Studies on Piecewise-Linear Approximations of Piecewise-C1 Mappings in Fixed Points and Complementarity Theory

Abstract

A PC1-mapping is a continuous mapping from a subset P of Rn into Rn, where P is partitioned into n-dimensional compact convex polyhedra. such that the restriction to each polyhedron is continuously differentiable. Based on the classical results on PL (piecewise linear) approximations of PC1-mappings given by Whitehead and an extension of the inverse function theorem to PC1-mappings, the following are investigated under nonsingularity conditions on piecewise linearizations of PC1-mappings with the use of their partial derivatives and conditions on the diameter and thickness of simplices on which PL approximations are affine: (1) One-to-one correspondence between solutions of a system of nonlinear equations and its PL approximations. (2) Monotone convergence of the continuous deformation method. (3) A lower bound of the convergence speed of the continuous deformation method. (4) Conditions which characterize local uniqueness of solutions to the nonlinear complementarity problem. (5) One-to-one correspondence between solutions of the nonlinear complementarity problem and their PL approximations. (6) Monotone convergence of the extended Lemke's method for PL approximations of the nonlinear complementarity problem.