Abstract

This paper considers a generalization of the “max-cut-polytope” \(\hbox {conv}\{\ xx^T\mid x\in {\mathbb {R}}^n, \ \ |x_k| = 1 \ \hbox {for} \ 1\le k\le n\}\) in the space of real symmetric \(n\times n\)-matrices with all-one diagonal to a complex “unit modulus lifting” \(\hbox {conv}\{xx^*\mid x\in {\mathbb {C}}^n, \ \ |x_k| = 1 \ \hbox {for} \ 1\le k\le n\}\) in the space of complex Hermitian \(n\times n\)-matrices with all-one diagonal. The unit modulus lifting arises in applications such as digital communications and shares similar symmetry properties as the max-cut-polytope. Set-completely positive representations of both sets are derived and the relation of the complex unit modulus lifting to its semidefinite relaxation is investigated in dimensions 3 and 4. It is shown that the unit modulus lifting coincides with its semidefinite relaxation in dimension 3 but not in dimension 4. In dimension 4 a family of deep valid cuts for the unit modulus lifting is derived that could be used to strengthen the semidefinite relaxation. It turns out that the deep cuts are also implied by a second lifting that could be used alternatively. Numerical experiments are presented comparing the first lifting, the second lifting, and the unit modulus lifting for \(n=4\).

中文翻译：

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最大切割多面体和单位模量提升的集合完全正表示和切割

摘要

本文考虑了{ max-cut-polytope“ \（\ hbox {conv} \ {\ xx ^ T \ mid x \ in {\ mathbb {R}} ^ n，\ \ | x_k | = 1 \ \ hbox中为{} \ 1 \文件ķ\了N \} \）在实对称的空间\（N \ n次\）与所有酮对角线到一个复杂的“单位模量提升” -matrices \（\ hbox中{CONV} \ {XX ^ * \中间X \在{\ mathbb {C}} ^ N，\ \ | X_K | = 1 \ \ hbox中{为} \ 1 \文件ķ\了N \} \）在复Hermitian \（n \ times n \）的空间对角为一的矩阵。单位模量提升出现在诸如数字通信之类的应用中，并且具有与max-cut-polytope相似的对称性。推导了两个集合的集合完全正表示，并研究了维数3和4的复数单位模量提升与其半确定松弛的关系。结果表明，单位模量提升与其在维度3上的半确定松弛一致，但在维度4。在维度4中，得出了用于单位模量提升的深层有效切口系列，可用于增强半确定松弛。事实证明，第二种提升也可以暗示深切。给出了数值实验，比较了第一提升，第二提升和单位模量提升。\（n = 4 \）。