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

Abstract

The constrained convex minimization problem is to find a point $x^{\ast }$ with the property that $x^{\ast }\in C$, and $h(x^{\ast })=\text{min}$ $h(x)$, $\forall x\in C$, where $C$ is a nonempty, closed, and convex subset of a real Hilbert space $H$, $h(x)$ is a real-valued convex function, and $h(x)$ 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 bifunction $F_1(x,y)$ defined as $F_1(x,y)=h(y)-h(x)$. And we ensure the equilibrium problem for $F_1(x,y)$ 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.