Conic Programming Reformulations of Two-Stage Distributionally Robust Linear Programs over Wasserstein Balls

TL;DR

This paper demonstrates that certain two-stage distributionally robust linear programs with Wasserstein ambiguity sets can be reformulated exactly or approximately as conic programs, enabling more efficient solutions.

Contribution

It shows exact or approximate conic reformulations of two-stage distributionally robust linear programs with Wasserstein ambiguity sets, extending tractability and solution methods.

Findings

Distributionally robust LPs with Wasserstein sets can be reformulated as copositive or linear programs.

Exact reformulations are possible for 2-Wasserstein balls with complete recourse.

Approximate reformulations are available for problems with expensive recourse.

Abstract

Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions. Specifically, when the ambiguity set constitutes a 2-Wasserstein ball centered at a discrete distribution, then the distributionally robust linear program is equivalent to a copositive program (if the problem has complete recourse) or can be approximated arbitrarily closely by a sequence of copositive programs (if the problem has sufficiently expensive recourse). These results directly extend to the classical robust setting and motivate strong tractable approximations of two-stage problems based on semidefinite approximations of the copositive cone. We also demonstrate that the two-stage distributionally robust optimization problem is equivalent to a tractable linear program when the ambiguity set constitutes a 1-Wasserstein ball centered at a discrete distribution and there are no support constraints.