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The Linearized Bregman Method via split feasibility problems: Analysis and Generalizations

SIAM JOURNAL ON IMAGING SCIENCES 2014年第2期7卷 1237-1262页

作者： Lorenz, Dirk A. Schoepfer, Frank Wenger, Stephan TU Braunschweig Inst Anal & Algebra D-38092 Braunschweig Germany Carl von Ossietzky Univ Oldenburg Inst Math D-26111 Oldenburg Germany TU Braunschweig Inst Comp Graph D-38092 Braunschweig Germany

The linearized Bregman method is a method to calculate sparse solutions to systems of linear equations. We formulate this problem as a split feasibility problem, propose an algorithmic framework based on Bregman projections, and prove a general convergence result for this framework. Convergence of the linearized Bregman method will be obtained as a special case. Our approach also allows for several generalizations such as other objective functions, incremental iterations, incorporation of non-Gaussian noise models, and box constraints.

关键词： linearized Bregman method split feasibility problems Bregman projections sparse solutions

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On descent-projection method for solving the split feasibility problems

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JOURNAL OF GLOBAL OPTIMIZATION 2012年第3期54卷 627-639页

作者： Bnouhachem, Abdellah Noor, Muhammad Aslam Khalfaoui, Mohamed Sheng Zhaohan COMSATS Inst Informat Technol Dept Math Islamabad Pakistan Nanjing Univ Sch Management Sci & Engn Nanjing 210093 Jiangsu Peoples R China Ibn Zohr Univ ENSA Agadir Morocco King Saud Univ Coll Sci Dept Math Riyadh 11451 Saudi Arabia

Let Omega and C be nonempty, closed and convex sets in R (n) and R (m) respectively and A be an real matrix. The split feasibility problem is to find with Many problems arising in the image reconstruction can be formulated in this form. In this paper, we propose a descent-projection method for solving the split feasibility problems. The method generates the new iterate by searching the optimal step size along the descent direction. Under certain conditions, the global convergence of the proposed method is proved. In order to demonstrate the efficiency of the proposed method, we provide some numerical results.

关键词： Variational inequalities split feasibility problems Convergence Iterative methods

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Global and linear convergence of alternated inertial methods for split feasibility problems

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REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2021年第2期115卷 1-26页

作者： Shehu, Yekini Dong, Qiao-Li Liu, Lu-Lu Zhejiang Normal Univ Dept Math Jinhua 321004 Zhejiang Peoples R China Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

The focus of this paper is to introduce algorithms with alternated inertial step to solve split feasibility problems. We obtain global convergence of the sequences of iterates generated by the proposed methods under some appropriate conditions. When the split feasibility problem satisfies some bounded linear regularity property, we show that the generated sequences converge linearly. As far as we know, no linear convergence result has been obtained before now for algorithms with inertial steps to solve split feasibility problems in the literature. Our numerical experiments which include a real-world application to jointly constrained Nash equilibrium model show that our methods outperform some inertial methods and other related methods for split feasibility problems in the literature.

关键词： Alternated inertial CQ methods split feasibility problems Linear convergence Signal processing

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Viscosity Approximation Methods for a Class of Generalized split feasibility problems with Variational Inequalities in Hilbert Space

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2019年第8期40卷 902-923页

作者： Tian, Ming Jiang, Bing-Nan Civil Aviat Univ China Coll Sci Tianjin Peoples R China

Strong convergence theorem of viscosity approximation methods for nonexpansive mapping have been studied. We also know that CQ algorithm for solving the split feasibility problem (SFP) has a weak convergence result. In this paper, we use viscosity approximation methods and some related knowledge to solve a class of generalized SFP's with monotone variational inequalities in Hilbert space. We propose some iterative algorithms based on viscosity approximation methods and get strong convergence theorems. As applications, we can use algorithms we proposed for solving split variational inequality problems (SVIP), split constrained convex minimization problems and some related problems in Hilbert space.

关键词： Iterative algorithms split feasibility problems Strong convergence variational inequalities viscosity approximation

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Auxiliary Principle Technique for Solving split feasibility problems

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APPLIED MATHEMATICS & INFORMATION SCIENCES 2013年第1期7卷 221-227页

作者： Noor, Muhammad Aslam Noor, Khalida Inayat COMSATS Inst Informat Technol Dept Math Islamabad Pakistan

Let K and C be nonempty, closed and convex sets in R-n and R-m respectively and A be an m x n real matrix. The split feasibility problem is to find u is an element of K with Au is an element of C. Many problems arising in the image reconstruction can be formulated in this form. In this paper, we use the auxiliary principle technique to suggest and analyze some new iterative algorithms for solving the split feasibility problems. Our new algorithms include the previously known ones as special cases. We also study the convergence criteria of these algorithms under some weaker conditions. In this respect, our results present a refinement and improvement of the previously known results.

关键词： Variational inequalities split feasibility problems convergence iterative methods auxiliary principle technique

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Weak convergence theorems for split feasibility problems on zeros of the sum of monotone operators and fixed point sets in Hilbert spaces

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Fixed Point Theory and Applications 2016年第1期2017卷 1-17页

作者： Suwannaprapa, Montira Petrot, Narin Suantai, Suthep Department of Mathematics Faculty of Science Naresuan University Phitsanulok 65000 Thailand Department of Mathematics Faculty of Science Chiang Mai University Chiang Mai 50200 Thailand

In this paper, we consider a type of split feasibility problem by focusing on the solution sets of two important problems in the setting of Hilbert spaces that are the sum of monotone operators and fixed point problems. By assuming the existence of solutions, we provide a suitable algorithm for finding a solution point. Some important applications and numerical experiments of the considered problem and constructed algorithm are also discussed. © 2017, The Author(s).

