Mathematical Programming

, Volume 147, Issue 1–2, pp 155–170 | Cite as

On the bilinearity rank of a proper cone and Lyapunov-like transformations

  • M. Seetharama Gowda
  • Jiyuan Tao
Full Length Paper Series A


A real square matrix \(Q\) is a bilinear complementarity relation on a proper cone \(K\) in \(\mathbb{R }^n\) if
$$\begin{aligned} x\in K, s\in K^*,\,\,\text{ and }\,\,\langle x,s\rangle =0\Rightarrow x^{T}Qs=0, \end{aligned}$$
where \(K^*\) is the dual of \(K\). The bilinearity rank of \(K\) is the dimension of the linear space of all bilinear complementarity relations on \(K\). In this article, we continue the study initiated by Rudolf et al. (Math Prog Ser B 129:5–31, 2011). We show that bilinear complementarity relations are related to Lyapunov-like transformations that appear in dynamical systems and in complementarity theory and further show that the bilinearity rank of \(K\) is the dimension of the Lie algebra of the automorphism group of \(K\). In addition, we correct a result of Rudolf et al., compute the bilinearity ranks of symmetric and completely positive cones, and state some Schur-type results for Lyapunov-like transformations.


Proper cone Primal-dual linear program Complementarity   Lyapunov-like transformation Bilinearity rank Symmetric cone  Completely positive cone 

Mathematics Subject Classification

17A15 17B40 47L07 90C33 90C46 93C05 



The authors wish to thank Professor Bit-Shun Tam, Tamkang University, Taiwan, for allowing us to include his proof of Corollary 3. Thanks are also due to the referees for their constructive comments and suggestions.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Maryland, Baltimore CountyBaltimoreUSA
  2. 2.Department of Mathematics and StatisticsLoyola University MarylandBaltimoreUSA

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