Global optimization method with dual Lipschitz constant estimates for problems with non-convex constraints

Abstract

This paper considers the constrained global optimization problems, in which the functions are of the “black-box” type and satisfy the Lipschitz condition. The algorithms for solving the problems of this class require the use of adequate estimates of the a priori unknown Lipschitz constants for the problem functions. A novel approach presented in this paper is based on a simultaneous use of two estimates of the Lipschitz constant: an overestimated and an underestimated one. The upper estimate provides the global convergence, whereas the lower one reduces the number of trials necessary to find the global optimizer with the required accuracy. The considered algorithm for solving the constrained problems does not use the ideas of the penalty function method; each constraint of the problem is accounted for separately. The convergence conditions of the proposed algorithm are formulated in the corresponding theorem. The results of the numerical experiments on a series of multiextremal problems with non-convex constraints demonstrating the efficiency of the proposed scheme of dual Lipschitz constant estimates are presented.