It has long been known that variational inequality problems can be reformulated as nonsmooth equations. Recently, locally high-order convergent Newton methods for nonsmooth equations have been well established via the concept of semismoothness. When the constraint set of the variational inequality problem is a rectangle, several locally convergent Newton methods for the reformulated nonsmooth e...

The object of this paper is to establish an existence result for nonlinear inequalities for not necessarily pseudomonotone maps. As a consequence of our result, we give some existence results for variational and variational-like inequalities. Keywords– Nonlinear inequalities, variational inequality problems, variational-like inequality problems, fixed points.

It is well known that the gradient-projection algorithm plays an important role in solving constrained convex minimization problems. In this paper, based on Xu’s method [Xu, H. K.: Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl. 150, 360-378(2011)], we use the idea of regularization to establish implicit and explicit iterative methods for finding the approximate ...

The analysis of the connections among generalized systems, Vector Optimization Problems and Variational Inequalities allows to deepen the study of the properties of the Minty Vector Variational Inequality. In particular, under suitable regularity assumptions,the equivalence between Minty Vector Variational Inequality and Stampacchia Vector Variational Inequality is shown.

Abstract In this article, we study a class of pseudomonotone split variational inequality problems (VIPs) with non-Lipschitz operator. We propose new inertial extragradient method self-adaptive step sizes for finding the solution to aforementioned problem in framework Hilbert spaces. Moreover, prove strong convergence result proposed algorithm without prior knowledge operator norm and under mil...