Application of the Monte‐Carlo Method to Nonlinear Stochastic Optimization with Linear Constraints

Abstract

We consider a problem of nonlinear stochastic optimization with linear constraints. The method of e-feasible solution by series of Monte-Carlo estimators has been developed for solving this problem avoiding “jamming” or “zigzagging”. Our approach is distinguished by two peculiarities: the optimality of solution is tested in a statistical manner and the Monte-Carlo sample size is adjusted so as to decrease the total amount of Monte-Carlo trials and, at the same time, to guarantee the estimation of the objective function with an admissible accuracy. Under some general conditions we prove by the martingale approach that the proposed method converges a.s. to the stationary point of the problem solved. As a counterexample the maximization of the probability of portfolio desired return is given, too.