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Accelerated Mann and cq algorithms for finding a fixed point of a nonexpansive mapping

FIXED POINT THEORY AND APPLICATIONS 2015年第1期2015卷 1-12页

作者： Dong, Qiao-Li Yuan, Han-bo 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

The purpose of this paper is to present accelerations of the Mann and cq algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerate the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the Mann and cq algorithms. Under certain assumptions, we show that the new algorithms converge to a fixed point of a nonexpansive mapping. Finally, we show the efficiency of the accelerated Mann algorithm by numerically comparing with the Mann algorithm. A numerical example is provided to illustrate that the acceleration of the cq algorithm is ineffective.

关键词： nonexpansive mapping convex minimization problem Picard algorithm Mann algorithm cq algorithm conjugate gradient method steepest descent method

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On the convergence analysis of the gradient-cq algorithms for the split feasibility problem

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NUMERICAL algorithmS 2020年第3期84卷 997-1017页

作者： Kesornprom, Suparat Pholasa, Nattawut Cholamjiak, Prasit Univ Phayao Sch Sci Phayao 56000 Thailand

In this work, based on the gradient method and the relaxed cq algorithm introduced by Lopez et al. (Inverse Probl.28, 085004,2012), we introduce two gradient-cq algorithms for solving the split feasibility problem in the framework of Hilbert spaces. The main advantage of the proposed method is not only that the variable stepsizes depending on the information from the current iterate not the operator norm are chosen but also that the metric projection onto half-spaces, which is convenient to be calculated, is taken into account. Then both weak and strong convergence are proved under some mild conditions. Finally, numerical experiments in signal processing reveal that the proposed algorithm is effective and outruns those of Yang, Lopez et al., Gibali et al., and others.

关键词： Split feasibility problem cq algorithm Hilbert space Gradient method

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Inertial relaxedcqalgorithms for solving a split feasibility problem in Hilbert spaces

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NUMERICAL algorithmS 2021年第3期87卷 1075-1095页

作者： Sahu, D. R. Cho, Y. J. Dong, Q. L. Kashyap, M. R. Li, X. H. Banaras Hindu Univ Inst Sci Dept Math Varanasi 221005 Uttar Pradesh India Gyeongsang Natl Univ Dept Math Educ Jinju 52828 South Korea Univ Elect Sci & Technol China Sch Math Sci Chengdu 611731 Sichuan Peoples R China Key Lab Adv Signal Proc Tianjin 300300 Peoples R China Coll Sci Tianjin 300300 Peoples R China Civil Aviat Univ China Tianjin 300300 Peoples R China

The split feasibility problem is to find a pointx*with the property thatx*is an element of CandAx*is an element of Q, whereCandQare nonempty closed convex subsets of real Hilbert spacesXandY, respectively, andAis a bounded linear operator fromXtoY. The split feasibility problem models inverse problems arising from phase retrieval problems and the intensity-modulated radiation therapy. In this paper, we introduce a new inertial relaxedcqalgorithm for solving the split feasibility problem in real Hilbert spaces and establish weak convergence of the proposedcqalgorithm under certain mild conditions. Our result is a significant improvement of the recent results related to the split feasibility problem.

关键词： Split feasibility problem cq algorithm Inertial technique Self-adaptive algorithm Weak convergence

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Linear convergence of cq algorithms and applications in gene regulatory network inference

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INVERSE PROBLEMS 2017年第5期33卷 1-1页

作者： Wang, Jinhua Hu, Yaohua Li, Chong Yao, Jen-Chih Zhejiang Univ Technol Dept Math Hangzhou 310032 Zhejiang Peoples R China Shenzhen Univ Coll Math & Stat Shenzhen 518060 Peoples R China Zhejiang Univ Sch Math Sci Hangzhou 310027 Zhejiang Peoples R China China Med Univ Ctr Gen Educ Taichung 40402 Taiwan Kaohsiung Med Univ Res Ctr Nonlinear Anal & Optimizat Kaohsiung 807 Taiwan

In the present paper, we consider the varying stepsize cq algorithm for solving the split feasibility problem in Hilbert spaces, investigate the linear convergence issue and explore an application in systems biology. In particular, we introduce a notion of bounded linear regularity property for the split feasibility problem, and use it to establish the linear convergence property for the varying stepsize cq algorithm when using some suitable types of stepsizes, which covers most types of stepsizes used in the literature of cq algorithms. We also provide some mild sufficient conditions for ensuring this bounded linear regularity property, and then conclude the linear convergence rate of the varying stepsize cq algorithm for many application cases. To the best of our knowledge, this is the first work to study the linear convergence rate of cq algorithms. In the aspect of application, we consider the gene regulatory network inference arising in systems biology, which is formulated as a group Dantzig selector and then cast into a split feasibility problem. The numerical study on gene expression data of mouse embryonic stem cell shows that the varying stepsize cq algorithm is applicable to gene regulatory network inference in the sense that it obtains a reliable solution matching with biological standards.

关键词： split feasibility problem cq algorithm Hilbert spaces linear convergence application in systems biology

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TWO SELF-ADAPTIVE cq algorithmS FOR THE SPLIT FEASIBILITY PROBLEM WITH MULTIPLE OUTPUT SETS

JOURNAL OF NONLINEAR FUNCTIONAL ANALYSIS

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JOURNAL OF NONLINEAR FUNCTIONAL ANALYSIS 2024年 2024卷 1-19页

作者： Zhang, Yaxuan Qin, Liqian Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

The split feasibility problem with multiple output sets is to find a point x* is an element of boolean AND(t)(i=1) C-i such that A(j)x* is an element of Q(j), j = 1, 2, ..., r where C-i subset of H and Q(j) subset of H-j are nonempty, convex, and closed subsets, H and H-j are Hilbert spaces, and A(j) : H -> H-j are linear and bounded operators. In this paper, we present two self-adaptive ball-relaxed cq algorithms. Under mild conditions, we establish strong convergence and provide numerical experiments to illustrate the effectiveness of the proposed algorithms.

