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Some iterative methods for finding fixed points and for solving constrained convex minimization problems

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2011年第16期74卷 5286-5302页

作者： Ceng, L. -C. Ansari, Q. H. Yao, J. -C. Kaohsiung Med Univ Ctr Gen Educ Kaohsiung 80708 Taiwan Aligarh Muslim Univ Dept Math Aligarh 202002 Uttar Pradesh India Sci Comp Key Lab Shanghai Univ Shanghai 200041 Peoples R China Shanghai Normal Univ Dept Math Shanghai 200234 Peoples R China

The present paper is divided into two parts. In the first part, we introduce implicit and explicit iterative schemes for finding the fixed point of a nonexpansive mapping defined on the closed convex subset of a real Hilbert space. We establish results on the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping. Such a fixed point is also a solution of a variational inequality defined on the set of fixed points. In the second part, we propose implicit and explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our schemes converge strongly to a solution of the constrained convex minimization problem. Such a solution is also a solution of a variational inequality defined over the set of fixed points of a nonexpansive mapping. The results of this paper extend and improve several results presented in the literature in the recent past. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Iterative schemes Variational inequality Fixed point constrained convex minimization Nonexpansive mapping

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A convergence analysis result for constrained convex minimization problem (vol 64, pg 2587, 2015)

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OPTIMIZATION 2015年第12期64卷 2599-2599页

作者： Shehu, Yekini Univ Nigeria Dept Math Nsukka Nigeria

Our purpose in this paper is to obtain strong convergence result for approximation of solution to constrained convex minimization problem using a new iterative scheme in a real Hilbert space. Furthermore, we give nume... 详细信息

关键词： gradient-projection method constrained convex minimization strong convergence 47H09 47J25

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A General Iterative Method for Solving constrained convex minimization Problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第1期162卷 202-207页

作者： Tian, Ming Li, Min-Min Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

It is well known that the gradient-projection algorithm plays an important role in solving minimization problems. In this paper, we will use the idea of regularization to establish a general method so that the sequence generated by the general method can be strongly convergent to a minimizer of constrained convex minimization problems, which solves a variational inequality under suitable conditions.

关键词： Variational inequality Regularization algorithm constrained convex minimization Fixed point Averaged mapping Nonexpansive mappings

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General iterative scheme based on the regularization for solving a constrained convex minimization problem

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2013年第1期2013卷 1-15页

作者： Tian, Ming Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

It is well known that the regularization method plays an important role in solving a constrained convex minimization problem. In this article, we introduce implicit and explicit iterative schemes based on the regularization for solving a constrained convex minimization problem. We establish results on the strong convergence of the sequences generated by the proposed schemes to a solution of the minimization problem. Such a point is also a solution of a variational inequality. We also apply the algorithm to solve a split feasibility problem.

关键词： averaged mapping gradient-projection algorithm constrained convex minimization regularization split feasibility problem variational inequality

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Iterative algorithms based on the viscosity approximation method for equilibrium and constrained convex minimization problem

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FIXED POINT THEORY AND APPLICATIONS 2012年第1期2012卷 1-17页

作者： Tian, Ming Liu, Lei Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

The gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. Based on the viscosity approximation method, we combine the GPA and averaged mapping approach to propose implicit and explicit composite iterative algorithms for finding a common solution of an equilibrium and a constrained convex minimization problem for the first time in this paper. Under suitable conditions, strong convergence theorems are obtained. MSC: 46N10, 47J20, 74G60.

关键词： iterative algorithm equilibrium problem constrained convex minimization variational inequality

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Iterative methods for constrained convex minimization problem in Hilbert spaces

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FIXED POINT THEORY AND APPLICATIONS 2013年第1期2013卷 1-18页

作者： Tian, Ming Huang, Li-Hua Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

In this paper, based on Yamada's hybrid steepest descent method, a general iterative method is proposed for solving constrained convex minimization problem. It is proved that the sequences generated by proposed implicit and explicit schemes converge strongly to a solution of the constrained convex minimization problem, which also solves a certain variational inequality.

