Optimal moment bounds under multiple shape constraints

Consider the problem of computing the optimal lower and upper bound for the expected value E[?(X)], where X is an uncertain random probability variable. This paper studies the case in which the density of X is restricted by multiple shape constraints, each imposed on a different subset of the domain. We derive (closed) convex hull representations that allow us to reduce the optimization problem to a class of generating measures that are composed of convex sums of local probability measures. Furthermore, the notion of mass constraints is introduced to spread out the probability mass over the entire domain. A generalization to mass uncertainty is discussed as well.