On Multivariate Discrete Moment Problems: Generalization of the Bivariate Min Algorithm for Higher Dimensions

Abstract

The objective of 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 with a discrete finite support where the probability distribution is unknown, but some of the moments are given. The moments may be binomial, power, or of a more general type. The MDMP can be formulated as a linear programming problem with a very ill-conditioned coefficient matrix. Hence, the LP problem can be solved with difficulty or cannot be solved at all. The central results of the field of the MDMP concern the structure of the dual feasible bases. These bases, on one hand, provide us with bounds without any numerical difficulties. On the other hand, they can be used as an initial basis of the dual simplex method. That results in shorter running time and better numerical stability because the first phase can be skipped. This paper introduces a new type of MDMP, where the bivariate moments up to a certain order m consisting of the first variable and further univariate moments up to the order $m_j$, $j=1,\ldots,s$, are given. Then we generalize the bivariate Min Algorithm of Mádi-Nagy and Prékopa [Math. Oper. Res., 29 (2004), pp. 229–258] for higher dimensions, which gives numerous dual feasible bases of the MDMP. By the aid of this, on one hand, we can give useful bounds for MDMPs with higher dimensional random vectors even if the usual solvers cannot give acceptable results. On the other hand, applying our algorithm for the binomial MDMP, we can give better bounds for probabilities of Boolean functions of event sequences than the recent bounds in the literature. These results are illustrated by numerical examples.