Convergent Lifted Lasserre Hierarchy of SDPs for Minimizing Expectation of Piecewise Polynomial Loss over Wasserstein Balls

TL;DR

This paper introduces a convergent hierarchy of SDP relaxations for minimizing expected piecewise polynomial losses over Wasserstein balls, advancing distributionally robust optimization with theoretical and practical insights.

Contribution

It develops a new lifted positivity certificate for piecewise polynomials and proves finite convergence of the hierarchy under convexity, enhancing optimization over Wasserstein balls.

Findings

Hierarchy converges asymptotically and finitely under convexity.

Numerical experiments demonstrate practical effectiveness.

New SOS representations for positive piecewise polynomials.

Abstract

This paper investigates the minimization of the expectation of piecewise polynomial loss functions over Wasserstein balls. This optimization problem often appears as a key sub-problem of distributionally robust optimization problems. We establish the asymptotic convergence of a hierarchy of semi-definite programming (SDP) relaxations, providing a framework for approximating the optimal values of these inherently infinite-dimensional optimization problems. A central foundational contribution is the development of a new lifted positivity certificate: we demonstrate that piecewise polynomials positive over Archimedean basic semi-algebraic sets admit a structured system of sum-of-squares (SOS) representations. Furthermore, we prove that the proposed hierarchy achieves finite convergence under suitable conditions when the defining polynomials are convex. The practical utility and versatility of this approach are demonstrated via numerical experiments in revenue estimation and portfolio optimization.