On homotopy-smoothing methods for box-constrained variational inequalities

A variational inequality problem with a mapping g : R-n --> R-n and lower a nd upper bounds on variables can be reformulated as a system of nonsmooth e quations F(x) = 0 in R-n. Recently, several homotopy methods, such as inter ior point and smoothing methods, have been employed to solve the problem. A ll of these methods use parametric functions and construct perturbed equati ons to approximate the problem. The solution to the perturbed system consti tutes a smooth trajectory leading to the solution of the original variation al inequality problem. The methods generate iterates to follow the trajecto ry. Among these methods Chen-Mangasarian and Gabriel-More proposed a class of smooth functions to approximate F. In this paper, we study several prope rties of the trajectory defined by solutions of these smooth systems. We pr opose a homotopy-smoothing method for solving the variational inequality pr oblem, and show that the method converges globally and superlinearly under mild conditions. Furthermore, if the involved function g is an affine funct ion, the method finds a solution of the problem in finite steps. Preliminar y numerical results indicate that the method is promising.