关键词： iterative algorithm constrained convex minimization nonexpansive mapping fixed point variational inequality

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Methods for solving constrained convex minimization problems and finding zeros of the sum of two operators in Hilbert spaces

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2015年第1期2015卷 1-27页

作者： Tian, Ming Jiao, Si-Wen Liou, Yeong-Cheng Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China Civil Aviat Univ China Tianjin Key Lab Adv Signal Proc Tianjin 300300 Peoples R China Cheng Shiu Univ Dept Informat Management Kaohsiung 833 Taiwan Kaohsiung Med Univ Ctr Fundamental Sci Kaohsiung 807 Taiwan

In this paper, let H be a real Hilbert space and let C be a nonempty, closed, and convex subset of H. We assume that , where is an alpha-inverse-strongly monotone mapping, is a maximal monotone operator, the domain of B is included in C. Let U denote the solution set of the constrained convex minimization problem. Based on the viscosity approximation method, we use a gradient-projection algorithm to propose composite iterative algorithms and find a common solution of the problems which we studied. Then we regularize it to find a unique solution by gradient-projection algorithm. The point which we find solves the variational inequality , . Under suitable conditions, the constrained convex minimization problem can be transformed into the split feasibility problem. Zeros of the sum of two operators can be transformed into the variational inequality problem and the fixed point problem. Furthermore, new strong convergence theorems and applications are obtained in Hilbert spaces, which are useful in nonlinear analysis and optimization.

关键词： iterative method fixed point constrained convex minimization maximal monotone operator resolvent inverse-strongly monotone mapping strict pseudo-contraction variational inequality

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ON THE RELAXED HYBRID-EXTRAGRADIENT METHOD FOR SOLVING constrained convex minimization PROBLEMS IN HILBERT SPACES

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TAIWANESE JOURNAL OF MATHEMATICS 2013年第3期17卷 911-936页

作者： Ceng, L. C. Chou, C. Y. Shanghai Normal Univ Dept Math Shanghai 200234 Peoples R China Sci Comp Key Lab Shanghai Univ Shanghai 200234 Peoples R China Natl Dong Hwa Univ Dept Appl Math Hualien 97401 Taiwan

In 2006, Nadezhkina and Takahashi [N. Nadezhkina, W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16(4) (2006), 1230-1241.] introduced an iterative algorithm for finding a common element of the fixed point set of a nonexpansive mapping and the solution set of a variational inequality in a real Hilbert space via combining two well-known methods: hybrid and extragradient. In this paper, motivated by Nadezhkina and Takahashi's hybrid-extragradient method we propose and analyze a relaxed hybrid-extragradient method for finding a solution of a constrained convex minimization problem, which is also a common element of the solution set of a variational inclusion and the fixed point set of a strictly pseudocontractive mapping in a real Hilbert space. We obtain a strong convergence theorem for three sequences generated by this algorithm. Based on this result, we also construct an iterative algorithm for finding a solution of the constrained convex minimization problem, which is also a common fixed point of two mappings taken from the more general class of strictly pseudocontractive mappings.

关键词： constrained convex minimization Variational inclusion Variational inequality Nonexpansive mapping Inverse strongly monotone mapping Maximal monotone mapping Strong convergence

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Regularized gradient-projection methods for equilibrium and constrained convex minimization problems

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2013年第1期2013卷 1-22页

作者： Tian, Ming Huang, Li-Hua Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

In this article, based on Marino and Xu's method, an iterative method which combines the regularized gradient-projection algorithm (RGPA) and the averaged mappings approach is proposed for finding a common solution of equilibrium and constrained convex minimization problems. Under suitable conditions, it is proved that the sequences generated by implicit and explicit schemes converge strongly. The results of this paper extend and improve some existing results.

关键词： iterative method constrained convex minimization equilibrium fixed point variational inequality

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Iterative algorithms with the regularization for the constrained convex minimization problem and maximal monotone operators

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OPTIMIZATION 2017年第10期66卷 1623-1646页

作者： Khuangsatung, Wongvisarut Kangtunyakarn, Atid Rajamangala Univ Technol Thanyaburi Fac Sci & Technol Dept Math & Comp Sci Pathum Thani Thailand King Mongkuts Inst Technol Ladkrabang Fac Sci Dept Math Bangkok Thailand

In this paper, weprove a strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the set of solutions of a finite family of variational inclusion problems in Hilbert space. A strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the solution sets of a finite family of zero points of the maximal monotone operator problem in Hilbert space is also obtained. Using our main result, we have some additional results for various types of non-linear problems in Hilbert space.

关键词： constrained convex minimization resolvent operator maximal monotone operators

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