[Paper Review] On Linear Optimization over Wasserstein Balls

Man–Chung Yue, Daniel Kühn+1 more|arXiv (Cornell University)|Apr 15, 2020

Risk and Portfolio Optimization23 references4 citations

TL;DR

This paper establishes the weak compactness of Wasserstein balls under mild moment conditions and provides easily verifiable necessary and sufficient conditions for the existence of optimal solutions in linear optimization over these balls. It further characterizes the sparsity of optimal solutions when the reference measure is discrete, offering a self-contained, concise, and rigorous analysis that improves upon prior work in distributionally robust optimization and machine learning.

ABSTRACT

Wasserstein balls, which contain all probability measures within a pre-specified Wasserstein distance to a reference measure, have recently enjoyed wide popularity in the distributionally robust optimization and machine learning communities to formulate and solve data-driven optimization problems with rigorous statistical guarantees. In this technical note we prove that the Wasserstein ball is weakly compact under mild conditions, and we offer necessary and sufficient conditions for the existence of optimal solutions. We also characterize the sparsity of solutions if the Wasserstein ball is centred at a discrete reference measure. In comparison with the existing literature, which has proved similar results under different conditions, our proofs are self-contained and shorter, yet mathematically rigorous, and our necessary and sufficient conditions for the existence of optimal solutions are easily verifiable in practice.

Motivation & Objective

• To establish weak compactness of Wasserstein balls under minimal assumptions, ensuring the existence of optimal solutions in distributionally robust optimization.
• To derive necessary and sufficient conditions for the finiteness and attainability of optimal values in linear optimization over Wasserstein balls.
• To characterize the sparsity of optimal solutions when the reference measure is discrete, particularly identifying the minimal number of atoms required.
• To provide a self-contained, concise, and mathematically rigorous proof that is easier to verify than prior approaches.
• To support data-driven optimization with rigorous statistical guarantees by clarifying the analytical foundations of Wasserstein ball formulations.

Proposed method

• Proves weak compactness of the Wasserstein ball $\mathcal{B}_r(\nu)$ using Prokhorov’s theorem and tightness arguments, assuming the reference measure $\nu$ has finite $p$-th moment.
• Demonstrates that the objective functional $\int_X f(x) \, d\mu(x)$ is weakly upper semi-continuous on the feasible set, enabling the application of compactness arguments.
• Uses the triangle inequality and properties of the Wasserstein distance to show that all measures in $\mathcal{B}_r(\nu)$ have uniformly bounded $p$-th moments when $\nu$ does.
• Applies Prokhorov’s theorem to conclude weak compactness of the feasible set, which ensures the existence of optimal solutions when the optimal value is finite.
• Characterizes optimal solution sparsity by showing that, under the given conditions, optimal solutions are supported on at most $N+1$ atoms when the reference measure is discrete.
• Relies on measure-theoretic tools and infinite-dimensional linear programming results from appendices to support the main theorems without invoking complex duality or optimality conditions.

Experimental results

Research questions

• RQ1 Under what conditions is the Wasserstein ball $\mathcal{B}_r(\nu)$ weakly compact?
• RQ2 What are the necessary and sufficient conditions for the optimal value of linear optimization over a Wasserstein ball to be finite and attainable?
• RQ3 What is the minimal number of atoms required to support an optimal solution when the reference measure is discrete?
• RQ4 How can the existence and sparsity of optimal solutions be established with minimal assumptions and maximal mathematical clarity?
• RQ5 Can a self-contained and concise proof be constructed for existence and sparsity results in Wasserstein-based optimization, avoiding reliance on complex duality theory?

Key findings

• The Wasserstein ball $\mathcal{B}_r(\nu)$ is weakly compact whenever the reference measure $\nu$ has a finite $p$-th moment.
• The optimal value of the linear optimization problem (3) is finite if and only if the integral $\int_X [f(x)]_+ \, d\nu(x)$ is finite.
• When the optimal value is finite, an optimal solution exists, and the feasible set is weakly compact due to tightness and weak closedness.
• If the reference measure $\nu$ is discrete with $N$ atoms, then there exists an optimal solution supported on at most $N+1$ atoms.
• The necessary and sufficient conditions for the existence of optimal solutions are easily verifiable in practice, unlike previous results that required more complex assumptions.
• The proof is self-contained, shorter, and more accessible than prior work, avoiding reliance on advanced duality theory or the Richter-Rogosinski theorem.