Box-TDI多面体的离散凸Min-Max公式
A Discrete Convex Min-Max Formula for Box-TDI Polyhedra
摘要：
证明了一个求整值可分离散凸函数极小值的极小-极大公式，其中极小值取在盒全对偶积分（box-TDI）多面体的整元集合上.该定理的一个变体使用了共轭函数的概念（非线性最优化中的一个基本概念），但我们也提供了另一个版本，避免了共轭，其精神在概念上更接近组合优化中经典极小-极大定理的标准式。所提出的框架提供了一个统一的背景下，可分离的凸最小化的整元的交集的两个完整的基多面体，子模流，L-凸集，和多面体定义的全么模（TU）矩阵。作为一个意想不到的应用程序，我们展示了如何广泛的一类逆组合最优化问题可以覆盖这个新的框架。
英文摘要：
A min-max formula is proved for the minimum of an integer-valued separable discrete convex function where the minimum is taken over the set of integral elements of a box total dual integral (box-TDI) polyhedron. One variant of the theorem uses the notion of conjugate function (a fundamental concept in non-linear optimization) but we also provide another version that avoids conjugates, and its spirit is conceptually closer to the standard form of classic min-max theorems in combinatorial optimization. The presented framework provides a unified background for separable convex minimization over the set of integral elements of the intersection of two integral base-polyhedra, submodular flows, L-convex sets, and polyhedra defined by totally unimodular (TU) matrices. As an unexpected application, we show how a wide class of inverse combinatorial optimization problems can be covered by this new framework.
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