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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 XS + n such that Y=L(x)+QS + n and and X,Y=0, where S n (S + n ) denote the set of symmetric (positive semidefinite) matrices, L:S n S n is a linear transformation, QS n and X,Y 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).


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