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2014 A General Approximation Method for a Kind of Convex Optimization Problems in Hilbert Spaces
Ming Tian, Li-Hua Huang
J. Appl. Math. 2014: 1-9 (2014). DOI: 10.1155/2014/156073


The constrained convex minimization problem is to find a point x with the property that xC, and h(x)=min h(x), xC, 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 F1(x,y) defined as F1(x,y)=h(y)h(x). And we ensure the equilibrium problem for F1(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.


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Ming Tian. Li-Hua Huang. "A General Approximation Method for a Kind of Convex Optimization Problems in Hilbert Spaces." J. Appl. Math. 2014 1 - 9, 2014.


Published: 2014
First available in Project Euclid: 2 March 2015

zbMATH: 07131361
MathSciNet: MR3198359
Digital Object Identifier: 10.1155/2014/156073

Rights: Copyright © 2014 Hindawi


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