Quasi-Newton Method For Variational Inequality

The main work of this thesis includes two parts.First,based on derivative-free line search proposed by Li and Fukushima,we present a new regularized smoothing quasi-Newton method for the box constrained variational inequality problems.We use only one smoothing approximation function Chen-Harker-Kanzow-Smale function and Robinson's normal equation to reformulate box constrained variational inequality as an equivalent smoothing equations to solve the problem. Nonsingularity of Jocabi matrix and boundedness of level set are proved under the assumption that F is P₀-function and regularity.Under approximate conditions,global convergence and superlinear convergence are proved.Numerical results are given which indicate that the algorithm is practically efficient.Also,we present a new smoothing quasi-Newton method for KKT system of the variational inequality problems.Make use of Chen-Harker-Kanzow-Smale to reformulate KKT system of variational inequality as an equivalent smoothing equations.Based on derivative-free line search for semismooth equations proposed by Li and Fukushima, we improve on the line search to make the thought and theoretical analysis of algorithm perfect.Under approximate conditions,global convergence is proved.