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

TL;DR

This paper introduces a cutting surface algorithm for semi-infinite convex programming and applies it to develop a novel distributionally robust optimization method that handles moment uncertainty with a hierarchy of risk levels.

Contribution

The paper presents a new cutting surface algorithm for semi-infinite convex problems and extends it to solve complex distributionally robust optimization problems with moment constraints.

Findings

The algorithm effectively solves semi-infinite convex problems with non-differentiable constraints.

It creates a hierarchy of risk-averse to risk-neutral optimization problems.

The combined method outperforms traditional approaches in complex distributional uncertainty scenarios.

Abstract

We present and analyze a central cutting surface algorithm for general semi-infinite convex optimization problems, and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments. Moments of arbitrary order, as well as non-polynomial moments can be included in the formulation. We show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk-aversion, with classic robust optimization at one end of the spectrum, and stochastic programming at the other. Although our primary motivation is to solve distributionally robust optimization problems with moment uncertainty, the cutting surface method for general semi-infinite convex programs is also of independent interest. The proposed method is 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 our 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. After the rate of convergence 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 central 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.