Classes of Functions and Feasibility Conditions in Nonlinear Complimentarity Problems

Abstract

Given a mapping $F$ from real Euclidean n-space into itself, we investigate the connection between various known classes of functions and the nonlinear complementarity problem: Find and $x^{*} \geq 0$ such that $ F x^{*} \geq 0$ and is orthogonal to $x^{*}$. In particular, we study the extent to which the existence of a $u \geq 0$ with $F $u \geq 0$ (feasible point) implies the existence of a solution to the nonlinear complementarity problem, and extend, to nonlinear mappings, known results in the linear complementarity problem on P-matrices, diagonally dominant matrices with nonnegative diagonal elements, matrices with off-diagonal non-positive entries, and positive semidefinite matrices.