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Distributionally Robust Optimization with Polynomial Robust Constraints

Jiawang Nie, Suhan Zhong

TL;DR

This work addresses DRO problems where robust constraints are polynomials and the ambiguity set is defined via moments. It converts the nonlinear robust constraint into a linear conic form using a Moment-SOS relaxation, reducing the problem to semidefinite programming; when the problem is SOS-convex, the relaxation is exact and yields a global optimizer. For nonconvex instances, the paper provides sufficient conditions (e.g., rank and flat-truncation criteria) under which the relaxation remains tight, and it offers a practical heuristic and a deterministic SOS-based alternative to obtain candidate solutions. Numerical experiments demonstrate the method’s efficiency and its ability to recover global optima in a range of DRO settings, including portfolio optimization and mean–variance frameworks. Overall, the approach offers a tractable, theoretically grounded pathway to globally solve a broad class of polynomial DROs with polynomial robust constraints and moment-based ambiguity sets.

Abstract

This paper studies distributionally robust optimization (DRO) with polynomial robust constraints. We give a Moment-SOS relaxation approach to solve the DRO. This reduces to solving linear conic optimization with semidefinite constraints. When the DRO problem is SOS-convex, we show that it is equivalent to the linear conic relaxation and it can be solved by the Moment-SOS algorithm. For nonconvex cases, we also give concrete conditions such that the DRO can be solved globally. Numerical experiments are given to show the efficiency of the method.

Distributionally Robust Optimization with Polynomial Robust Constraints

TL;DR

This work addresses DRO problems where robust constraints are polynomials and the ambiguity set is defined via moments. It converts the nonlinear robust constraint into a linear conic form using a Moment-SOS relaxation, reducing the problem to semidefinite programming; when the problem is SOS-convex, the relaxation is exact and yields a global optimizer. For nonconvex instances, the paper provides sufficient conditions (e.g., rank and flat-truncation criteria) under which the relaxation remains tight, and it offers a practical heuristic and a deterministic SOS-based alternative to obtain candidate solutions. Numerical experiments demonstrate the method’s efficiency and its ability to recover global optima in a range of DRO settings, including portfolio optimization and mean–variance frameworks. Overall, the approach offers a tractable, theoretically grounded pathway to globally solve a broad class of polynomial DROs with polynomial robust constraints and moment-based ambiguity sets.

Abstract

This paper studies distributionally robust optimization (DRO) with polynomial robust constraints. We give a Moment-SOS relaxation approach to solve the DRO. This reduces to solving linear conic optimization with semidefinite constraints. When the DRO problem is SOS-convex, we show that it is equivalent to the linear conic relaxation and it can be solved by the Moment-SOS algorithm. For nonconvex cases, we also give concrete conditions such that the DRO can be solved globally. Numerical experiments are given to show the efficiency of the method.
Paper Structure (11 sections, 6 theorems, 155 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 11 sections, 6 theorems, 155 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relaxations for the polynomial robust constraint
  4. Conic representation of robust constraints
  5. Moment relaxations for functions in $x$
  6. A Moment-SOS Algorithm
  7. The SOS-convex case
  8. The non-SOS-convex case
  9. A heuristic trick
  10. Numerical Experiments
  11. Conclusions and discussions

Key Result

Theorem 3.2

Suppose (eq:Kdual) holds and $\mathbb{E}_{\mu}[h(x,\xi)]$ is SOS-concave in $x$ for each $\mu\in \mathcal{M}$. Then, (eq:momrel) is a tight relaxation for (md:ecDRO), in the following sense: $x$ is an optimizer of (md:ecDRO) if and only if $(x, w)$ is an optimizer of (eq:momrel).

Figures (1)

  • Figure 1: A flowchart to represent Algorithm \ref{['def:alg']} and each approximation step of (\ref{['md:ecDRO']})

Theorems & Definitions (22)

  • Example 3.1
  • Theorem 3.2
  • proof
  • Example 3.3
  • Theorem 3.4
  • proof
  • Example 3.5
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • ...and 12 more