Stability and Continuity in Robust Linear and Linear Semi-Infinite Optimization

TL;DR

This paper investigates the stability of robust linear and semi-infinite optimization problems under uncertainty set perturbations, establishing Lipschitz continuity of optimal values and solution sets with explicit bounds.

Contribution

It provides new theoretical results on the Lipschitz continuity and stability of solutions in robust linear and semi-infinite optimization under uncertainty.

Findings

Optimal value is Lipschitz continuous with respect to uncertainty set changes.

Explicit Lipschitz constants are derived for the stability bounds.

Solution sets exhibit closedness and upper semi-continuity under perturbations.

Abstract

We consider the stability of Robust Optimization problems with respect to perturbations in their uncertainty sets. We focus on Linear Optimization problems, including those with a possibly infinite number of constraints, also known as Linear Semi-Infinite Optimization (LSIO) problems, and consider uncertainty in both the cost function and constraints. We prove Lipschitz continuity of the optimal value and {\epsilon}-approximate optimal solution set with respect to the Hausdorff distance between uncertainty sets and with an explicit Lipschitz constant that can be calculated. In addition, we prove closedness and upper semi-continuity for the optimal solution set mapping with respect to the uncertainty set.