Genetic Algorithm Difficulty and the Modality of Fitness Landscapes

Abstract

We assume that the modality (i.e., number of local optima) of a fitness landscape is related to the difficulty of finding the best point on that landscape by evolutionary computation (e.g., hillclimbers and genetic algorithms (GAs)). We first examine the limits of modality by constructing a unimodal function and a maximally multimodal function. At such extremes our intuition breaks down. A fitness landscape consisting entirely of a single hill leading to the global optimum proves to be harder for hillclimbers than GAs. A provably maximally multimodal function, in which half the points in the search space are local optima, can be easier than the unimodal, single hill problem for both hillclimbers and GAs. Exploring the more realistic intermediate range between the extremes of modality, we construct local optima with varying degrees of “attraction” to our evolutionary algorithms. Most work on optima and their basins of attraction has focused on hills and hillclimbers, while some research has explored attraction for the GA's crossover operator. We extend the latter results by defining and implementing maximal partial deception in problems with k arbitrarily placed global optima. This allows us to create functions, such as the minimum distance function fmdG, with k isolated global optima and multiple local optima attractive to both crossover and hillclimbers. The function fmdG seems to be a powerful new tool for generalizing deception and relating hillclimbers (and Hamming space) to GAs and crossover.