A General Iterative Method for Constrained Convex Minimization Problems in Hilbert Spaces

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 minimizer of a constrained convex minimization problem and prove that the sequences generated by our methods converge strongly to a solution of the constrained convex minimization problem. Such a solution is also a solution of a variational inequality defined over the set of the solutions of the constrained convex minimization problem. As application, we apply our main result to solve the split feasibility problem in Hilbert spaces. Key–Words: Variational inequality; Regularization algorithm; Constrained convex minimization; Fixed point; Averaged mapping; Nonexpansive mappings.