Convergence and Error Bound of a Method for Solving Variational Inequality Problems via the Generalized D-Gap Function

The variational inequality problem (VIP) can be reformulated as an unconstrained minimization problem through the generalized D-gap function. Recently, a hybrid Newton-type method was proposed by Peng and Fukushima for minimizing a special form of the generalized D-gap function. In this paper, the hybrid Newton-type algorithm is extended to minimize the general form g of the generalized D-gap function. It is shown that the algorithm has nice convergence properties. Under some reasonable conditions, it is proved that the algorithm is locally and globally convergent. Moreover, it is proved that the function g has bounded level sets for strongly monotone VIP. An error bound of the algorithm is obtained.