关键词： Ball-relaxation cq algorithm Inverse problem Split feasibility problem with multiple output sets Self-adaptive step-size

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A novel projection method for split feasibility problems with applications to compressive sensing

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COMPUTATIONAL & APPLIED MATHEMATICS 2023年第4期42卷 1-20页

作者： Anh, Pham Ky Vinh, Nguyen The Vietnam Natl Univ Univ Sci Hanoi Vietnam Univ Transport & Commun Dept Math Anal 3 Cau Giay St Hanoi Vietnam

In this paper, by combining the hyperplane projection technique and gradient projection method with Polyak's stepsizes, we propose a novel projection algorithm to solve the split feasibility problem in Hilbert spaces. Unlike the existing projection methods, our method is designed such that the next iterate is closer to the solution set than the previous one. The weak and strong convergence of new algorithms under standard assumptions are established. We examine the performance of our method on the sparse signal recovery problem. The reported numerical results demonstrate the efficiency of the proposed method.

关键词： Split feasibility problem Hyperplane projection method Gradient projection method cq algorithm Hilbert space Weak convergence Polyak's stepsize

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A NEW algorithm FOR SOLVING THE MULTIPLE-SETS SPLIT FEASIBILITY PROBLEM IN BANACH SPACES

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2014年第1期35卷 99-110页

作者： Wang, Fenghui Luoyang Normal Univ Dept Math Luoyang 471022 Peoples R China

In this article, we propose a new algorithm for solving the multiple-sets split feasibility problem in Banach spaces. Under some mild conditions we establish the norm convergence of the proposed algorithm. One advantage of the proposed algorithm is that its norm convergence does not need the weak-to-weak continuity of the duality mapping.

关键词： Bregman projection cq algorithm Duality mapping Metric projection Split feasibility problem Uniformly convex Uniformly smooth

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Best proximity points of (EP)-operators with qualitative analysis and simulation

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MATHEMATICS AND COMPUTERS IN SIMULATION 2021年 187卷 215-230页

作者： Usurelu, Gabriela Ioana Turcanu, Teodor Univ Politehn Bucuresti Dept Math & Informat Bucharest 060042 Romania

In this paper, we analyze a Thakur three-step iterative process adapted for the context of non-self-mappings. Based on this iteration, we state and prove the existence of best proximity points for the recently introduced class of (EP)-non-self-mappings. Under certain assumptions, we study the convergence of the considered Thakur process to a best proximity point for the same class of operators. Moreover, we design a cq-type algorithm which strongly converges to a best proximity point of such kind of mappings. In addition, we present the cq-variant of the proposed algorithm that strongly converges to a best proximity pair. Some examples and numerical simulations sustain the efficiency of our new algorithms. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Best proximity point Non-self mapping cq algorithm P-property Hybrid algorithm

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General splitting methods with linearization for the split feasibility problem

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JOURNAL OF GLOBAL OPTIMIZATION 2021年第4期79卷 813-836页

作者： Dong, Qiao-Li He, Songnian Rassias, Michael Th. Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China Univ Zurich Inst Math CH-8057 Zurich Switzerland Moscow Inst Phys & Technol Dolgoprudnyi 141700 Russia

In this article, we introduce a general splitting method with linearization to solve the split feasibility problem and propose a way of selecting the stepsizes such that the implementation of the method does not need any prior information about the operator norm. We present the constant and adaptive relaxation parameters, and the latter is "optimal" in theory. These ways of selecting stepsizes and relaxation parameters are also practised to the relaxed splitting method with linearization where the two closed convex sets are both level sets of convex functions. The weak convergence of two proposed methods is established under standard conditions and the linear convergence of the general splitting method with linearization is analyzed. The numerical examples are presented to illustrate the advantage of our methods by comparing with other methods.

关键词： Split feasibility problem The general splitting method with linearization Relaxed splitting method with linearization cq algorithm Linear convergence

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Modified Hybrid Steepest Method for the Split Feasibility Problem in Image Recovery of Inverse Problems

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2017年第4期38卷 507-522页

作者： Sitthithakerngkiet, Kanokwan Deepho, Jitsupa Kumam, Poom King Mongkuts Univ Technol North Bangkok Fac Sci Appl Dept Math Nonlinear Dynam Anal Res Ctr Bangkok Thailand King Mongkuts Univ Technol North Bangkok Fac Sci Energy & Environm Rayong Thailand King Mongkuts Univ Technol Thonburi Dept Math KMUTT Fixed Point Res Lab Fac Sci Bangkok Thailand King Mongkuts Univ Technol Thonburi KMUTT Fixed Point Theory & Applicat Res Grp Theoret & Computat Sci Ctr Fac Sci Bangkok Thailand China Med Univ China Med Univ Hosp Dept Med Res Taichung Taiwan

In this paper, we regard the cq algorithm as a fixed point algorithm for averaged mapping, and also try to study the split feasibility problem by the following hybrid steepest method;where {(n)}(0,1). It is noted that Xu's original iterative method can conclude only weak convergence. Consequently, we obtain the sequence {x(n)} generated by our iteration method converges strongly to , where is the unique solution of the variational inequality Our result extends and improves the result of Xu, as shown in the literature, from weak to strong convergence theorems. Finally, in the last section, numerical examples for study behavior convergence analysis of this algorithm are obtained.

关键词： cq algorithm fixed point hybrid steepest method inverse problems split feasibility problem variational inequality viscosity method 47H10 35A15 35D40 65K10 49J40

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