On Homotopy-Smoothing Methods for Box-Constrained Variational Inequalities

A variational inequality problem with a mapping $g:\Re^n \to \Re^n$ and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations $F(x)=0$ in $\Re^n$. Recently, several homotopy methods, such as interior point and smoothing methods, have been employed to solve the problem. All of these methods use parametric functions and construct perturbed equations to approximate the problem. The solution to the perturbed system constitutes a smooth trajectory leading to the solution of the original variational inequality problem. The methods generate iterates to follow the trajectory. Among these methods Chen--Mangasarian and Gabriel--Moré proposed a class of smooth functions to approximate F. In this paper, we study several properties of the trajectory defined by solutions of these smooth systems. We propose a homotopy-smoothing method for solving the variational inequality problem, and show that the method converges globally and superlinearly under mild conditions. Furthermore, if the involved function g is an affine function, the method finds a solution of the problem in finite steps. Preliminary numerical results indicate that the method is promising.