Numerical Solution Of Variational Inequality Problem And Its Related Theory

This paper studies numerical algorithms for variational inequality problem. A projection algorithm and a smoothing Newton algorithm are given. Both two algorithms are convergent under proper conditions.In chapter one, variational inequality problem and its relationship with nonlinear complementarity problem are given. Some relevant definitions are introduced.In chapter two, a new projection algorithm is given which is a reformation of an existing algorithm. The algorithm employs a new descent direction and a new step-size rule which ensures the step-size of each iteration is greater than some positive constant. It is proved that the iteration sequence produced by the algorithm converges to a solution of variational inequality under the condition that the mapping is pseudo-monotone. Numerical experiments show the efficiency of the algorithm.In chapter three, a new smoothing approximation function is given first. Variational inequality is reformulated as an equivalent system of smooth equations based on the smoothing approximation function. A new smoothing Newton algorithm is proposed and the algorithm is convergent. Numerical results show the algorithm is effective.