Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

TLDR

Robust optimization problems involving uncertain probability vectors arise frequently in inventory control and finance, where terms depend on moments or expected utilities of random variables. This study investigates robust linear optimization with uncertainty regions defined by φ‑divergences such as chi‑squared, Hellinger, and Kullback–Leibler. The authors formulate these problems by treating φ‑divergence uncertainty as confidence sets for probability vectors and extend the framework to nonlinear optimization variables. They prove that the robust counterpart is tractable for most common φ choices, and demonstrate its applicability through asset‑pricing and multi‑item newsvendor examples. Accepted by Gérard P.

Abstract

In this paper we focus on robust linear optimization problems with uncertainty regions defined by ϕ-divergences (for example, chi-squared, Hellinger, Kullback–Leibler). We show how uncertainty regions based on ϕ-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with ϕ-divergence uncertainty is tractable for most of the choices of ϕ typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach. This paper was accepted by Gérard P. Cachon, optimization.