Let ${\mathcal{K}}^n$ be the second-order cone in ${\mathbb{R}}^n$, where n ≥ ~3. Given a vector $q\in {\mathbb{R}}^n$ and an n × n matrix G, the second order cone linear complementarity problem SOLCP(G, q) is to find a vector $x\in {\mathbb{R}}^n$ such that $x\in {\mathcal{K}}^n,y:=Gx+q\in {\mathcal{K}}^n\text{and}{x}^{T}y=0.$The matrix G is said to have the Q-property if SOLCP(G, q) has a solution for all $q\in {\mathbb{R}}^n$. An n × n matrix G is called a cone automorphism if $G{\mathcal{K}}^n={\mathcal{K}}^n$. In this paper, we obtain a simple characterization for the Q-property of a cone automorphism. This says that G has the Q-property if and only if zero is the only solution to SOLCP(G, 0).

中文翻译：

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二阶锥线性互补问题中锥自同构Q性质的刻画

让${\mathcal{钾}}^n$是二阶锥${\mathbb{R}}^n$，其中n  ≥ ~3。给定一个向量$q\in {\mathbb{R}}^n$和一个n  ×  n矩阵G，二阶锥线性互补问题 SOLCP( G , q ) 是找到一个向量$X\in {\mathbb{R}}^n$这样$X\in {\mathcal{钾}}^n,是:=GX+q\in {\mathcal{钾}}^n\text{和}{X}^{吨}是=\mathrm{0。}$如果 SOLCP( G , q ) 对所有问题都有解，则称矩阵G具有 Q 属性$q\in {\mathbb{R}}^n$. n  ×  n矩阵G称为锥自同构，如果$G{\mathcal{钾}}^n={\mathcal{钾}}^n$. 在本文中，我们获得了锥自同构的 Q 性质的简单表征。这表示G具有 Q 性质当且仅当零是 SOLCP( G , 0) 的唯一解。