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

Henri Lefebvre, Anirudh Subramanyam

TL;DR

This work identifies a flaw in the original sufficient conditions for obtaining a closed‑form optimal Lagrange multiplier in two‑stage robust optimization with binary uncertainty, demonstrated through a counterexample. It furnishes corrected, stronger conditions under which the closed‑form multiplier $u(\mathbf{x})-\ell(\mathbf{x})$ remains optimal and proves a corresponding inner‑problem integrality result via total unimodularity, while also deriving a polynomial‑time bound $\bar{\lambda}$ for more general data. To handle instances that do not satisfy the conditions, the authors develop practical algorithmic modifications to the Benders and column‑and‑constraint generation frameworks, including exact procedures for problems with indicator constraints and for general mixed‑integer second‑stage problems, with post‑hoc checks to ensure rigor. Computational experiments on 378 benchmarks show the modifications incur only modest overhead and largely preserve the original method’s computational advantages, confirming the approach’s practical viability and suggesting warm‑start strategies if suboptimality is detected. Overall, the paper reconciles theoretical guarantees with practical performance, enabling reliable, efficient solutions for a broad class of two‑stage robust optimization problems with binary uncertainty.

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.

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

TL;DR

This work identifies a flaw in the original sufficient conditions for obtaining a closed‑form optimal Lagrange multiplier in two‑stage robust optimization with binary uncertainty, demonstrated through a counterexample. It furnishes corrected, stronger conditions under which the closed‑form multiplier remains optimal and proves a corresponding inner‑problem integrality result via total unimodularity, while also deriving a polynomial‑time bound for more general data. To handle instances that do not satisfy the conditions, the authors develop practical algorithmic modifications to the Benders and column‑and‑constraint generation frameworks, including exact procedures for problems with indicator constraints and for general mixed‑integer second‑stage problems, with post‑hoc checks to ensure rigor. Computational experiments on 378 benchmarks show the modifications incur only modest overhead and largely preserve the original method’s computational advantages, confirming the approach’s practical viability and suggesting warm‑start strategies if suboptimality is detected. Overall, the paper reconciles theoretical guarantees with practical performance, enabling reliable, efficient solutions for a broad class of two‑stage robust optimization problems with binary uncertainty.

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.

Paper Structure

This paper contains 10 sections, 6 theorems, 39 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Counterexample
  4. Correct Sufficient Conditions
  5. Absence of Sufficient Conditions
  6. Computational Complexity
  7. Algorithmic Modifications
  8. Modifications for Problem \ref{['eq:two_stage_ro_indicator']}
  9. Modifications for Problem \ref{['eq:two_stage_ro_general']}
  10. Impact on Computational Performance and Practical Considerations

Key Result

Theorem 1

Suppose that the following conditions are satisfied in problem eq:two_stage_ro_general. Then, for any feasible first-stage decision $\bm{x} \in \mathcal{X}$; that is, for which $\sup \{\mathcal{Q}(\bm{x}, \bm{\xi}) : \bm{\xi} \in \Xi \} < +\infty$, we have that where $u(\bm{x})$ is any finite upper bound on $\sup \{\mathcal{Q}(\bm{x}, \bm{\xi}) : \bm{\xi} \in \Xi \}$ and $\ell(\bm{x})$ is any fi

Figures (1)

  • Figure 1: Plot of $\max_{\xi\in\Xi} \mathcal{L}(x,\xi,\lambda)$ versus $\lambda$ for the counterexample.

Theorems & Definitions (13)

  • Remark 1
  • Theorem 1
  • proof
  • Example 1: Interdicted Combinatorial Problems
  • Theorem 2
  • Lemma 1: buchheim2023bilevel
  • Theorem 3
  • proof : Proof of Theorem \ref{['theorem:polynomial_time_lambda']}
  • Theorem 4
  • proof
  • ...and 3 more