有序优化中线性松弛的强度
On the Strength of Linear Relaxations in Ordered Optimization
AI总结：
本文研究有序优化中线性松弛的整数性条件，发现排序增加组合复杂性，松弛强度取决于与经典问题的接近度，实验验证了条件预测价值。
摘要：
我们研究了有序优化问题（其中排序是决策过程的一部分）的混合整数线性规划公式的凸松弛产生整数最优解的条件，从而在多项式时间内精确求解该问题。我们的分析识别了影响松弛整数性的输入数据的结构性质。我们表明，纳入有序组件引入了额外的组合复杂性层，这使得经典（非有序）设置中观察到的精确性失效。特别是，对于某些有序问题，如最小-最大情形，线性松弛永远无法恢复整数解。这些结果阐明了排序引入的内在难度，并揭示松弛的强度关键取决于有序问题与其经典对应问题的“接近程度”：更接近非有序情形的问题倾向于具有更紧的松弛，而那些偏离较远的问题则表现出显著更弱的行为。在基准实例上的计算实验证实了整数性条件的预测价值，并展示了精确松弛对有序选址问题的实际意义。
英文摘要：
We study the conditions under which the convex relaxation of a mixed-integer linear programming formulation for ordered optimization problems, where sorting is part of the decision process, yields integral optimal solutions. Thereby solving the problem exactly in polynomial time. Our analysis identifies structural properties of the input data that influence the integrality of the relaxation. We show that incorporating ordered components introduces additional layers of combinatorial complexity that invalidate the exactness observed in classical (non-ordered) settings. In particular, for certain ordered problems such as the min--max case, the linear relaxation never recovers the integral solution. These results clarify the intrinsic hardness introduced by sorting and reveal that the strength of the relaxation depends critically on the ``proximity'' of the ordered problem to its classical counterpart: problems closer to the non-ordered case tend to admit tighter relaxations, while those further away exhibit substantially weaker behavior. Computational experiments on benchmark instances confirm the predictive value of the integrality conditions and demonstrate the practical implications of exact relaxations for ordered location problems.
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