Optimization by Monte Carlo Methods

Abstract

One category of global optimization problems is complicated by being combinatorial in nature, another is by having a large number of local optima. Monte Carlo methods are well-suited to attack such problems. Two such methods are simulated annealing and genetic algorithms. Both are inspired by the natural world. In simulated annealing, the objective plays the role of energy in a thermal process to be minimized. Points of the solution space are selected by a permutation of the last solution tried and a Metropolis acceptance discipline. An annealer is an instance of a time-varying Markov Chain and is not guaranteed to converge unless, by Hajek’s theorem, if an artificially supplied temperature is lowered slowly enough. In genetic algorithms, the objective plays the role of fitness of artificial organisms which are modeled as potential solutions. A “colony” of such organisms is evolved by a permutation process, called mutation, and a melding of two organisms called mating. A genetic algorithm is a regular Markov Chain and has an invariant distribution so that eventually all potential solutions will be tried. These methods are demonstrated by the application to several problems: the Traveling Sales Man problem, the permanent problem, and a function optimization problem.