Existence and Limiting Behavior of Trajectories Associated with P0-equations

Abstract

Given a continuous P0-function F : Rn → Rn, we describe a method of constructing trajectories associated with the P0-equation F(x) = 0. Various well known equation-based reformulations of the nonlinear complementarity problem and the box variational inequality problem corresponding to a continuous P0-function lead to P0-equations. In particular, reformulations via (a) the Fischer function for the NCP, (b) the min function for the NCP, (c) the fixed point map for a BVI, and (d) the normal map for a BVI give raise to P0-equations when the underlying function is P0. To generate the trajectories, we perturb the given P0-function F to a P-function F(x, ε); unique solutions of F(x, ε) = 0 as ε varies over an interval in (0, ∞) then define the trajectory. We prove general results on the existence and limiting behavior of such trajectories. As special cases we study the interior point trajectory, trajectories based on the fixed point map of a BVI, trajectories based on the normal map of a BVI, and a trajectory based on the aggregate function of a vertical nonlinear complementarity problem.