Optimization Letters

, Volume 10, Issue 1, pp 11–17 | Cite as

An improved bound for the Lyapunov rank of a proper cone

Original Paper

Abstract

Given a proper cone \(K\) in \(\mathbb {R}^{n}\) with its dual \(K^{*}\), the complementarity set of \(K\) is \(C\left( K\right) := \left\{ \left( \mathbf {x},\mathbf {s}\right) : \mathbf {x} \in K, \mathbf {s} \in K^{*}, \left\langle \mathbf {x},\mathbf {s} \right\rangle = 0 \right\} \). A matrix \(\mathbf {A}\) on \(\mathbb {R}^{n}\) is said to be Lyapunov-like on \(K\) if \(\left\langle \mathbf {A}\mathbf {x},\mathbf {s} \right\rangle = 0\) for all \(\left( \mathbf {x},\mathbf {s}\right) \in C\left( K\right) \). The set of all such matrices forms a vector space whose dimension \(\beta \left( K\right) \) is called the Lyapunov rank of \(K\). 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 that \(\beta \left( K\right) \) is bounded above by \(\left( n-1\right) ^{2}\), thereby improving the previously known bound of \(n^{2}-n\). We also show that when \(\beta \left( K\right) \ge n\), the complementarity set \(C\left( K\right) \) can be expressed in terms of \(n\) linearly-independent Lyapunov-like matrices.

Keywords

Complementarity Lyapunov rank Lyapunov-like matrix Perfect cone Second-order cone 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Maryland, Baltimore CountyBaltimoreUSA

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