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We present and analyze a central cutting surface algorithm for generalsemi-infinite convex optimization problems, and use it to develop a novelalgorithm for distributionally robust optimization problems in which theuncertainty set consists of probability distributions with given bounds ontheir moments. Moments of arbitrary order, as well as non-polynomial momentscan be included in the formulation. We show that this gives rise to a hierarchyof optimization problems with decreasing levels of risk-aversion, with classicrobust optimization at one end of the spectrum, and stochastic programming atthe other. Although our primary motivation is to solve distributionally robustoptimization problems with moment uncertainty, the cutting surface method forgeneral semi-infinite convex programs is also of independent interest. Theproposed method is applicable to problems with non-differentiable semi-infiniteconstraints indexed by an infinite-dimensional index set. Examples comparingthe cutting surface algorithm to the central cutting plane algorithm ofKortanek and No demonstrate the potential of our algorithm even in the solutionof traditional semi-infinite convex programming problems whose constraints aredifferentiable and are indexed by an index set of low dimension. After the rateof convergence analysis of the cutting surface algorithm, we extend theauthors' moment matching scenario generation algorithm to a probabilisticalgorithm that finds optimal probability distributions subject to momentconstraints. The combination of this distribution optimization method and thecentral cutting surface algorithm yields a solution to a family ofdistributionally robust optimization problems that are considerably moregeneral than the ones proposed to date.

A cutting surface algorithm for semi-infinite convex programming with an application to moment robust optimization