Orthounimodal Distributionally Robust Optimization: Representation, Computation and Multivariate Extreme Event Applications

TL;DR

This paper introduces orthounimodality (OU) as a shape constraint in distributionally robust optimization (DRO), providing a nonparametric method for multivariate extreme event estimation with valid confidence bounds.

Contribution

It characterizes OU distributions, develops a Choquet representation, and reduces complex moment problems into finite-dimensional variational problems for improved extreme event analysis.

Findings

OU-based DRO offers statistically valid confidence bounds.

The method is competitive with traditional extreme value techniques.

Numerical results demonstrate practical effectiveness in multivariate settings.

Abstract

This paper studies a basic notion of distributional shape known as orthounimodality (OU) and its use in shape-constrained distributionally robust optimization (DRO). As a key motivation, we argue how such type of DRO is well-suited to tackle multivariate extreme event estimation by giving statistically valid confidence bounds on target extremal probabilities. In particular, we explain how DRO can be used as a nonparametric alternative to conventional extreme value theory that extrapolates tails based on theoretical limiting distributions, which could face challenges in bias-variance control and other technical complications. We also explain how OU resolves the challenges in interpretability and robustness faced by existing distributional shape notions used in the DRO literature. Methodologically, we characterize the extreme points of the OU distribution class in terms of what we call OU sets and build a corresponding Choquet representation, which subsequently allows us to reduce OU-DRO into moment problems over infinite-dimensional random variables. We then develop, in the bivariate setting, a geometric approach to reduce such moment problems into finite dimension via a specially constructed variational problem designed to eliminate suboptimal solutions. Numerical results illustrate how our approach gives rise to valid and competitive confidence bounds for extremal probabilities.