learning step, and w j (t + 1) and w j (t) are the weight vectors after and before updating, respectively. α(t) is learning ratio. The SOM achieves the detailed approximation of the probability density of the input distribution by continuous updating the weight vector. In the proposed strategy, the chromosomes of the present generation are used as input vectors of SOM, and weight vectors after learning are employed as the chromosomes of the next generation. The procedure of the proposed strategy employing the SOM is summarized as follows. 0: Weight vectors are initialized by learning of an ordinary SOM, in which randomly generated vectors are used as input vectors. 1: New chromosomes are produced from the randomly-selected parents by using the crossover and mutation strategies. 2: One chromosome is selected from the present population, and it is applied to the input layer. 3: The Euclidean distance x w j is calculated, and the unit which has the minimum distance is defined as the winner unit. 4: The weight vectors of all units in the competitive layer are updated by: w j (t + 1) = w j (t) + fx h(fx, d j ) (1 fw j(t)) (x w j (t)), (2) where fx and fw j (t) are the fitness values of chromosome and those of weight vector, respectively. Fitness value is normalized from 0 to 1. d j is distance between the j-th unit and the winner unit in the competitive layer. h(fx, d j ) is a coefficient and is represented by: ( 2 ) dj h(fx, d j ) = exp 2fx 2. (3) In the Eq.(2), the weight vector is attracted to the chromosome, when fx is large and fw j(t) is small. Adversely, the weight vector is not attracted to the chromosomes, when fx is small and fw j (t) is large. The Eq.(3) means that a lot of weight vectors are attracted to the chromosome, when fx is large. The Eq.(2) facilitates that the weight vectors with row fitness values are attracted to the chromosomes with high fitness values. 5: 2 to 4 are repeated until that all inputs are selected. 6: 1 to 5 are repeated until the given conditions are satisfied. The continuous update by using Eq.(2) can generate new chromosomes which are different from the present chromosomes based on the distribution of the fitness values. This reproduction can preserve the genetic diversity and achieve the effective search. 4. Simulation Results In order to verify the effectiveness of proposed strategy, the following two optimization problems are achieved. 29

4 Reproduction Strategy based on SOM for Real-coded GAs R. Kubota, T. Yamakawa, and K. Horio x i X = ( x 1,,,, x M ) 1 i M Input Layer W 1 1 j -2 j -1 j j +1 j +2 N Competitive Layer Figure 3. Construction of SOM. Table 1. Simulation Results. Problem 1 Problem 2 # of generations CPU Time Proposed sec. RWS sec. Proposed sec. RWS sec. 30 Problem 1: Minimize kx 4 k + GAUSS(0, 1), 1.27 x k k=1 25 Problem 2: Minimize l=1 1, x k (x k a kl ) 6 k=1 The parameters of RcGA are: population size and the number of the unit in the competitive layer N = 100, crossover probability P c = 0.3, mutation probability P m = BLX-α crossover [2] and uniform mutation are used in crossover and mutation strategy, respectively. a kl is decided using random value. Table 1 shows simulation results. Ten runs are executed and the average value is calculated to suppress dependencies on all stochastic effects. The proposed strategy achieves faster search than the RWS with respect to the number of generations and CPU time. These results show that the RcGA employing the proposed strategy generates many kinds of chromosomes with high fitness values by preserving the genetic diversity of the population and achieves the effective search. 5. Conclusions In this paper, we proposed the reproduction strategy using the SOM for RcGA to preserve the genetic diversity of the population. The novel continuous updating based on the fitness of the chromosomes can achieve the preserving the genetic diversity and the searching the optimum solution, and overcome the drawbacks of the ordinary RcGA based on the RWS. The results of the test problems showed the effectiveness of the proposed algorithm. Acknowledgements: This work was partially supported by the 21th Century COE (Center of Excellence) Program in Kyushu Institute of Technology, entitled World of brain computing interwoven out of animals and robots, and was also supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (A), 2002,

5 Neural Information Processing - Letters and Reviews Vol. 5, No. 2, November 2004 References [1] L. Davis, The Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York, [2] L. J. Eshleman and J. D. Schaffer, Real-Coded Genetic Algorithms and Interval-Schemata, Foundations of Genetic Algorithms 2, pp , [3] D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Publishing Company Inc., [4] I. Ono, S. Kobayashi and K. Yoshida, Global and Multi-objective Optimization for Lens Design by Real-coded Genetic Algorithms, SPIE Proc. Vol. 3482, International Optical Design Conference 1998, pp , [5] I. Ono and S. Kobayashi, A Real-coded genetic Algorithm for Function Optimization Using Unimodal Normal Distribution Crossover, Proc. 7th International Conference on Genetic Algorithms, pp , [6] T. Kohonen, Self-Organized formation of topologically correct feature maps, Biological Cybernetics, Vol. 43, pp , [7] T. Kohonen, Self-Organizing Maps, Springer-Verlag, Ryosuke Kubota was born in He received the B.E. and the M.E. degrees in control engineering and science from Kyushu Institute of Technology, Japan, in 2000 and 2002, respectively. He is presently a Ph.D. candidate at Kyushu Institute of Technology, Japan. His current research interests include machine learning, evolutional computing and bioinformatics. Takeshi Yamakawa was born in He received the B.E. degree in electronics engineering in 1969 from Kyushu Institute of Technology and the M.E. degree in electronics engineering in 1971 from Tohoku University, both in Japan. He received the Ph.D. degree for his studies on electrochemical devices in 1974 from Tohoku University, Japan. He was an Associate Professor at Kumamoto University. He joined the faculty of Computer Science and Systems Engineering, Kyushu Institute of Technology (KIT), Iizuka, Japan in April 1989 as a full professor. He is a professor of the Graduate School of Life Science and Systems Engineering, KIT since He is also the chairman of Fuzzy Logic Systems Institute (FLSI), the foundation established by himself in His main research interest lies on hardware implementations of fuzzy systems, fuzzy neural networks, and chaotic systems, and also on the micromachining and the micro-electrophoresis device. He holds 11 patents in U.S.A., 4 patents in Europe, 1 patent in Australia and 1 patent in Taiwan, and he has also applied for more than 80 patents in Japan. He received the International MOISIL Prize and the Gold Medal in the Fuzzy Systems Engineering in 1994, Distinguished Service Prize from Biomedical Fuzzy Systems Association in 1998, Doctorate Honoris Causa from University of Pitesti, Romania in 1999, Lifetime Achievement Award from WAC in 2000, Gabor Award from IJCNN 2001, Fellowship Award from Japan Society for Fuzzy Theory and Intelligent Informatics in Dr. Yamakawa is a member of IFSA and other 9 academic institutes. He is acting as a member of editorial board and a regional editor of 19 international professional journals. He contributes more than 50 international conferences as an organizer or a member of organizing/programming committee. 31

6 Reproduction Strategy based on SOM for Real-coded GAs R. Kubota, T. Yamakawa, and K. Horio Keiichi Horio was born in He received the B.E., the M.E. and the Ph.D. degrees from Kyushu Institute of Technology (KIT), Japan in 1996, 1998 and 2001, respectively. He is an assistant professor of the Graduate School of Life Science and Systems Engineering, KIT, Japan since He was a Research Fellow of Japan Society for Promotion of Science at KIT. He is a member of IEICE and Japan Society for Fuzzy Theory and Intelligent Informatics (SOFT) 32