Distributionally Robust Optimization under Mean-Covariance Ambiguity Set and Half-Space Support for Bivariate Problems

TL;DR

This paper develops a distributionally robust optimization framework for bivariate problems with mean-covariance ambiguity and half-space support, deriving closed-form bounds and analyzing inventory control implications.

Contribution

It introduces a novel approach to handle mean-covariance ambiguity with half-space support, providing closed-form solutions and insights into inventory management under uncertainty.

Findings

Closed-form tight bounds for the inner problem in six cases

Optimal distributions identified via primal-dual approach

Covariance information significantly impacts inventory decisions

Abstract

In this paper, we study a bivariate distributionally robust optimization problem with mean-covariance ambiguity set and half-space support. Under a conventional type of objective function widely adopted in inventory management, option pricing, and portfolio selection, we obtain closed-form tight bounds of the inner problem in six different cases. Through a primal-dual approach, we identify the optimal distributions in each case. As an application in inventory control, we first derive the optimal order quantity and the corresponding worst-case distribution, extending the existing results in the literature. Moreover, we show that under the distributionally robust setting, a centralized inventory system does not necessarily reduce the optimal total inventory, which contradicts conventional wisdom. Furthermore, we identify two effects, a conventional pooling effect, and a novel shifting effect, the combination of which determines the benefit of incorporating the covariance information in the ambiguity set. Finally, we demonstrate through numerical experiments the importance of keeping the covariance information in the ambiguity set instead of compressing the information into one dimension.