HOMOTOPY CONTINUATION METHODS FOR NONLINEAR COMPLEMENTARITY-PROBLEMS

A complementarity problem with a continuous mapping f from R(n) into itself can be written as the system of equations F(x,y) = 0 and (x,y) greater-than-or-equal-to 0. Here F is the mapping from R2n into itself defined by F(x,y) = (x1y1, x2y2,...,x(n)y(n),y-f(x)). Under the assumption that the mapping f is a P0-function, we study various aspects of homotopy continuation methods that trace a trajectory consisting of solutions of the family of systems of equations F(x,y) = t(a,b) and (x,y) greater-than-or-equal-to 0 until the parameter t greater-than-or-equal-to 0 attains 0. Here (a,b) denotes a 2n-dimensional constant positive vector. We establish the existence of a trajectory which leads to a solution of the problem, and then present a numerical method for tracing the trajectory. We also discuss the global and local convergence of the method.