Original paper
Soft Computing 7 (2003) 506–515 Ó Springer-Verlag 2003 DOI 10.1007/s00500-002-0235-1
A novel approach in parameter adaptation and diversity maintenance for genetic algorithms Y.-Y. Wong, K.-H. Lee, K.-S. Leung, C.-W. Ho
506 Abstract In this paper, we propose a probabilistic ruledriven adaptive model (PRAM) for parameter adaptation and a repelling approach for diversity maintenance in genetic algorithms. PRAM uses three parameter values and a set of greedy rules to adapt the value of the control parameters automatically. The repelling algorithm is proposed to maintain the population diversity. It modifies the fitness value to increase the survival opportunity of chromosomes with rare alleles. The computation overheads of repelling are reduced by the lazy repelling algorithm, which decreases the frequency of the diversity fitness evaluations. From experiments with commonly used benchmark functions, it is found that the PRAM and repelling techniques outperform other approaches on both solution quality and efficiency. Keywords Adaptive genetic algorithm, Diversity control, Rule-driven adaptive model
chromosome with probability pm . The parameters pc and pm are known as the crossover rate and mutation rate respectively. In order to optimize the efficiency and effectiveness, GAs must maintain a balance between the exploitation of beneficial aspects of existing solutions (by crossover) in order to improve them, and the exploration of the solution space (by mutation) so as to increase the probability of finding the optimal solution. This balance is determined by the crossover rate and the mutation rate. In this paper, we propose a probabilistic rule-driven adaptive model (PRAM) [16] to adapt these parameters automatically. PRAM uses three different parameter values on the population for the adaptation. These values will be adjusted automatically during the search by a set of rules according to the fitness improvement gained. Maintaining the population diversity helps to prevent premature convergence and maintain multiple solutions in a GA. In this paper, a simple repelling algorithm is proposed for diversity maintenance. The repelling algorithm samples a population to a ‘‘representative’’. Population diversity is maintained by driving the population away from the representative. The fitness of a chromosome is modified to include the ‘‘diversity fitness’’, which is inversely proportional to the similarity between the chromosome and the representative. A lazy repelling algorithm is proposed to further reduce the computational overheads. The diversity fitness of a chromosome is evaluated only if the population has been changed significantly. The rest of this paper is organized as follows. Section 2 presents the PRAM model, which automatically adapts the crossover rate and the mutation rate of a GA. Section 3 introduces the repelling algorithm and the lazy repelling algorithm to maintain the population diversity. In Sect. 4, we evaluate the performance of a GA that combines the techniques of PRAM and lazy repelling. Finally, this paper is concluded in Sect. 5.
1 Introduction In general, solving a global optimization problem, like resource allocation, pattern recognition, and machine learning, involves searching on a space of all possible solutions. Classical approaches like exhaustive searching or heuristic methods perform well for a small search space, whereas Genetic Algorithms (GAs) [17, 7] outperform these approaches for a large and complex search space. GAs are stochastic search algorithms modeling genetic inheritance and Darwinian strife for survival. It can explore various kinds of huge, varied search spaces effectively and robustly. GAs optimize a given objective function by searching with a population of candidate solutions (chromosomes) encoded by strings of genes. In each generation, genetic operations like selection, crossover, and mutation are operated to evolve the population. Crossover is performed with probability pc between two selected chromosomes by exchanging parts of their encoding to produce offspring. Mutation operates by inverting the value of each gene in a 2 Adaptations on the crossover rate and mutation rate Crossover and mutation are two important operations in Published online: 14 July 2003 GAs. Tuning the control parameters pc and pm to maintain Y.-Y. Wong (&) a balance between exploitation and exploration has direct Computing Services Centre, City University of Hong Kong, impact on the effectiveness and the efficiency of the search. Tat Chee Ave., Kln, Hong Kong Too much exploitation will lead to premature converE-mail:
[email protected] gence, whereas too much exploration will degenerate to a random search. Traditionally, users need to tune these K.-H. Lee, K.-S. Leung, C.-W. Ho control parameters manually for each problem, which is Department of Computer Science and Engineering, time-consuming and error prone. Moreover, the optimal Chinese University of Hong Kong, Shatin, Hong Kong
parameter values may be changing during the progress of the GA. In order to avoid these problems, we propose a novel probabilistic rule-driven model to adapt the control parameters automatically during the search.
2.1 Related work in parameter adaptation Methods of adaptation on control parameters can be classified into four approaches [14]. The static approach uses a fixed parameter value throughout the evolution. Researchers have proposed different sets of parameter values that promise good performance on a range of problems. For examples, De Jong [3] proposed to set pm ¼ 0:001 and pc ¼ 0:6. Formulas pm ¼ 0:01 and pm ¼ 1=b are recommended by Grefenstette [11] and Mu¨hlenbein [21], respectively. Ba¨ck [2] proposed the formula pm ¼ 1:75=ðn b1=2 Þ based on the result of Schaffer et al. [23]. In the above formulas, n represents the population size and b represents the bit-length of a chromosome. The dynamic deterministic approach alters the parameter values based on some deterministic rules without using feedback. For example, Hinterding and Michalewicz [14] suggested the equation pm ¼ 0:1 0:09ðg=GÞ, where g is the generation number from 1 to G. In the dynamic self-adaptive approach, parameters are encoded in the chromosomes and evolved together with the chromosomes. For example, Ba¨ck [2] and Smith and Fogarty [26] used this approach to adapt the mutation rate. Hinterding [13] applied this approach to alter the mutation strength of Gaussian mutation. Finally, the dynamic adaptive approach modifies the parameter values based on the feedback from the GA. For example, the adaptive genetic algorithm (AGA) [27] adapts pc and pm simultaneously. When the population tends to converge to a local optimum, pc and pm are increased to perform random jumps. When the population is scattered in the solution space, pc and pm are decreased. The strength of Gaussian mutation and the population size are adapted concurrently in the self-adaptive genetic algorithm (SAGA) [15].
2.2 The probabilistic rule-driven adaptive model PRAM [16] adapts the crossover rate, the mutation rate, or both, during the search to determine appropriate balance between exploration and exploitation. Taking a similar approach as SAGA [15], PRAM uses three parameter values (v0 ; v1 ; v2 with v0
