Explicit Reformulation of Discrete Distributionally Robust Optimization Problems

TL;DR

This paper introduces a simplified discrete distributionally robust optimization (DDRO) approach that reduces complex problems to a single convex program, improving computational efficiency and practical applicability in real-world uncertain systems.

Contribution

It proposes a novel DDRO reformulation using two types of uncertainty balls, transforming the problem into a single-layer convex program with practical guidance for parameter selection.

Findings

Achieved up to 3% variation in mean hitting time.

Reduced CVaR by up to 13%.

Demonstrated improved tractability in a patrol-agent design problem.

Abstract

Distributionally robust optimization (DRO) is an effective framework for controlling real-world systems with various uncertainties, typically modeled using distributional uncertainty balls. However, DRO problems often involve infinitely many inequality constraints, rendering exact solutions computationally expensive. In this study, we propose a discrete DRO (DDRO) method that significantly simplifies the problem by reducing it to a single trivial constraint. Specifically, the proposed method utilizes two types of distributional uncertainty balls to reformulate the DDRO problem into a single-layer smooth convex program, significantly improving tractability. Furthermore, we provide practical guidance for selecting the appropriate ball sizes. The original DDRO problem is further reformulated into two optimization problems: one minimizing the mean and standard deviation, and the other minimizing the conditional value at risk (CVaR). These formulations account for the choice of ball sizes, thereby enhancing the practical applicability of the method. The proposed method was applied to a distributionally robust patrol-agent design problem, identifying a Pareto front in which the mean and standard deviation of the mean hitting time varied by up to 3% and 14%, respectively, while achieving a CVaR reduction of up to 13%.