A Smoothing Newton Method For Variational Inequalities

This paper focuses on studying the smoothing Newton method for solving variational inequalities. By using the entropy function, a new computable smoothing function of the point to convex set projection operator is introduced. Unlike introducing m (m is the number of constraints) additional variables in exsiting computable smoothing function of the projection operator, only one additional variable is required in our method. It is proved that the presented smoothing function is monotone, nonexpansive, uniformly convergent and its gradient is symmetric, positive semidefinite. Based on this smoothing function, a new version of the smoothing Newton method for solving variational inequalities is presented. The method is suitable for the problem that the variational function is denned only on the feasible region. The method is globally convergent under the conditions that the function is monotone and Slater constraint qualification is satisfied, and also locally superlinearly convergent if the problem is CD-regular at the solution and linear independence constraint is satisfied. Numerical results are presented and show the algorithm is feasible and effective.The smoothing function H(x, ,z) of the K-K-T conditions of the variational inequality is also constructed by using the entropy function and the nonsingularity of H (x, , z)and V H(x, , z) is proved. The related Broyden-like quasi-Newton method is presented and the global convergence of the method is proved.