Partially Relaxed Cocoercive Variational Inequalities and Auxiliary Problem Principle

Let T : K → H be a mapping from a nonempty closed convex subset K of a finitedimensional Hilbert space H into H . Let f : K → R be proper, convex, and lower semicontinuous on K and let h : K → R be continuously Frećhet-differentiable on K with h′ (gradient of h), α-strongly monotone, and β-Lipschitz continuous on K . Then the sequence {xk} generated by the general auxiliary problem principle converges to a solution x∗ of the variational inequality problem (VIP) described as follows: find an element x∗ ∈ K such that 〈T(x∗),x− x∗〉+ f (x)− f (x∗) ≥ 0 for all x ∈ K .