超越手工推导的不等式：二元多项式优化中割平面生成的决策图方法
Beyond Hand-Derived Inequalities: Decision Diagrams for Cut Generation in Binary Polynomial Optimization
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中文总结 AI 辅助
本文针对二元多项式优化的割平面生成问题，提出决策图方法，获多线性多面体扩展形式等成果，实验表明该方法可加速分支定界过程。
中文摘要 AI 辅助
我们研究二元多项式优化（BPO）的割平面生成问题，其可行域是超图的多线性集合。针对该集合的强不等式（如双链接、花形和奇β-环），传统方法是针对固定支撑模式手工推导得到。本文提出一种决策图（DD）方法：对于任意选定的支撑，该方法在局部多线性多面体中分离出一个面定义割，并将其提升回原问题。我们利用一种新颖的紧凑DD编码，该编码基于递归公式，仅表示顶点变量，并在状态表示中隐式编码超边变量。通过该DD编码，我们还得到：（i）多线性多面体的扩展形式；（ii）通过有效反链得到的宽度表征，该表征揭示了一类新的多项式可解超图；（iii）生成割的面性证明。我们探索了三种不同的割生成支撑策略，分别为重用、扩展或划分局部结构为截面超图。实验结果表明，与现有方法相比，我们的基于DD的方法在根节点处用更少的割实现了更大的间隙闭合，进而显著加速了分支定界过程。
英文摘要
We study cutting-plane generation for binary polynomial optimization (BPO), whose feasible region is the multilinear set of a hypergraph. Strong inequalities for this set---such as two-links, flowers, and odd $β$-cycles---are classically hand-derived for fixed support patterns. Instead, we propose a decision-diagram (DD) approach: for any chosen support, it separates a facet-defining cut in the local multilinear polytope and lifts it back to the original problem. We utilize a novel compact DD encoding based on a recursive formulation that represents only the vertex variables and implicitly encodes the hyperedge variables inside the state representation. From this DD encoding, we also obtain: (i) an extended formulation of the multilinear polytope, (ii) a width characterization via valid antichains that uncovers a new polynomially solvable class of hypergraphs, and (iii) a certificate for the facetness of the generated cuts. We explore three different support strategies for cut generation that reuse, expand, or partition local structures into section hypergraphs. Our empirical results show that our DD-based methodologies achieve a larger gap closure at the root node with fewer cuts than existing procedures and, in turn, markedly accelerate branch-and-bound procedures.
