Mathematical Analysis of Optimization Methods Using Search Distributions

We show that UMDA transforms the discrete optimization problem f(x) into a continuous one deened by the average tness W (p). For proportionate selection, UMDA performs gradient ascent in this landscape. For functions with highly correlated variables UMDA has to be extended to an algorithm FDA which uses more complex search distributions. FDA also transforms the discrete optimization problem into a continuous one deened by W (p), where W (p) now depends on the factorization. The diierence between UMDA and FDA are discussed for a deceptive function. 1 Univariate Marginal Distribution Algorithm Let x = (x 1 ; : : : ; x n) denote a binary vector. We consider the optimization problem x opt = argmax f(x). Let p(x; t) denote the probability of x in a population of vectors at generation t. We denote by X i variable names, whereas x i is used for assignments. So p(X 1 =x 1) is the marginal probability of the rst variable having value x 1 and will be abbreviated to p(x 1) if the context allows. p(x i jx j) := p(X i =x i jX j =x j) = p(x i ; x j)=p(x j) denotes the conditional probability. We have shown that genetic algorithms using random-ized recombination/crossover can be approximated by an algorithm that keeps the population in linkage equilibrium 2]. This can be done by computing the uni-variate marginal frequencies from the selected points. This method is used by the Univariate Marginal Distribution Algorithm (UMDA). UMDA formally needs 2n parameters, the marginal distributions p(x i). We consider the average tness f(t) := P x f(x)p(x) as a function which depends on p(x i). To emphasize this dependency we write Theorem 1. 4] For innnite populations and proportionate selection the diierence equations for the gene frequencies used by UMDA are given by p i (t + 1) = p i (t) + p i (t)(1 ? p i (t)) @ ~ W @pi ~ W (t) (2) The above equation completely describes the dynamics of UMDA with proportionate selection. Mathematically UMDA performs gradient ascent in the landscape deened by ~ W. When the tness function has highly correlated variables , UMDA may not be able to optimize it 6]. In this case an algorithm that uses more complex probability distributions is needed. The FDA (Factorized Distribution Algorithm) 5] uses the theory of Bayesian networks to sample points with …