On the convergence of cutting-plane methods for robust optimization with ellipsoidal uncertainty sets

TL;DR

This paper proves that a class of cutting-plane algorithms for robust optimization with ellipsoidal uncertainty sets converges in a finite number of steps, addressing a gap in theoretical guarantees.

Contribution

It establishes finite convergence of cutting-plane methods for ellipsoidal uncertainty sets, extending previous results limited to polyhedral sets.

Findings

Proves finite convergence of the algorithm for ellipsoidal sets

Extends theoretical guarantees beyond polyhedral uncertainty sets

Supports the effectiveness of cutting-plane methods in robust optimization

Abstract

Recent advances in cutting-plane strategies applied to robust optimization problems show that they are competitive with respect to problem reformulations and interior-point algorithms. However, although its application with polyhedral uncertainty sets guarantees convergence, finite termination when using ellipsoidal uncertainty sets is not theoretically guaranteed. This paper demonstrates that the cutting-plane algorithm set out for ellipsoidal uncertainty sets in its more general form also converges in a finite number of steps.