# An improved bound for the Lyapunov rank of a proper cone

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## 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## References

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