An Accelerated Newton Method for Equations with Semismooth Jacobians and Nonlinear Complementarity Problems: Extended Version

We discuss local convergence of Newton’s method to a singular solution x∗ of the nonlinear equations F (x) = 0, for F : IR → IR. It is shown that an existing proof of Griewank, concerning linear convergence to a singular solution x∗ from a starlike domain around x∗ for F twice Lipschitz continuously differentiable and x∗ satisfying a particular regularity condition, can be adapted to the case in which F ′ is only strongly semismooth at the solution. Further, under this regularity assumption, Newton’s method can be accelerated to produce fast linear convergence to a singular solution by overrelaxing every second Newton step. These results are applied to a nonlinear-equations formulation of the nonlinear complementarity problem (NCP) whose derivative is strongly semismooth when the function f arising in the NCP is sufficiently smooth. Conditions on f are derived that ensure that the appropriate regularity conditions are satisfied for the nonlinear-equations formulation of the NCP at x∗.