### (DP 1996-07) A Note on the Parametric Linear Complementarity Problem

#### Abstract

Consider the parametric linear complementarily problem, w = Mz + q + ¦Ëp, w ¡Ý 0, z ¡Ý 0, where p ¡Ù 0, 0 ¡Ù q ¡Ý 0, and ¦Ë ¡Ý 0. We show that a necessary condition for every complementary map z(¦Ë) to be isotone for every nonzero q ¡Ý 0 and every p is that M be either a P-matrix or P1*-matrix (M є P1*iff M є P1Q and det(M) = 0). Cottle's necessary and sufficient conditions for strong and uniform isotonicity for P-matrices are restated for P1*-matrices.

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