Approximate Newton Methodsfor Nonsmooth Equations 1

We develop general approximate Newton methods for solving Lips-chitz continuous equations by replacing the iteration matrix with a consistently approximated Jacobian thereby reducing the computation in the generalized Newton method. Locally superlinear convergence results are presented under moderate assumptions. To construct a consistently approximated Jacobian, we introduce two main methods: the classic diierence approximation method and the-generalized Jacobian method. The former can be applied to problems with speciic structures while the latter is expected to work well for general problems. Numerical tests show the two methods are eecient. Finally, a norm-reducing technique for the global convergence of the generalized Newton method is brieey discussed. 1 The authors are grateful to the referee and D. Q. Mayne for their insightful comments and constructive suggestions. Thanks go to L. Qi and M. Kojima for providing some helpful references and S. Leyyer for many helpful discussions.