Partially relaxed cocoercive variational inequalities and auxiliary problem principle (Q1774442)

Summary: Let \(T:K\to H\) be a mapping from a nonempty closed convex subset \(K\) of a finite-dimensional Hilbert space \(H\) into \(H\). Let \(f:K\to\mathbb{R}\) be proper, convex, and lower semi-continuous on \(K\) and let \(h:K\to\mathbb{R}\) be continuously Fréchet-differentiable on \(K\) with \(h'\) (gradient of \(h)\) \(\alpha\)-strongly monotone and \(\beta\)-Lipschitz continuous on \(K\). Then the sequence \(\{x^k\}\) generated by the general auxiliary problem principle converges to a solution \(x^*\) of the variational inequality problem described as follows: find an element \(x^*\in K\) such that \(\langle T(x^*),x-x^*\rangle+f(x)-f(x^*)>0\) for all \(x\in K\).