Some New Results for the Semidefinite Linear Complementarity Problem
M. Seetharama Gowda, Yoon Song · SIAM Journal on Matrix Analysis and Applications · 2002
In this paper, we present some new results for the semidefinite linear complementarity problem (SDLCP). In the first part, we introduce the concepts of (i) nondegeneracy for a linear transformation $L:{\cal S}^n \rightarrow {\cal S}^n$ and (ii) the locally-star-like property of a solution point of an SDLCP(L,Q) for $Q\in {\cal S}^n$, and we relate them to the finiteness of the solution set of SDLCP(L,Q) as Q varies in ${\cal S}^n$. In the second part, we show that for positive stable matrices A 1 ,. . ., A k , the linear transformation L:=L A 1 \circ L_{A_2} \circ \cdots \circ L_{A_k} $ has the Q-property where L A i (X):= A i X + XA i T . A similar result is proved for the transformation $S:=S_{A_1} \circ S_{A_2} \circ \cdots \circ S_{A_k}$, where each A i is Schur stable and $S_{A_i}(X):=X-A_iXA_i^{T}$. We relate these results to the simultaneous stability of a finite set of matrices.