## Journal of Applied Mathematics

- J. Appl. Math.
- Volume 2014 (2014), Article ID 156073, 9 pages.

### 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.

#### Article information

**Source**

J. Appl. Math., Volume 2014 (2014), Article ID 156073, 9 pages.

**Dates**

First available in Project Euclid: 2 March 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.jam/1425305691

**Digital Object Identifier**

doi:10.1155/2014/156073

**Mathematical Reviews number (MathSciNet)**

MR3198359

#### Citation

Tian, Ming; Huang, Li-Hua. A General Approximation Method for a Kind of Convex Optimization Problems in Hilbert Spaces. J. Appl. Math. 2014 (2014), Article ID 156073, 9 pages. doi:10.1155/2014/156073. https://projecteuclid.org/euclid.jam/1425305691