Research On The Algorithms For Variational Inequalities Problems

The variational inequality, which is a generalization of the NCP, provides a broad unifying setting for the study of optimization and equilibrium problems and serves as the main computational framework for the practical solution of a host of continuum problems in the mathematical sciences. It has a wide range of important applications in engineering optimization, economic and traffic equilibriums and has heavy influence on the various mathematical field and computer sciences and so on. Due to the connections with the real-life economics, it is always a hot issue in finding efficient and robust methods for solving variational inequality problems for mathematicians and economists. In this paper, we focus on the study of methods for solving the KKT-conditions of the variational inequality.Firstly, the origin and historical development of variational inequality have been introduced. Then it analyzes the existing methods for solving the variational inequality problem. Also it presents the basic conceptions and related mathematical knowledge. Secondly, based on the optimization technique, this paper studies two new NCP-functions which are used to reformulate the KKT-conditions as semismooth system of equations which are further formulated as an unconstrained minimization problem. The semismooth Newton method is given and can be proved globally convergent and local superlinear convergent. The numerical results indicate that the algorithm is efficient. Moreover, considering a new smoothing function, the KKT-conditions are equivalent to a smoothing nonlinear system of equations, which can be solved by non-interior smoothing method under some mild assumptions. It also shows the algorithm is well-defined and convergent. The numerical experiment proves that the method is promising. Finally, the strengths and weakness of the algorithms are concluded.