Application of the Solution of the Univariate Discrete Moment Problem for the Multivariate Case

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 presented in Prékopa (1990). 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 also can be consider as an (ill-conditioned) LP. The central results of MDMP concern the structure of the dual feasible bases, 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 DMP cannot be developed. However, there exists a method in Mádi-Nagy (2005), 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 method of DMP for solving those independent subproblems. The efficiency of this new method will be illustrated by numerical examples.