发表机构
Antai College of Economics and Management, Shanghai Jiao Tong University; Mitch Daniels School of Business, Purdue University（上海交通大学安泰经济与管理学院; 普渡大学米切尔·丹尼尔斯商学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对析取子模函数提出首个强多项式分离算法，并刻画其凸包络，应用于库存与0-1二次优化，循环不等式可大幅缩小松弛间隙。
AI 中文摘要
本文考虑析取子模函数的凸包络——这类函数在超立方体的面上具有格族子模性——并针对两个面析取的情形，构造了第一个强多项式时间分离算法。子模函数（其凸包络由Lovász扩展刻画）在构造组合与非线性优化问题的松弛中扮演基础角色。然而，除使用实际不可行的椭球算法外，析取子模函数的包络此前未被探索。我们的算法通过三步推导：将析取函数表示为两个扩展子模函数的最小值，引入变量提升技术，并在提升空间中构造次线性包络。本文还做出其他若干贡献。首先，我们为每个子模函数具有线性规划表述的情形提供析取公式。其次，我们推导了相交子模函数的闭式次线性包络刻画，为多产品库存销售最大化问题带来新的结构洞见。第三，我们在原始变量空间中完整刻画了定义在循环图上的双线性函数的凸包络。最后，我们通过计算表明，循环不等式对完全图和Hadamard图平均缩小约60%的间隙，对完全二部图缩小超过30%的间隙，对稀疏图（如仙人掌图和Halin图）缩小超过80%的间隙。所得松弛也比先前的扩展空间公式更易求解。
英文摘要
This paper considers convex envelopes of disjunctive submodular functions---functions that are lattice family submodular over faces of a hypercube---and constructs the first strongly polynomial algorithm for their separation when there are two facial disjunctions. Submodular functions, whose convex envelopes are characterized by the Lovász extension, have occupied a fundamental role in constructing relaxations for combinatorial and nonlinear optimization problems. However, disjunctive submodular function envelopes have not been explored besides the use of ellipsoid algorithm, which remains practically intractable. Our algorithm is derived in three steps by expressing the disjunctive function as a minimum of two extended submodular functions, introducing a variable lifting technique, and constructing the sublinear envelope in the lifted space. The paper also makes several other contributions. First, we provide a disjunctive formulation for the case where each submodular function admits a linear programming formulation. Second, we derive the closed-form sublinear envelope characterization for intersecting submodular functions, yielding new structural insights into a multi-product inventory sales maximization problem. Third, we fully characterize the convex envelope of a bilinear function defined over a cycle graph in the original variable space. Finally, we show computationally that the cycle inequalities close approximately 60\% of the gap for complete and Hadamard graphs, over 30\% of the gap for complete bipartite graphs, and over 80\% of the gap for sparse graphs such as cactus and Halin graphs. The resulting relaxations are also more efficient to solve than previous extended space formulations.
