混合整数双层线性规划标准松弛的紧致性
On the Tightness of Standard Relaxations for Mixed-Integer Bilevel Linear Programs
- University of Zurich（苏黎世大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明混合整数双层线性规划中标准单层松弛界在多项式时间或MILP预言机下无法一致改进，除非P=NP或多项式层级坍缩，表明其计算紧致性。
AI中文摘要：
求解混合整数双层线性规划（MIBLP）的精确算法通常依赖于收敛到最优值的上下界序列。这些过程通常使用单层松弛（SLR）初始化，该松弛通过省略跟随者的最优性条件并求解所得的单层优化问题获得。在本文中，我们研究对于广泛类别的MIBLP，所得的标准界是否允许在相同计算复杂度范围内计算出一致改进。对于纯连续双层线性规划，我们证明，除非$P = NP$，否则基于SLR的下界及其相关上界都不能在多项式时间内得到一致改进，即使对于最小-最大问题类别也是如此。然后，我们在多项式层级不坍缩的假设下，将此分析扩展到纯整数最小-最大双层线性规划类别。首先，我们证明SLR的连续松弛不允许一致的多项式时间可计算改进。接着，我们证明SLR本身及其相关上界都不允许通过具有混合整数线性规划（MILP）预言机的多项式时间算法进行一致改进。重要的是，这排除了基于迭代MILP的方法（包括割平面法和分解算法）的一致改进。总体而言，我们的结果表明，基于SLR的界在复杂度理论意义上，在其自然计算范围内无法系统性地改进。
英文摘要：
Exact algorithms for solving mixed-integer bilevel linear programs (MIBLPs) typically rely on sequences of lower and upper bounds that converge to the optimal value. These procedures are commonly initialized using the single-level relaxation (SLR), obtained by omitting the follower's optimality condition and solving the resulting single-level optimization problem. In this paper, we investigate whether, for broad classes of MIBLPs, the resulting standard bounds admit uniform improvements that can be computed within the same computational complexity regime. For pure continuous bilevel linear programs, we show that, unless $P = NP$, neither the SLR-based lower bound nor its associated upper bound can be uniformly improved in polynomial time, even for the class of min-max problems. We then extend this analysis to the class of pure integer min-max bilevel linear programs under the assumption that the polynomial hierarchy does not collapse. First, we show that the continuous relaxation of the SLR admits no uniform polynomial-time computable improvement. We then prove that neither the SLR itself nor its associated upper bound admits a uniform improvement by a polynomial-time algorithm with access to a mixed-integer linear programming (MILP) oracle. Importantly, this rules out uniform improvements by iterative MILP-based approaches, including cutting-plane-based and decomposition algorithms. Overall, our results demonstrate that the SLR-based bounds are, in a complexity-theoretic sense, unimprovable systematically within their natural computational regimes.
