An improved bound for the Lyapunov rank of a proper cone
An improved bound for the Lyapunov rank of a proper cone
复制标题
真锥李亚普诺夫等级的改进界限
DOI：
10.1007/s11590-015-0903-6
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发表时间：
2016
影响因子：
1.6
通讯作者：
M. Seetharama Gowda
中科院分区：
文献类型：
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作者：
Michael Orlitzky;M. Seetharama Gowda
给定一个适当的圆锥及其对偶，互补集为$$C\Left(K\Right)：=\Left(\mathbf{x}，\mathbf{S}\right)：\Mathbf{x}\in K，\Mathbf{S}\in K^{*}，\Left\Lange\Mathbf{x}，\mathbf{S}\Range=0\right\$$.据说，所有人都像李亚普诺夫一样。所有这类矩阵的集合形成一个向量空间，其维度称为Lyapunov秩of。这个数在圆锥优化和互补理论中很有用，因为它与表示互补集所需的线性无关双线性关系的个数有关。本文是鲁道夫等人发起的研究的继续。(数学课程系列B 129：5-31,2011)，并在高达和陶氏数学课程(数学课程147：155-170,2014)中继续学习。通过回答Gowda和Tao(数学程序147：155-170,2014)中提出的几个问题，我们证明了它在上界，从而改进了以前已知的界。我们还证明了当，互补集可以用线性早期独立的类Lyapunov矩阵来表示。
Given a proper coneinwith its dual, the complementarity set ofis $$C\left( K\right) := \left\{ \left( \mathbf {x},\mathbf {s}\right) : \mathbf {x} \in K, \mathbf {s} \in K^{*}, \left\langle \mathbf {x},\mathbf {s} \right\rangle = 0 \right\} $$. A matrixonis said to beLyapunov-likeoniffor all. The set of all such matrices forms a vector space whose dimensionis called theLyapunov rankof. This number is useful in conic optimization and complementarity theory, as it relates to the number of linearly-independent bilinear relations needed to express the complementarity set. This article is a continuation of the study initiated in Rudolf et al. (Math Program Ser B 129:5–31, 2011) and further pursued in Gowda and Tao (Math Program 147:155–170, 2014). By answering several questions posed in Gowda and Tao (Math Program 147:155–170, 2014), we show thatis bounded above by, thereby improving the previously known bound of. We also show that when, the complementarity setcan be expressed in terms oflinearly-independent Lyapunov-like matrices.