Smoothing Newton Methods with Global and Local Superlinear Convergence

In a series of papers—specifically, Xiaojun Chen, Liqun Qi and Defeng Sun, “Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities”, Mathematics of Computation, 67 (1998), pp. 519-540, Liqun Qi, Defeng Sun and Guanglu Zhou, “A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities”, Mathematical Programming, 87 (2000), 1–35, and Defeng Sun, “A regularization Newton method for solving nonlinear complementarity problems”, Applied Mathematics and Optimization, 40 (1999), 315-339—we have developed globally convergent smoothing Newton methods that achieve local superlinear (or quadratic) convergence for solving semismooth equations under mild conditions. These methods extend the classical Newton methods for smooth equations to a broader class of problems.