Explicit Reformulation of Discrete Distributionally Robust Optimization Problems

TL;DR

This paper addresses discrete distributionally robust optimization (DDRO) under distributional uncertainty by introducing two tractable uncertainty sets: a weighted L2 ball and a density-ratio (DR) ball. It reformulates the min–max DDRO into a single-layer smooth convex program and links ball sizes to familiar risk measures: the L2 ball corresponds to minimizing the mean plus a multiple of the standard deviation, while the DR ball corresponds to CVaR minimization with beta = d/(1+d). Theoretical results establish strong duality and convexity, and interpretability results show how to choose ball sizes to navigate the trade-off between performance and risk. Numerical experiments on a patroller-agent design problem demonstrate Pareto fronts in mean versus variability and significant CVaR improvements, validating the practical usefulness of the approach for discrete, uncertainty-rich control tasks.

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%.

Figures (4)

• Figure 1: Overview of the proposed method.
• Figure 2: Graphs of the four topologies in almeida2004recent. In the figure, the black dots represent the nodes and the solid lines represent the edges.
• Figure 3: Results of the expectation and the standard deviation of mean hitting time. The solid lines represents the expectation of mean hitting time $\mathbb{E}_{p_0(i)}\left[J(\boldsymbol{x},\,i)\right]$, whereas the dashed lines represent the standard deviation $\sqrt{\mathbb{V}_{p_0(i)}\left[J(\boldsymbol{x},\,i)\right]}$.
• Figure 4: Pareto front of the expectation and the standard deviation of mean hitting time. The circles represent Pareto-optimal solutions that the proposed method have found. The horizontal axis is the expectation of mean hitting time $\mathbb{E}_{p_0(i)}\left[J(\boldsymbol{x},\,i)\right]$, while the vertical axis is the standard deviation $\sqrt{\mathbb{V}_{p_0(i)}\left[J(\boldsymbol{x},\,i)\right]}$.

Theorems & Definitions (35)

• Remark 1: Difficulty in solving DDRO problems
• Remark 2: Motivation for Using Weighted L2 and Density-Ratio Balls
• Theorem 3: Reformulation of DDRO Problems with Weighted L2 Balls
• Remark 4: Solvability of DDRO Problems with Weighted L2 balls
• Theorem 5: Reformulation of DDRO Problems with Density-Ratio Balls
• Remark 6: Solvability of DDRO Problems with DR balls
• Theorem 7: Expectation and Standard Deviation Minimization
• Remark 8: Weight Parameters and Size of Weighted L2 Balls
• Remark 9: Pareto Front and Optimality
• Corollary 10: Pareto-optimal solutions to Expectation and Standard Deviation Minimization