Letter to the Editor—A Monte Carlo Method for the Approximate Solution of Certain Types of Constrained Optimization Problems

Martin Pincus
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Martin Pincus: Polytechnic Institute of Brooklyn, Brooklyn, New York

Operations Research, 1970, vol. 18, issue 6, 1225-1228

Abstract: This paper considers the problem of minimizing a function F ( x 1 , …, x n ) over a closed, bounded region S in n -dimensional space under the assumption that there exists a unique minimizing point ( z 1 , …, z n )ϵ S . In a previous paper I represented the coordinates of the minimizing point as the limit of a ratio of integrals. The same type of ratio appears, in a different context, in statistical mechanics where a Monte Carlo method has been developed, by Metropolis et al., for its numerical evaluation. The purpose of this paper is to point out the connection of Metropolis's method with the above type of minimisation problem. The idea of the method is to associate with the minimization problem a Markov chain whose sample averages converge with probability one to (approximately) the minimizing point ( z 1 , …, z n ). The Markov chain should be easily realizable on a computer. An estimate of the error from sampling over a finite time period is given.

Date: 1970
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