On stability of fixed points of limit models of univariate marginal distribution algorithm and factorized distribution algorithm

Aims to study the advantages of using higher order statistics in estimation distribution of algorithms (EDAs). We study two EDAs with two-tournament selection for discrete optimization problems. One is the univariate marginal distribution algorithm (UMDA) using only first-order statistics and the other is the factorized distribution algorithm (FDA) using higher order statistics. We introduce the heuristic functions and the limit models of these two algorithms and analyze stability of these limit models. It is shown that the limit model of UMDA can be trapped at any local optimal solution for some initial probability models. However, degenerate probability density functions (pdfs) at some local optimal solutions are unstable in the limit model of FDA. In particular, the degenerate pdf at the global optimal solution is the unique asymptotically stable point in the limit model of FDA for the optimization of an additively decomposable function. Our results suggest that using higher order statistics could improve the chance of finding the global optimal solution.