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Convergence rate of the relaxed cq algorithm under Holderian type error bound property

OPTIMIZATION 2024年第4期73卷 1285-1301页

作者： Zhang, Lufang Wang, Jinhua Li, Chong Yang, Xiaoqi Zhejiang Univ Sci & Technol Sch Sci Hangzhou Peoples R China Hangzhou Normal Univ Dept Math Peoples Republ China Hangzhou Peoples R China Zhejiang Univ Sch Math Sci Hangzhou Peoples R China Hong Kong Polytech Univ Dept Appl Math Hong Kong Peoples R China

The relaxed cq algorithm is one of the most important algorithms for solving the split feasibility problem. We study the issue of strong convergence of the relaxed cq algorithm in Hilbert spaces together with estimates on the convergence rate. Under a kind of Holderian type bounded error bound property, strong convergence of the relaxed cq algorithm is established. Furthermore, qualitative estimates on the convergence rate is presented. In particular, for the case when the involved exponent is equal to 1, the linear convergence of the relaxed cq algorithm is established. Finally, numerical experiments are performed to show the convergence property of the relaxed cq algorithm for the compressed sensing problem.

关键词： Convergence rete analysis relaxed cq algorithm split feasibility problem Holderian type error bound

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Strong convergence of a relaxed cq algorithm for the split feasibility problem

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2013年第1期2013卷 1-11页

作者： He, Songnian Zhao, Ziyi 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 split feasibility problem (SFP) is finding a point in a given closed convex subset of a Hilbert space such that its image under a bounded linear operator belongs to a given closed convex subset of another Hilbert space. The most popular iterative method is Byrne's cq algorithm. Lopez et al. proposed a relaxed cq algorithm for solving SFP where the two closed convex sets are both level sets of convex functions. This algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, their algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed cq algorithm such that the strong convergence is guaranteed. Our result extends and improves the corresponding results of Lopez et al. and some others.

关键词： split feasibility problem relaxed cq algorithm Hilbert space strong convergence bounded linear operator

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An inertial relaxed cq algorithm with an application to the LASSO and elastic net

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OPTIMIZATION 2021年第5-6期70卷 1101-1119页

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

The relaxed cq algorithm is a very efficient algorithm for solving the split feasibility problem (SFP) whenever the convex subsets involved are level subsets of given convex functions. It approximates the original convex subset by a sequence of half-spaces that overcomes the difficulties for calculating the projection onto original convex subsets. In this paper, we propose a new inertial relaxed algorithm in which we approximate the original convex subset by a sequence of closed balls instead of half spaces. Moreover, we construct a new variable step-size that does not need any prior information of the norm. We then establish the weak convergence of the proposed algorithm under two different assumptions. Experimental results in the LASSO and elastic net methods show that our algorithm has a better performance than other relaxed algorithms.

关键词： Split feasibility problem relaxed cq algorithm variable step-size

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Solving the multiple-set split feasibility problem and the equilibrium problem by a new relaxed cq algorithm

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RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO 2020年第3期69卷 1131-1148页

作者： Kankam, Kunrada Srinak, Pittaya Cholamjiak, Prasit Pholasa, Nattawut Univ Phayao Sch Sci Mae Ka 56000 Phayao Thailand

In this paper, we introduce a new kind of a relaxed cq algorithm to find the solution of the multiple-set split feasibility problem and the equilibrium problem in a Hilbert space. We prove weak and strong convergence theorems to the proposed algorithm under some mild conditions. Finally, we provide some numerical experiments to show the efficiency and the implementation of our method.

关键词： Multiple-set split feasibility problem Equilibrium problem relaxed cq algorithm Hilbert space Weak and strong convergence Bounded linear operator

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ON THE INERTIAL relaxed cq algorithm IN HILBERT SPACES

JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS

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JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS 2021年第4期5卷 549-559页

作者： LI, Haiying Wang, Fenghui Yu, Hai Henan Normal Univ Dept Math Xinxiang 453007 Henan Peoples R China Luoyang Normal Univ Dept Math Luoyang 471934 Peoples R China

In this paper, we study the inertial relaxed cq algorithm for solving a split feasibility problem in Hilbert spaces. For this algorithm, we establish two convergence theorems under two different conditions. The first condition is weaker than the existing condition, and the second condition is completely different from the existing one. Moreover the preliminary numerical experiment indicates that our proposed algorithms converge faster than the existing algorithms.

