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On semidefinite linear complementarity problems. (English) Zbl 0987.90082
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).

90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
93D05Lyapunov and other classical stabilities of control systems
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