On semidefinite linear complementarity problems
type: Item
instance of: scholarly article
title: On semidefinite linear complementarity problems
zbMATH Open document ID: 0987.90082
author: M. Seetharama Gowda
author: Yoon J. Song
published in: Mathematical Programming. Series A. Series B
publication date: 2002-06-17T00:00:00Z
review text: The paper deals with the SemiDefinite Linear Complementarity Problem (SDLCP\((L,S^n_+)\): find a matrix \(X \in S^n_+\) such that \(Y=L(x)+Q \in S^n_+\) and and \(\langle X,Y\rangle=0\), where \(S^n\) (\(S^n_+\)) denote the set of symmetric (positive semidefinite) matrices, \(L: S^n \rightarrow S^n\) is a linear transformation, \(Q \in S^n\) and \(\langle X,Y\rangle\) denotes the trace of the matrix \(XY\). In this paper several LCP related concepts are extended to SDLCP. The well known LCP properties (\(R_0\) \(Q_0\)) are extended to linear transformations, as well as semi-monotone , strictly semi-monotone, column sufficiency, cross commutative, and variations of the P-property (so called \(P_1\) and \(P_2\)) -properties). The authors consider in particular these properties for Lyapunov transformation \(L_A=AX+XA^T.\) It is shown the equivalence between some of these properties for \(L_A\), and it is also proven that the P-property (the Q-property) is equivalent to \(A\) being a positive stable (i.e., real parts of eigenvalues of \(A\) are positive). As a special case, a theorem of Lyapunov is deduced. As shown by a counterexample, the P-property of \(L\) does not imply the uniqueness of a solutions in the SDLCP. So, in order to address the unique solvability of SDLCP the author introduces the globally uniquely solvable property (GUS-property) of a linear transformation \(L\). It is proven that \(L_A\) has the GUS property iff A is positive stable (i.e., and positive semidefinite).
reviewed by: S. M. Allende-Alonso
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 93D05
zbMATH DE Number: 1538620
zbMATH Keywords: semidefinite linear complementarity problem
zbMATH Keywords: P-property
zbMATH Keywords: GUS-property
zbMATH Keywords: Lyapunov theorem
MaRDI profile type: Publication
DOI: 10.1007/S101070000152
description: scientific article; zbMATH DE number 1538620