A General Approximation Method for a Kind of Convex Optimization Problems in Hilbert Spaces

Abstract

The constrained convex minimization problem is to find a pointx∗with the property thatx∗∈C, andh(x∗)=min  h(x),∀x∈C, whereCis a nonempty, closed, and convex subset of a real Hilbert spaceH,h(x)is a real-valued convex function, andh(x)is not Fréchet differentiable, but lower semicontinuous. In this paper, we discuss an iterative algorithm which is different from traditional gradient-projection algorithms. We firstly construct a bifunctionF1(x,y)defined asF1(x,y)=h(y)−h(x). And we ensure the equilibrium problem forF1(x,y)equivalent to the above optimization problem. Then we use iterative methods for equilibrium problems to study the above optimization problem. Based on Jung’s method (2011), we propose a general approximation method and prove the strong convergence of our algorithm to a solution of the above optimization problem. In addition, we apply the proposed iterative algorithm for finding a solution of the split feasibility problem and establish the strong convergence theorem. The results of this paper extend and improve some existing results.