Nonlinear complementarity as unconstrained optimization

Several methods for solving the nonlinear complementarity problem (NCP) are developed. These methods are generalizations of the recently proposed algorithms of Mangasarian and Solodov (Ref. 1) and are based on an unconstrianed minimization formulation of the nonlinear complementarity problem. It is shown that, under certain assumptions, any stationary point of the unconstrained objective function is already a solution of NCP. In particulr, these assumptions are satisfied by the mangasarian and Soolodov implicit Lagranian functioin. Furthermore, a special Newton-type method is suggested, and conditions for its local quadratic convergence are given. Finally, some preliminary numerical results are presented.The author would like to thank Dr. Oswald Knoth (Leipzig) for pointing out that the equivalence of Lemma 2.2. is not true for complementarity problems which have no solutions. He is also grateful to the anonymous referencees for their helpful comments.