Distributionally robust optimization with generalized total variation ambiguity sets

Abstract

This paper introduces a data-driven framework for distributionally robust optimization (DRO), founded on a new class of ambiguity sets termed generalized total variation (GTV) sets. In contrast to traditional DRO approaches, the proposed scheme constructs ambiguity sets whose geometry incorporates sample size, support, confidence level, empirical distribution, and cost function structure. Under this framework, we develop two tractable solution methods (first-order, gradient-based), each offering finite-sample statistical guarantees. The first-order approach employs sequential convex programming to construct a solution, followed by a linear program to determine a high-confidence upper bound (i.e., generalization bound) on the solution’s unknown true risk. The gradient-based approach, applicable when the ambiguity set has a smoothly curved boundary, utilizes gradient information to establish a high-confidence upper bound through a sequence of convex programs, all linear except for the final step. We prove that both methods produce statistically consistent risk estimates. Then, we empirically validate the framework on two applications: a synthetic two-item Newsvendor problem and a real-world portfolio optimization problem using S&P 500 asset returns. Results demonstrate that for finite support problems, GTV ambiguity sets can deliver generalization bounds that are as tight as, or tighter than, those from popular alternatives such as Wasserstein and total variation ambiguity sets. We thus highlight the practical benefits of incorporating several types of information into ambiguity set construction, offering improved robustness-performance tradeoffs for data-driven decision-making under uncertainty.