Partially relaxed cocoercive variational inequalities and auxiliary problem principle
Ram U. Verma · International Journal of Stochastic Analysis · 2004
Let T : K → H be a mapping from a nonempty closed convex subset K of a finite‐dimensional 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.