The Implicit Complementarity Problem: Part II.

Abstract : Given an n by n matrix A,n-vector b and a mapping m from R superscript n into R superscript n, the implicit complementary problem is to find a vector x in R superscript n so that Ax + b or = 0, X or = m(x) and (Ax + b) (transposed) (x - m(x)) = 0. This is the first of two papers in which we study this complementarity problem via an implicitly defined mapping F which depends on a given splitting of the matrix A. In the present paper, we derive sufficient conditions for the problem to have a unique solution for each vector b and study the problem in connection with a least-element theory. (Author)