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A Novel Relaxed Method for the Split Feasibility Problem in Hilbert Spaces

BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY 2025年第2期51卷 1-16页

作者： Zhan, Wanrong Yu, Hai Luoyang Normal Univ Dept Math Luoyang 471934 Peoples R China

In this paper, we propose a novel relaxed method in order to solve the split feasibility problem in Hilbert spaces. Our algorithms involve a new strategy where the projection onto the half-space is replaced by the projection onto the intersection of two half-spaces. In addition, our algorithms do not require any prior information about the operator norm. We also establish the weak convergence and strong convergence of the proposed algorithms under standard conditions. Finally, the results of numerical experiments indicate that the proposed algorithm is effective in the LASSO problem.

关键词： Split feasibility problem Half-space Intersection of two half-spaces cq algorithm

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A new relaxed method for the split feasibility problem in Hilbert spaces

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OPTIMIZATION 2024年第5期73卷 1417-1432页

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

In this paper, we introduce a new relaxed method for solving the split feasibility problem in Hilbert spaces. In our method, the projection to the halfspace is replaced by the one to the intersection of two halfspaces. We give convergence of the sequence generated by our method under some suitable assumptions. Finally, we give a numerical example for illustrating the efficiency and implementation of our algorithms in comparison with existing algorithms in the literature.

关键词： Split feasibility problem cq algorithm relaxed cq algorithm projection

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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 Inertial Self-adaptive Gradient algorithm for the Split Feasibility Problem and an Application to the Sparse Recovery Problem

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Acta Mathematica Sinica,English Series 2023年第12期39卷 2489-2506页

作者： Nguyen The VINH Pham Thi HOAI Le Anh DUNG Yeol Je CHO Department of Mathematical Analysis University of Transport and Communications3 Cau Giay StreetHanoiVietnam Department of Applied Mathematics School of Applied Mathematics and InformaticsHano University of Science and Technology1 Dai Co Viet RoadHanoiVietnam Department of Mathematics and Informatics Hanoi University of Education136 Xuan ThuyCau GiayHanoiVietnam Department of Mathematics Education Gyeongsang National UniversityJinju52828Republicof Korea

In this paper,by combining the inertial technique and the gradient descent method with Polyak's stepsizes,we propose a novel inertial self-adaptive gradient algorithm to solve the split feasi-bility problem in Hilbert spaces and prove some strong and weak convergence theorems of our method under standard *** examine the performance of our method on the sparse recovery prob-lem beside an example in an infinite dimensional Hilbert space with synthetic data and give some numerical results to show the potential applicability of the proposed method and comparisons with related methods emphasize it further.

关键词： Split feasibility problem cq algorithm Hilbert space sparse recovery problem

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Two inertial-type algorithms for solving the split feasibility problem

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OPTIMIZATION 2023年第10期72卷 2661-2678页

作者： Yao, Zhongsheng Vong, Seakweng Univ Macau Dept Math Ave Univ Macau Peoples R China

In this paper, we study two inertial-type algorithms with a relaxed splitting method for solving the split feasibility problem in Hilbert spaces. Weak convergence is established without assuming conditions such as Sigma(infinity)(k=1) theta(k)parallel to x(k) - x(k-1)parallel to(2) < infinity and Sigma(infinity)(k=1) parallel to x(k) - x(k-1)parallel to(2) < infinity, where theta(k) is the inertial factor. < Compared with existing results where inertial factors are usually less than one, the inertial factor in the first algorithm can be taken as nonnegative numbers greater than one. The efficiency and advantage of our algorithms are illustrated by numerical experiments.

关键词： Split feasibility problem inertial factor relaxed splitting method cq algorithm

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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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Second order self-adaptive dynamical system for sparse signal reconstruction and applications to image recovery

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APPLIED MATHEMATICS AND COMPUTATION 2023年第1期451卷

作者： Che, Haitao Liu, Kaiping Chen, Haibin Yan, Hong Weifang Univ Sch Math & Informat Sci Weifang 261061 Shandong Peoples R China Qufu Normal Univ Sch Management Sci Rizhao 276800 Shandong Peoples R China Ctr Intelligent Multidimens Data Anal Shatin Hong Kong Sci Pk Hong Kong Peoples R China City Univ Hong Kong Dept Elect Engn Hong Kong Peoples R China

In this article, we consider sparse signal reconstruction problems by an alternative sec-ond order self-adaptive dynamical system . By split feasibility problem of sparse signal re-construction, we introduce a new second order self-adaptive dynamical system. Then, we prove that the proposed system has a unique solution under reasonable conditions. Fur-thermore, it is shown that the corresponding orbit of the system always converges. Finally, all kinds of numerical results on synthetic data and data from practical problems verify the efficiency of the proposed approach .(c) 2023 Elsevier Inc. All rights reserved.

关键词： Dynamical system Split feasibility problem cq algorithm Self-adaptive

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The ball-relaxed cq algorithms for the split feasibility problem

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OPTIMIZATION 2018年第10期67卷 1687-1699页

作者： Yu, Hai Zhan, Wanrong Wang, Fenghui Luoyang Normal Univ Dept Math Luoyang 471022 Peoples R China

The split feasibility problem (SFP) is to find x is an element of C so that Ax is an element of Q, where C and Q are non-empty closed convex subsets in Hilbert spaces H-1 and H-2, respectively, and A is a linear bounded operator from H-1 to H-2. Byrne proposed an iterative method called the cq algorithm that involves the orthogonal projections onto C and Q. However, the projections onto C and Q might be hard to be implemented in general. In this paper, we propose a ball-relaxed projection method for the SFP. Instead of half spaces, we replace C and Q in the proposed algorithm by two properly chosen closed balls C-k(b) and Q(k)(b). Since the projection onto the closed ball has closed form, the proposed algorithm is thus easy to be implemented. Under some mild conditions, we establish the weak convergence of the proposed algorithm to a solution of the SFP. As an application, we obtain new algorithms for solving the split equality problem. Preliminary numerical experiments show the efficiency of the proposed method.

关键词： Split feasibility problem cq algorithm variable stepsize projection

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Linear convergence of gradient projection algorithm for split equality problems

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OPTIMIZATION 2018年第12期67卷 2347-2358页

作者： Shi, Luo-Yi Ansari, Qamrul Hasan Yao, Jen-Chih Wen, Ching-Feng Tianjin Polytech Univ Dept Math Tianjin Peoples R China Aligarh Muslim Univ Dept Math Aligarh Uttar Pradesh India King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran Saudi Arabia China Med Univ Ctr Gen Educ Taichung Taiwan Kaohsiung Med Univ Ctr Fundamental Sci Kaohsiung 80708 Taiwan Kaohsiung Med Univ Res Ctr Nonlinear Anal & Optimizat Kaohsiung 80708 Taiwan Kaohsiung Med Univ Hosp Dept Med Res Kaohsiung 80708 Taiwan

In this paper, we consider the varying stepsize gradient projection algorithm (GPA) for solving the split equality problem (SEP) in Hilbert spaces, and study its linear convergence. In particular, we introduce a notion of bounded linear regularity property for the SEP, and use it to establish the linear convergence property for the varying stepsize GPA. We provide some mild sufficient conditions to ensure the bounded linear regularity property, and then conclude the linear convergence rate of the varying stepsize GPA. To the best of our knowledge, this is the first work to study the linear convergence for the SEP.

关键词： Split equality problems gradient projection algorithm cq algorithm linear convergence

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