An improved bound for the Lyapunov rank of a proper cone

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.