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

Abstract

We provide a correction to the sufficient conditions under which closed-form expressions for the optimal Lagrange multiplier are provided in [4]. 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.

Methodology and applications of robust optimization

“…RO problems are often difficult to solve, especially in the presence of recourse and nonconvexities. Subramanyam (2022) addresses the case of two-stage RO with binary uncertainties. Problems of this type are typically solved using a Benders decomposition or column-and-constraint generation approach; both involve solving subproblems to determine worst-case parameter realizations.…”

“…These subproblems are formulated as mixed-integer optimization problems with bilinear terms that need to be linearized, which leads to large and weak formulations that cause computational difficulties. Subramanyam (2022) proposes an alternative Lagrangian dual method that circumvents these difficulties by moving all uncertainties to the objective function and proves that the resulting formulations satisfy strong duality properties. Results from computational experiments demonstrate that the proposed method can provide up to two orders of magnitude speedups over existing methods in computing worst-case parameter realizations.…”