Multimodal Function Optimization Using Local Ruggedness Information

In multimodal function optimization, niching techniques create diversification within the population, thus encouraging heterogeneous convergence. The key to the effective diversification is to identify the similarity among individuals. Without knowledge of the fitness landscape, it is usually determined by uninformative assumptions. In this article, we propose a method to estimate the sharing distance for niching and the population size. Using the Probably Approximately Correct (PAC) learning theory and the -cover concept, we prove a PAC neighborhood of a local optimum exists for a given population size. The PAC neighbor distance is further derived. Within this neighborbood, we uniformly sample the fitness landscape and compute its subspace fitness distance correlation (FDC) coefficients. An algorithm for estimating the granularity feature is described. The sharing distance and the population size are determined when above procedure converges. Experiments demonstrate that by using the estimated population size and sharing distance an Evolutionary Algorithm (EA) can correctly identify multiple optima. Introduction Evolutionary Algorithms have been successful in solving single-optimum problems, such as pattern recognition (Dasgupta & Michalewicz 1997) and image processing (Yuan, Zhang, & Buckles 2002). When optimizing complex problem with many local optima, EAs suffer from premature or slow convergence. To overcome difficulties imposed by multiple local optima, hybrid EAs incorporating local search are developed. Such techniques include clustering (Törn 1978), stochastic approximation (Liang, Yao, & Newton 1999) and parallel local search (Guo & Yu 2003). Meanwhile, niching or speciation is a particular mechanism that allows and maintains several subpopulations so that each optimum can attract a number of them (Mahfoud 1995). Among niching strategies, sharing is an approach that divides the fitness of an individual by the number of “similar” individuals. Determining the similarity among individuals is nontrivial and is usually based on user assumptions. In our previous work (Zhang, Yuan, & Buckles 2003), we found that population size is problem-dependent and can Copyright c © 2004, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. be estimated by analyzing the ruggedness of a fitness landscape. With the information on population size, we are able to determine the sharing distance as well. This is, however, infeasible for a free-form function optimization problem without an impractical number of samples. Hence, we propose an approximating approach that employs the Fitness Distance Correlation coefficient (Jones & Forrest 1995) as a measure of local ruggedness. FDC has been developed as a measure of problem difficulty for Genetic Algorithms. Generally, one FDC coefficient is unable to uncover the variation in ruggedness of a fitness landscape. Moreover, computing FDC requires the knowledge of the global optimum, which is usually unknown. Based on PAC learning theory, we divide the search space into subspaces. A few samples are drawn from each subspace and the largest one is assumed to be the “local” optimum1. Subspace FDC coefficients are computed and used as a guide for further subgrouping. The granularity of a fitness landscape is obtained by iterating this procedure until it converges. The rest of this article is organized as follows. In the next section, we briefly discuss the concept of sharing distance in niching techniques and PAC learning theory. In Section “PAC Neighborhood Distance” we prove that based on the initial population a neighborhood distance exists. The analytical result is developed using PAC learning theory. In Section “Granularity of A Fitness Landscape”, we present the concept of subspace FDC as a measure for subspace granularity. An iterative algorithm for the estimation of the overall granularity is described next. The sharing distance and population size are estimated in the next section. In Section “Experiments and Discussions”, we demonstrate the results on 1-dimensional and 2-dimensional functions. This article is closed with conclusions. Background Sharing Method Sharing (Goldberg & Richardson 1987) is a popular and successful niching method. It attempts to maintain a diverse population with members distributed among niches in a multimodal fitness landscape. To diversify its population, it reduces the fitness of an individual within a neighborhood deWithout loss of generality, we assume the optimization is to find the maximum. fined by the sharing function. This rewards individuals that uniquely exploit regions of the fitness landscape by discouraging redundant solutions. The shared fitness is defined as foi(dij) = fi ∑m j=1 sh(dij) (1) where fi is the raw fitness of individual oi, m is the number of individuals in the population, and dij is the distance between the ith and the jth individuals. The sharing function sh(·) reaches a maximum of 1 at zero, decreases monotonically with distance, and falls to zero for distances that are greater than σsh. For example, the triangular sharing function shown in Figure 1 is given by sh(dij) = { 1 − dij σsh if dij < σsh 0 otherwise (2) where sh(dij) measures the amount of sharing or similarity between two individuals. The parameter σsh is vital to the