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Existence and limiting behavior of trajectories associated with $${\mathbf P}_0$$-equations. (English) Zbl 1040.90563
Summary: Given a continuous $${\mathbf P}_0$$-function $$F: \mathbb{R}^n \rightarrow \mathbb{R}^n$$, we describe a method of constructing trajectories associated with the $${\mathbf P}_0$$-equation $$F(x) = 0$$. Various well known equation-based reformulations of thenonlinear complementarity problem and the box variational inequality problem corresponding to a continuous $${\mathbf P}_0$$-function lead to $${\mathbf P}_0$$-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 $${\mathbf P}_0$$-equations when the underlying function is $${\mathbf P}_0$$. To generate the trajectories, we perturb the given $${\mathbf P}_0$$-function for $$F$$ to a $${\mathbf P}$$-function $$F (x, \varepsilon)$$; unique solutions of $$F (x, \varepsilon) = 0$$ as $$\varepsilon$$ varies over an interval in $$(0, \infty)$$ 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.

##### MSC:
 90C51 Interior-point methods 65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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