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 ,~ ye, y -ffx)). 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. ) is defined to be the problem of finding a z E R2" such that z = (x, y) 2 0, y = ffx), and xiyi = 0 (i = 1,2,. . . , n). Under the nonnegativity condition z = (x, y) 0, the complementarity condition x, y, = 0 (i = 1,2,. . . , n) can be rewritten as the condition that the inner product x . y = xTy is equal to zero. We say that the CP[f] is linear if the mapping f is a linear mapping of the form Rx) = Mx + q for some n x n matrix M and q E Rn, and nonlinear otherwise. A feasible solution is a z = (x, y) E R2" satisfying the nonnegativity condition z = (x, y) 2 0 and the equality y = ffx). This paper studies homotopy continuation methods for nonlinear complementarity problems, which were originally developed for linear programs (Gonzaga [31, Kojima, Mizuno, and Yoshise [12]
Introduction. Let
