Mathematical Programming

, Volume 88, Issue 3, pp 575–587 | Cite as

On semidefinite linear complementarity problems

  • M. Seetharama Gowda
  • Yoon Song

Abstract.

Given a linear transformation L:?n→?n and a matrix Q∈?n, where ?n is the space of all symmetric real n×n matrices, we consider the semidefinite linear complementarity problem SDLCP(L,?n+,Q) over the cone ?n+ of symmetric n×n positive semidefinite matrices. For such problems, we introduce the P-property and its variants, Q- and GUS-properties. For a matrix ARn×n, we consider the linear transformation LA:?n→?n defined by LA(X):=AX+XAT and show that the P- and Q-properties for LA are equivalent to A being positive stable, i.e., real parts of eigenvalues of A are positive. As a special case of this equivalence, we deduce a theorem of Lyapunov.

Key words: semidefinite linear complementarity problem –P-property –GUS-property – Lyapunov 
Mathematics Subject Classification (2000): 90C33, 93D05 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • M. Seetharama Gowda
    • 1
  • Yoon Song
    • 2
  1. 1.Department of Mathematics & Statistics, University of Maryland, Baltimore County, Baltimore, Maryland 21250, USA, e-mail: gowda@math.umbc.edu, http://www.math.umbc.edu/˜gowdaUS
  2. 2.Department of Mathematics & Statistics, University of Maryland, Baltimore County, Baltimore, Maryland 21250, USA, e-mail: song@math.umbc.eduUS

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