框约束下对称多线性多项式的多面体分析
Polyhedral Analysis of Symmetric Multilinear Polynomials over Box Constraints
摘要：
众所周知，盒上多线性多项式的凸包络和凹包络都是多面体函数。用于这些包络的指数大小的扩展和投影公式也是已知的。我们考虑关于变量排列对称的多线性多项式的凸化问题。这样一个置换不变的结构自然意味着一个二次规模的扩展制定的信封，通过使用析取编程。优化和分离问题直接回答，而不使用此扩展。问题的对称性允许优化和分离问题直接回答，而不使用任何扩展。它还意味着置换核心刻面集的系数生成所有刻面。给出了一个有效不等式成为核心分面的必要条件和充分条件。这些条件适用于获得两类包络：对称超模函数和反射对称的多线性单项式，从而产生替代文献的证明。此外，我们使用重构线性化技术的结构，以完全表征位于每个面上的点集。
英文摘要：
It is well-known that the convex and concave envelope of a multilinear polynomial over a box are polyhedral functions. Exponential-sized extended and projected formulations for these envelopes are also known. We consider the convexification question for multilinear polynomials that are symmetric with respect to permutations of variables. Such a permutation-invariant structure naturally implies a quadratic-sized extended formulation for the envelopes through the use of disjunctive programming. The optimization and separation problems are answered directly without using this extension. The problem symmetry allows the optimization and separation problems to be answered directly without using any extension. It also implies that permuting the coefficients of a core set of facets generates all the facets. We provide some necessary conditions and some sufficient conditions for a valid inequality to be a core facet. These conditions are applied to obtain envelopes for two classes: symmetric supermodular functions and multilinear monomials with reflection symmetry, thereby yielding alternate proofs to the literature. Furthermore, we use constructs from the reformulation-linearization-technique to completely characterize the set of points lying on each facet.
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