Bounding the Expectation of a Saddle Function with Application to Stochastic Programming

Abstract

We previously obtained tight upper and lower bounds to the expectation of a saddle function of multivariate random variables using first and cross moments of the random variables without independence assumptions. These bounds are applicable when domains of the random vectors are compact sets in the euclidean space. In this paper, we extend the results to the case of unbounded domains, similar in spirit to the extensions by Birge and Wets in the pure convex case. The relationship of these bounds to a certain generalized moment problem is also investigated. Finally, for solving stochastic linear programs utilizing the above bounding procedures, a computationally more appealing order-cone decomposition scheme is proposed which behaves quadratically in the number of random variables. Moreover, the resulting upper and lower approximations are amenable to efficient solution techniques.