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.

中文翻译：

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基于逻辑的Benders分解在求解0-1最小-最大后悔优化问题中的有限最优收敛性

本文解决了区间数据不确定性下的一类问题，这些问题由具有区间成本的经典 0-1 优化问题的最小-最大遗憾版本组成。我们将它们称为区间 0-1 最小-最大后悔问题。此类问题的最新精确算法通过以 Benders 分解方式求解相应的混合整数线性规划公式来工作。通过经典 0-1 优化问题对应实例的解析，每个可能呈指数级的 Benders 切割都被动态分离。由于这些分离子问题可能是 NP-hard，除非 P = NP，否则并非所有子问题都可以通过线性规划建模。在这些情况下，不能以直接的方式保证上述算法的收敛。实际上，据我们所知，对于任何区间 0-1 最小-最大遗憾问题，它们的有限收敛性尚未得到明确证明。在这项工作中，我们通过基于逻辑的 Benders 分解框架的定义来正式描述这些算法，并证明它们在有限迭代次数中收敛到最佳解决方案。由于该框架适用于任何区间 0-1 min-max 后悔问题，其有限最优收敛性也适用于分离子问题是 NP-hard 的情况。