On the Finite Optimal Convergence of Logic-Based Benders’ Decomposition in Solving 0–1 Min-Max Regret Optimization Problems with Interval Costs

Abstract

This paper addresses a class of problems under interval data uncertainty composed of min-max regret versions of classical 0–1 optimization problems with interval costs. We refer to them as interval 0–1 min-max regret problems. The state-of-the-art exact algorithms for this class of problems work by solving a corresponding mixed integer linear programming formulation in a Benders’ decomposition fashion. Each of the possibly exponentially many Benders’ cuts is separated on the fly through the resolution of an instance of the classical 0–1 optimization problem counterpart. Since these separation subproblems may be NP-hard, not all of them can be modeled by means of linear programming, unless P = NP. In these cases, the convergence of the aforementioned algorithms are not guaranteed in a straightforward manner. In fact, to the best of our knowledge, their finite convergence has not been explicitly proved for any interval 0–1 min-max regret problem. In this work, we formally describe these algorithms through the definition of a logic-based Benders’ decomposition framework and prove their convergence to an optimal solution in a finite number of iterations. As this framework is applicable to any interval 0–1 min-max regret problem, its finite optimal convergence also holds in the cases where the separation subproblems are NP-hard.

L. Assunção—Partially supported by the Coordination for the Improvement of Higher Education Personnel, Brazil (CAPES).

L. Assunção—The author thanks Vitor A. A. Souza and Phillippe Samer for the valuable discussions throughout the conception of this work.