A method to find the best bounds in a multivariate discrete moment problem if the basis structure is given

Abstract

The multivariate discrete moment problem (MDMP) is to find the minimum and/or maximum of the expected value of a function of a random vector which has a discrete finite support. The probability distribution is unknown, but some of the moments are given. The MDMP has been initiated by Prékopa who developed a linear rogramming methodology to solve it. The central results in this respect concern the structure of the dual feasible bases. These bases provide us with bounds without any numerical difficulty (which is arising in the usual solution methods). In this paper we shortly summarize the properties of the above mentioned basis structures, and then we show a new method which allows us to get the basis corresponding to the best bound out of the known structures by optimizing independently on each variable. We illustrate the efficiency of this method by numerical examples.