Genetic crossover operator for partially separable functions

Partial separation is a mathematical technique that has been used in optimization for the last 15 years. On the other hand, genetic algorithms are widely used as global optimizers. This paper investigates how partial separability can be used in conjunction with GA. In the first part of this paper, a crossover operator designed to solve partially separable global optimization problems involving many variables is introduced. Then, a theoretical analysis is presented on a test case, along with practical experiments on fixed size populations, with different kinds of selection methods. Introduction This paper deals with minimization problems for which the function to minimize is the sum of many positive terms, each term only involving a subset of variables. Griewank and Toint gave first definition of such functions (i.e partially separable functions) about 15 years ago (Griewank and Toint 1982). Properties of partial separability are commonly used to solve local optimization problems. On the other hand, it is quite widely accepted that on real problems, when a crossover operator that makes sense can be defined, the efficiency of GA (compared to other stochastic optimization methods) is closely related to the use of the crossover operator: considering large size problems, the use of random crossover operator can penalize GAs performance. Classical crossover operators described in the literature (Goldberg 1989, Michalewicz 1992, Holland 1975) create two children from two parents chosen in the population. Initial operators on bit strings simply cut the two parents strings in two parts. The main drawback of these operators is that short schemes have a greater probability to survive than long ones. Consequently the encoding problem becomes very important. Multi-points crossovers and Gray codes are introduced to solve this problem. When using real variable coding, programmers very often use arithmetic crossover. However, none of these methods recognizes and favors good schemes. Heuristics are sometimes used to favor “good” crossovers and “good” mutations (Ravise et al. 1995) or try to reduce disruption of superior building blocks (Corcoran and Wainright 1996). We will show in this paper that taking advantage of the structure of partially separable functions to define a new crossover operator improves the converging speed and converging rate of a genetic algorithm. 1 Partial separability 1.1 Definition Partially separable problems considered in this paper have the followingcharacteristics: the function to minimize depends on variables , , .., ( large) and is a sum of positive functions involving only a subset of variables. Partially separable functions can be written: 1.2 Adapted crossover principle The crossover operator introduced in this article does not require any particular coding. The chromosome is directly coded with the variables of the problem (these variables may be real, integer,...). The intuitive idea is the following: for a completely separable problem, optimizing the global function can be done by optimizing each variable separately. This strategy is adapted to partially separable functions. When creating a child from two parents, the idea is to take for each variable the one that locally fits better (more or less , where controls the determinism of the operator). First, we define a local fitness for variable as follows: where is the set of such as is a a variable of and the number of variables of . Intuitively, the local fitness associated to a variable isolates its contribution to the global fitness1. Furthermore, it has the following good property: When minimizing , if: