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

N. D. Dizon, Q. Y. Huang, T. D. Chuong, G. Li, V. Jeyakumar

TL;DR

It is demonstrated that piecewise polynomials positive over Archimedean basic semi-algebraic sets admit a structured system of sum-of-squares (SOS) representations and it is proved that the proposed hierarchy achieves finite convergence under suitable conditions when the defining polynomials are convex.

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.

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

TL;DR

It is demonstrated that piecewise polynomials positive over Archimedean basic semi-algebraic sets admit a structured system of sum-of-squares (SOS) representations and it is proved that the proposed hierarchy achieves finite convergence under suitable conditions when the defining polynomials are convex.

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.
Paper Structure (14 sections, 9 theorems, 80 equations, 2 figures, 4 tables)

This paper contains 14 sections, 9 theorems, 80 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wasserstein Ambiguity
  4. Polynomials Positive on a Compact Basic Semi-algebraic Set
  5. Expectation Problems: Asymptotically Converging SDPs
  6. Positivity Representation for Piecewise Polynomials
  7. Converging Hierarchy of SDPs
  8. Piecewise Convex Polynomials and Finite Convergence
  9. Numerical Experiments
  10. Data-Driven Revenue Estimation
  11. Data-driven Portfolio Optimization
  12. Strengths and Limitations of the Computational Study
  13. Conclusion and Outlook
  14. Technical Details

Key Result

Lemma 2.1

For the problem problem:mmt_0, let $\Xi \subseteq \mathbb{R}^m$ be a compact set, $g : \mathbb{R}^m \to \mathbb{R}$ be a continuous function, and $\varepsilon > 0$. Then, $\min problem:mmt_0 = \max problem:mmtd$.

Figures (2)

  • Figure 1: Average maximum revenue (in solid line) computed over ten independent simulation runs. The shaded region represents the inter-quantile range between the 20th and 80th quantiles.
  • Figure 2: Average in-sample mean-CVaR risk calculated over ten independent simulation runs, as the Wasserstein radius $\varepsilon$ varies within the interval $[10^{-3}, 10]$.

Theorems & Definitions (20)

  • Lemma 2.1: Reduction to finite-dimensional program gao2023distributionally
  • proof
  • Proposition 2.3: Putinar’s Positivstellensatz
  • Proposition 2.4: Scheiderer's non-negativity representation
  • Definition 3.1: Piecewise polynomial
  • Theorem 3.2: Lifted Putinar's positivstellensatz for piecewise polynomials
  • proof
  • Remark 3.3
  • Theorem 3.4: Asymptotic convergence
  • proof
  • ...and 10 more