Application of the solution of the univariate discrete moment problem for the multivariate case

Abstract

The objective of the univariate discrete moment problem (DMP) is to find the minimum and/or maximum of the expected value of a function of a random variable which has a discrete finite support. The probability distribution is unknown, but some of the moments are given. This problem is an ill-conditioned LP, but it can be solved by the dual method developed by Prékopa. The multivariate discrete moment problem (MDMP) is the generalization of the DMP where the objective function is the expected value of a function of a random vector. The MDMP has also been initiated by Prékopa and it can also be considered as an (ill-conditioned) LP. The central results of the MDMP concern the structure of the dual feasible bases and provide us with bounds without any numerical difficulties. Unfortunately, in this case not all the dual feasible bases have been found hence the multivariate counterpart of the dual method of the DMP cannot be developed. However, there exists a method developed by Mádi-Nagy [7] which allows us to get the basis corresponding to the best bound out of the known structures by optimizing independently on each variable. In this paper we present a method using the dual algorithm of the DMP for solving those independent subproblems. The efficiency of this new method is illustrated by numerical examples.