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

- 124 Downloads
- 4 Citations

## 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

- 1.Cottle, R.W., Pang, J.-S., Stone, R.: The Linear Complementarity Problem. SIAM, Philadelphia (2009)CrossRefMATHGoogle Scholar
- 2.Gowda, M.S., Sznajder, R.: Some global uniqueness and solvability results for linear complementarity problems over symmetric cones. SIAM J. Optim.
**18**, 461–481 (2007)MathSciNetCrossRefMATHGoogle Scholar - 3.Gowda, M.S., Tao, J.: On the bilinearity rank of a proper cone and Lyapunov-like transformations. Math. Program.
**147**, 155–170 (2014)MathSciNetCrossRefMATHGoogle Scholar - 4.Gowda, M.S., Trott, D.: On the irreducibility, Lyapunov rank, and automorphisms of special Bishop–Phelps cones. J. Math. Anal. Appl.
**419**, 172–184 (2014)MathSciNetCrossRefMATHGoogle Scholar - 5.Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefMATHGoogle Scholar
- 6.Rudolf, G., Noyan, N., Papp, D., Alizadeh, F.: Bilinear optimality constraints for the cone of positive polynomials. Math. Program. Ser. B
**129**, 5–31 (2011)MathSciNetCrossRefMATHGoogle Scholar - 7.Schneider, H., Vidyasagar, M.: Cross-positive matrices. SIAM J. Numer. Anal.
**7**, 508–519 (1970)MathSciNetCrossRefMATHGoogle Scholar