Polynomial bases on the numerical solution of the multivariate discrete moment problem

Abstract

The multivariate discrete moment problem (MDMP) has been introduced by Prekopa. The objective of the MDMP is to find the minimum and/or maximum of the expected value of a function of a random vector with a discrete finite support where the probability distribution is unknown, but some of the moments are given. The MDMP can be formulated as a linear programming problem, however, the coefficient matrix is very ill-conditioned. Hence, the LP problem usually cannot be solved in a regular way. In the univariate case Prekopa developed a numerically stable dual method for the solution. It is based on the knowledge of all dual feasible bases under some conditions on the objective function. In the multidimensional case the recent results are also about the dual feasible basis structures. Unfortunately, at higher dimensions, the whole structure has not been found under any circumstances. This means that a dual method, similar to Prekopa’s, cannot be developed. Only bounds on the objective function value are given, which can be far from the optimum. This paper introduces a different approach to treat the numerical difficulties. The method is based on multivariate polynomial bases. Our algorithm, in most cases, yields the optimum of the MDMP without any assumption on the objective function. The efficiency of the method is tested on several numerical examples.