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 ) > 0. Here F is the mapping from R ~ " into itself defined by F(x, y) = ( x l y ,, x 2 y Z , . . . , x , ~ y e , y f f x ) ) . Under the assumption that the mapping f is a P,,-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 ) 8 0 until the parameter t > 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.