A Semi-Smooth Newton Method For Non-Smooth Equations

Non-smooth optimization is also called non-differential optimization. It's an important branch of the theory and method of optimization. Because of being not continuously differentiable, the theories and methods based on the concept of inte-gration of differential optimization no longer apply to non-differentiable problem. To promote the classic concept of differential, establish various generalized differen-tial concept and the corresponding optimal theories and methods, it is non-smooth optimization study. So far, we have no effective methods general form of non-smooth optimization problem. Only according to the special form of non-smooth optimiza-tion problem, respectively investigate. In various types of non-smooth optimization, convex programming and local Lipschitz programming have the biggest influence now, and the most widely accepted for non-smooth optimization problem.In this paper we use a kind of semi-smooth Newton algorithm which is of super-linear convergence to solve the variational inequalities with the second order cone. In certain conditions, the choice of descent direction, it's the focus of this paper.Firstly, transform the variational inequality into non-smooth variational in-equality equations, and suppose F(x) is C1 smooth. Assuming semi-smoothness it is shown that super-linearly convergent Newton methods can be globalized. Then in Chapter 3, the main part of paper, we calculate the directional derivative of F. The main method is the second cone decomposition, similarly with spectral decomposi-tion. In Chapter 4, we analysis the direction of directional derivative d can not be the descent direction, because it do not ensure the global super-linear convergence. But the d of G(x;d), which satisfies certain condition can be the descent direction. Then we introduce the concept of the quasi-directional derivative and given the expression of G(x;d) and prove that it is the quasi-directional derivative of F.