关键词： fixed point problems inverse strongly monotone operator maximal monotone operators split feasibility problems weak convergence theorems

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A strong convergence theorem for equilibrium problems and split feasibility problems in Hilbert spaces

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FIXED POINT THEORY AND APPLICATIONS 2014年第1期2014卷 36-36页

作者： Tang, Jinfang Chang, Shih-sen Yuan, Fei Yibin Univ Dept Math Yibin 644007 Sichuan Peoples R China Yunnan Univ Finance & Econ Coll Stat & Math Kunming 650221 Yunnan Peoples R China Staffordshire Univ Fac Comp Engn & Technol Stafford ST18 0AD Staffs England

The main purpose of this paper is to introduce an iterative algorithm for equilibrium problems and split feasibility problems in Hilbert spaces. Under suitable conditions we prove that the sequence converges strongly to a common element of the set of solutions of equilibrium problems and the set of solutions of split feasibility problems. Our result extends and improves the corresponding results of some others.

关键词： equilibrium problems split feasibility problems strong convergence bounded linear operator fixed point

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Variable KM-like algorithms for fixed point problems and split feasibility problems

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FIXED POINT THEORY AND APPLICATIONS 2014年第1期2014卷 211-211页

作者： Latif, Abdul Sahu, Daya R. Ansari, Qamrul H. King Abdulaziz Univ Dept Math Jeddah 21413 Saudi Arabia Banaras Hindu Univ Dept Math Varanasi 221005 Uttar Pradesh India Aligarh Muslim Univ Dept Math Aligarh 202002 Uttar Pradesh India King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran 31261 Saudi Arabia

The convergence analysis of a variable KM-like method for approximating common fixed points of a possibly infinitely countable family of nonexpansive mappings in a Hilbert space is proposed and proved to be strongly convergent to a common fixed point of a family of nonexpansive mappings. Our variable KM-like technique is applied to solve the split feasibility problem and the multiple-sets split feasibility problem. Especially, the minimum norm solutions of the split feasibility problem and the multiple-sets split feasibility problem are derived. Our results can be viewed as an improvement and refinement of the previously known results.

关键词： split feasibility problems fixed point problems regularized algorithms convergence analysis of algorithms

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An extragradient method for solving split feasibility and fixed point problems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2012年第4期64卷 633-642页

作者： Ceng, L-C Ansari, Q. H. Yao, J-C Aligarh Muslim Univ Dept Math Aligarh 202002 Uttar Pradesh India Shanghai Normal Univ Dept Math Shanghai 200234 Peoples R China Sci Comp Key Lab Shanghai Univ Shanghai 200234 Peoples R China King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran 31261 Saudi Arabia Kaohsiung Med Univ Ctr Gen Educ Kaohsiung 80708 Taiwan

The purpose of this paper is to introduce and analyze an extragradient method with regularization for finding a common element of the solution set Gamma of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. Combining the regularization method and the extragradient method due to Nadezhkina and Takahashi [N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006) 191-201], we propose an iterative algorithm for finding an element of Fix(S) boolean AND Gamma. We prove that the sequences generated by the proposed algorithm converge weakly to an element of Fix(S) boolean AND Gamma under mild conditions. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： split feasibility problems Fixed point problems Extragradient method Regularization Averaged mappings Maximal monotone mappings

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Mann type iterative methods for finding a common solution of split feasibility and fixed point problems

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POSITIVITY 2012年第3期16卷 471-495页

作者： Ceng, Lu-Chuan Ansari, Qamrul Hasan Yao, Jen-Chih Aligarh Muslim Univ Dept Math Aligarh 202002 Uttar Pradesh India Sci Comp Key Lab Shanghai Univ Shanghai Peoples R China King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran 31261 Saudi Arabia Kaohsiung Med Univ Ctr Gen Educ Kaohsiung 80708 Taiwan Shanghai Normal Univ Dept Math Shanghai 200234 Peoples R China

The purpose of this paper is to study and analyze three different kinds of Mann type iterative methods for finding a common element of the solution set I" of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. By combining Mann's iterative method and the extragradient method, we first propose Mann type extragradient-like algorithm for finding an element of the set;moreover, we derive the weak convergence of the proposed algorithm under appropriate conditions. Second, we combine Mann's iterative method and the viscosity approximation method to introduce Mann type viscosity algorithm for finding an element of the;moreover, we derive the strong convergence of the sequences generated by the proposed algorithm to an element of set under mild conditions. Finally, by combining Mann's iterative method and the relaxed CQ method, we introduce Mann type relaxed CQ algorithm for finding an element of the set . We also establish a weak convergence result for the sequences generated by the proposed Mann type relaxed CQ algorithm under appropriate assumptions.

关键词： split feasibility problems Fixed point problems Mann type iterative methods Extragradient method Viscosity approximation method Relaxed CQ method Nonexpansive mappings Averaged mappings Fixed points Projection

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