Linear complementarity problems with an invariant number of solutions

This paper studies the class INS of all realn × n matricesM for which the linear complementarity problem (q, M) has exactlyk solutions—k depending only onM—for all realn-vectorsq interior to the coneK(M) of vectors for which (q, M) has any solution at all. This generalizes the results in Cottle and Stone (1983) which deal with the subclassU in INS wherek equals one.After the first two sections of this paper, which introduce the problem and background material, we move on to examine necessary conditions for a matrixM to be in INS (Section 3) and sufficient conditions under whichM will be in INS (Section 4). Section 5 deals with the possible values whichk may have. Section 6 discusses related results concerning the geometry of linear complementarity problems. Finally, Section 7 deals with some known and new matrix classes which are in INS.