A Cutting Surface Algorithm for Semi-Infinite Convex Programming with an Application to Moment Robust Optimization

We first present and analyze a central cutting surface algorithm for general semi-infinite convex optimization problems, and use it to develop an algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments. The cutting surface algorithm is also applicable to problems with non-differentiable semi-infinite constraints indexed by an infinite-dimensional index set. Examples comparing the cutting surface algorithm to the central cutting plane algorithm of Kortanek and No demonstrate the potential of the central cutting surface algorithm even in the solution of traditional semi-infinite convex programming problems, whose constraints are differentiable, and are indexed by an index set of low dimension. Our primary motivation for the higher level of generality is to solve distributionally robust optimization problems with moment uncertainty. After the analysis of the cutting surface algorithm, we extend the authors’ moment matching scenario generation algorithm to a probabilistic algorithm that finds optimal probability distributions subject to moment constraints. The combination of this distribution optimization method and the cutting surface algorithm yields a solution to a family of distributionally robust optimization problems that are considerably more general than the ones proposed to date.