Correction to: A Lagrangian dual method for two-stage robust optimization with binary uncertainties

TL;DR

This paper corrects and clarifies the sufficient conditions for optimal Lagrange multipliers in a robust optimization method, ensuring the validity of related algorithms and discussing their computational implications.

Contribution

It provides corrected conditions and proofs for optimal Lagrange multipliers in a robust optimization framework, along with validated algorithm modifications.

Findings

Counterexample showing original conditions are insufficient

Corrected conditions with valid proofs

Polynomial-time computation of optimal Lagrange multipliers

Abstract

We provide a correction to the sufficient conditions under which closed-form expressions for the optimal Lagrange multiplier are provided in arXiv:2112.13138 [math.OC]. We first present a simple counterexample where the original conditions are insufficient, highlight where the original proof fails, and then provide modified conditions along with a correct proof of their validity. Finally, although the original paper discusses modifications to their method for problems that may not satisfy any sufficient conditions, we substantiate that discussion along two directions. We first show that computing an optimal Lagrange multiplier can still be done in polynomial time. We then provide complete and correct versions of the corresponding Benders and column-and-constraint generation algorithms in which the original method is used. We also discuss the implications of our findings on computational performance.