Some Results about the Isolated Calmness of a Mixed Variational Inequality Problem

It is well known that optimization problem model has many applications arising from matrix completion, image processing, statistical learning, economics, engineering sciences, and so on. And convex programming problem is closely related to variational inequality problem. The so-called alternative direction of multiplier method (ADMM) is an importance class of numerical methods for solving convex programming problem. When analyzing the rate of convergence of various ADMMs, an error bound condition is usually required. The error bound can be obtained when the isolated calmness of the inverse of the KKT mapping of the related problem holds at the given KKT point. This paper is to study the isolated calmness of the inverse KKT mapping onto the mixed variational inequality problem with nonlinear term defined by norm function and indicator function of a convex polyhedral set, respectively. We also consider the isolated calmness of the inverse KKT mapping onto classical variational inequality problem with equality and inequality constrains under strict Mangasarian-Fromovitz constraint qualification condition. The results obtained here are new and very interesting.