Journal of Applied Mathematics

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

Ming Tian and Li-Hua Huang

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Abstract

The constrained convex minimization problem is to find a point x with the property that x C , and h ( x ) = min    h ( x ) , x 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


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