We consider the multilinear polytope, defined as the convex hull of the feasible region of a lifted binary polynomial optimization problem. We define a relaxation in an extended space for this polytope, which we call the complete edge relaxation. The complete edge relaxation is stronger than several well-known relaxations of the multilinear polytope, including the standard linearization, the flower relaxation, and the intersection of all possible recursive McCormick relaxations. In addition, for fixed-degree binary polynomial optimization problems, the case of primary practical interest, the complete edge relaxation is of polynomial size and is computationally efficient in practice. We prove that the complete edge relaxation is an extension of the multilinear polytope if and only if the corresponding hypergraph is alpha-acyclic, the most general type of hypergraph acyclicity. This is in stark contrast with the widely-used standard linearization, which describes the multilinear polytope if and only if the hypergraph is Berge-acyclic, the most restrictive type of hypergraph acyclicity. Finally, we introduce a new class of facet-defining inequalities for the multilinear polytope of alpha-cycles of length three, which serve as the generalization of the well-known triangle inequalities for the Boolean quadric polytope.

翻译：我们考虑多重线性多面体，定义为提升二元多项式优化问题可行区域的凸包。我们在此多面体的扩展空间中定义了一种松弛，称之为完全边松弛。完全边松弛强于多重线性多面体的多个著名松弛，包括标准线性化、花形松弛以及所有可能的递归McCormick松弛的交集。此外，对于固定阶数的二元多项式优化问题（实际应用中的主要情形），完全边松弛具有多项式规模且在计算上高效。我们证明，完全边松弛是多重线性多面体的扩展当且仅当对应的超图是alpha-无环的——这是最广泛的超图无环性类型。这与广泛使用的标准线性化形成鲜明对比：后者描述多重线性多面体当且仅当超图是Berge-无环的——这是限制最严格的超图无环性类型。最后，我们为长度为三的alpha-环对应的多重线性多面体引入了一类新的定义面不等式，这些不等式推广了布尔二次多面体著名的三角不等式。