Moment Problems and Semidefinite Optimization

Abstract

Problems involving moments of random variables arise naturally in many areas of mathematics, economics, and operations research. How d o w e obtain optimal bounds on the probability that a random variable belongs in a set, given some of its moments? How d o w e price enancial derivatives without assuming any model for the underlying price dynamics, given only moments of the price of the underlying asset? How d o w e o b-tain stronger relaxations for stochastic optimization problems exploiting the knowledge that the decision variables are moments of random variables? Can we generate near optimal solutions for a discrete optimization problem from a semideenite relaxation by interpreting an optimal solution of the relaxation as a covariance matrix? In this paper, we demonstrate that convex, and in particular semideenite, optimization methods lead to interesting and often unexpected answers to these questions.