[Paper Review] Convergent Lifted Lasserre Hierarchy of SDPs for Minimizing Expectation of Piecewise Polynomial Loss over Wasserstein Balls

N. D. Dizon, Q. Y. Huang+3 more|arXiv (Cornell University)|Feb 24, 2026

Risk and Portfolio Optimization0 citations

TL;DR

The paper develops a lifted SOS-based SDP hierarchy to approximate and converge to the optimal value of distributionally robust optimization problems with piecewise polynomial loss under Wasserstein ambiguity, including finite convergence under convexity.

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.

Motivation & Objective

• Motivate and formulate the problem of minimizing the expectation of piecewise polynomial loss over Wasserstein balls in distributionally robust optimization.
• Develop a new lifted positivity certificate for piecewise polynomials on Archimedean semi-algebraic sets.
• Construct a convergent SDP hierarchy to approximate the infinite-dimensional DRO problem and prove asymptotic (and, under convexity, finite) convergence.

Proposed method

• Introduce piecewise polynomial loss and Wasserstein ambiguity set within a basic semi-algebraic support Xi.
• Derive a lifted sum-of-squares representation (positivity certificate) for piecewise polynomials over Xi (Theorem 3.2).
• Formulate a hierarchy of SDPs (AD_r) that provide increasingly tight relaxations for the expectation problem (EP).
• Prove asymptotic convergence of the SDP hierarchy to the DRO optimum (Theorem 3.4).
• Establish finite convergence under suitable convexity conditions using convex-concave minimax and Lagrange duality with Scheiderer’s SOS representation.
• Demonstrate applicability via numerical experiments in revenue estimation and mean-CVaR portfolio optimization.

Experimental results

Research questions

• RQ1 How to approximate the infinite-dimensional problem of minimizing the Wasserstein-ambiguous expectation of a piecewise polynomial loss?
• RQ2 Can a lifted SOS representation certify positivity of piecewise polynomials over Archimedean semi-algebraic sets?
• RQ3 Under what conditions does the SDP hierarchy converge asymptotically or finitely to the DRO optimum?
• RQ4 How does the framework perform on practical DRO problems such as revenue estimation and portfolio optimization?

Key findings

• A lifted positivity certificate shows piecewise polynomials positive on Archimedean basic semi-algebraic sets admit a structured SOS representation.
• An SDP hierarchy (AD_r) is constructed whose feasibility constrains yield converging approximations to the EP optimum.
• The hierarchy is proven to converge asymptotically to the true optimum as r → ∞ (under convexity of the defining polynomials, finite convergence occurs).
• The approach extends moment-based DRO to general piecewise polynomials beyond previous piecewise quadratic or SOS-convex settings.
• Numerical experiments illustrate applicability to revenue estimation with a piecewise polynomial utility and to mean-CVaR portfolio optimization.