楼层布局问题的超越SDP界：一个简单的组合思想
Beating the SDP bound for the floor layout problem: A simple combinatorial idea
摘要：
对于许多混合整数规划（MIP）问题，高质量的对偶边界可以通过先进的配方技术加上一个国家的最先进的MIP求解器，或通过半定规划（SDP）松弛层次。在本文中，我们介绍了一种替代的边界的方法，利用“组合内爆”的效果，解决部分的原始问题，并汇总这些信息，以获得一个全球性的双重约束。我们将这种技术应用到一维和二维的楼层布局问题，并将其与最先进的MIP求解器和SDP松弛产生的边界进行比较。具体来说，我们证明了通过所提出的技术获得的界限至少是通过SDP松弛获得的一样好，目前的计算结果，这些界限可以显着更强，更容易计算比这些替代策略，特别是对于非常困难的问题的情况下。
英文摘要：
For many mixed-integer programming (MIP) problems, high-quality dual bounds can be obtained either through advanced formulation techniques coupled with a state-of-the-art MIP solver, or through semidefinite programming (SDP) relaxation hierarchies. In this paper, we introduce an alternative bounding approach that exploits the "combinatorial implosion" effect by solving portions of the original problem and aggregating this information to obtain a global dual bound. We apply this technique to the one-dimensional and two-dimensional floor layout problems and compare it with the bounds generated by both state-of-the-art MIP solvers and by SDP relaxations. Specifically, we prove that the bounds obtained through the proposed technique are at least as good as those obtained through SDP relaxations, and present computational results that these bounds can be significantly stronger and easier to compute than these alternative strategies, particularly for very difficult problem instances.
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