Distributionally Robust Optimization with Moment Ambiguity Sets

TL;DR

This paper develops a new approach for distributionally robust optimization with moment-based ambiguity sets, reformulating the problem as a conic program and proposing a Moment-SOS relaxation method with proven convergence.

Contribution

It introduces a novel reformulation of DRO problems with polynomial objectives and constraints as conic programs and provides a Moment-SOS relaxation with convergence guarantees.

Findings

Reformulation of DRO as a linear conic optimization problem.

Development of a Moment-SOS relaxation method.

Numerical examples demonstrating the effectiveness of the approach.

Abstract

This paper studies distributionally robust optimization (DRO) when the ambiguity set is given by moments for the distributions. The objective and constraints are given by polynomials in decision variables. We reformulate the DRO with equivalent moment conic constraints. Under some general assumptions, we prove the DRO is equivalent to a linear optimization problem with moment and psd polynomial cones. A Moment-SOS relaxation method is proposed to solve it. Its asymptotic and finite convergence are shown under certain assumptions. Numerical examples are presented to show how to solve DRO problems.