Convergence of a non-interior smoothing method for variational inequality problems

The variational inequality problem can be reformulated as a system of equations. One can solve the reformulated equations to obtain a solution of the original problem. In this paper, based on a symmetric perturbed min function, we propose a new smoothing function, which has some nice properties. By which we propose a new non-interior smoothing algorithm for solving the variational inequality problem, which is based on both the non-interior continuation method and the smoothing Newton method. The proposed algorithm only needs to solve at most one system of equations at each iteration. In particular, we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions. The preliminary numerical results are reported.