关键词： Hilbert space relaxed cq algorithm Split feasibility problem Strong convergence

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A relaxed self-adaptive cq algorithm for the multiple-sets split feasibility problem

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OPTIMIZATION 2015年第9期64卷 1907-1918页

作者： He, Songnian Zhao, Ziyi Luo, Biao Civil Aviat Univ China Coll Sci Tianjin Peoples R China Civil Aviat Univ China Tianjin Key Lab Adv Signal Proc Tianjin Peoples R China

The multiple-sets split feasibility problem (MSFP) is to find a point belongs to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation belongs to the intersection of another family of closed convex sets in the image space. Many iterative methods can be employed to solve the MSFP. Jinling Zhao et al. proposed a modification for the cq algorithm and a relaxation scheme for this modification to solve the MSFP. The strong convergence of these algorithms are guaranteed in finite-dimensional Hilbert spaces. Recently Lopez et al. proposed a relaxed cq algorithm for solving split feasibility problem, this algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, this algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed self-adaptive cq algorithm for solving the MSFP where closed convex sets are level sets of some convex functions such that the strong convergence is guaranteed in the framework of infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results.

关键词： multiple-sets split feasibility problem relaxed cq algorithm Hilbert space strong convergence bounded linear operator

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Self-adaptive relaxed cq algorithms for solving split feasibility problem with multiple output sets

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2024年第2期70卷 1441-1469页

作者： Tong, Xiaolei Ling, Tong Shi, Luoyi Tiangong Univ Sch Math Sci Tianjin 300387 Peoples R China Tiangong Univ Sch Software Tianjin 300387 Peoples R China

In this paper, we propose two new self-adaptive relaxed cq algorithms to solve the split feasibility problem with multiple output sets, which involve the computation of projections onto half-spaces instead of the computation onto the closed convex sets. Our proposed algorithms with selection technique reduce the computation of projections. And then, as a generalization, we construct two new algorithms to solve the variational inequalities over the solution set of split feasibility problem with multiple output sets. More importantly, strong convergence of all proposed algorithms is proved under suitable conditions. Finally, we conduct numerical experiments to show the efficiency and accuracy of our algorithms compared to some existing results.

关键词： Split feasibility problem Multiple output sets Strong convergence relaxed cq algorithm Self-adaptive algorithm

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Modified relaxed cq methods for the split feasibility problems in Hilbert spaces with applications

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2023年第4期69卷 3067-3094页

作者： Ling, Tong Tong, Xiaolei Shi, Luoyi Tiangong Univ Sch Math Sci Tianjin 300387 Peoples R China Tiangong Univ Sch Software Tianjin 300387 Peoples R China

In this paper, four modified versions of relaxed cq algorithms are proposed for solving the split feasibility problems (SFP) in infinite-dimensional real Hilbert spaces. The methods are based on replacing the projection to the half-space with that to the intersection of two half-spaces. The convergence speed is accelerated as the algorithms make use of the previous half-spaces. The stepsize is determined dynamically without requiring any prior information about the operator norm. Furthermore, the proposed algorithms are proven to converge strongly to the minimum-norm solution of the SFP. As an application, we apply our results to signal recovery problems.

关键词： Split feasibility problem relaxed cq algorithm Self-adaptive Strong convergence

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Two relaxed cq Methods for the Split Feasibility Problem with Multiple Output Sets

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BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 2024年第2期47卷 1-25页

作者： Thuy, Nguyen Thi Thu Tung, Tran Thanh Hanoi Univ Sci & Technol Fac Math & Informat Hanoi Vietnam

In this paper, two relaxed cq algorithms with non-inertial and inertial steps are proposed for solving the split feasibility problems with multiple output sets (SFPMOS) in infinite-dimensional real Hilbert spaces. The step size is determined dynamically without requiring prior information about the operator norm. Furthermore, the proposed algorithms are proven to converge strongly to the minimum-norm solution of the SFPMOS. Some applications of our main results regarding the solution of the split feasibility problem are presented. Finally, we give two numerical examples to illustrate the efficiency and implementation of our algorithms in comparison with existing algorithms in the literature.

关键词： Split feasibility problems Multiple output sets relaxed cq algorithm Self-adaptive Strong convergence

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Ball-relaxed projection algorithms for multiple-sets split feasibility problem

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OPTIMIZATION 2022年第12期71卷 3571-3601页

作者： Taddele, Guash Haile Kumam, Poom Gebrie, Anteneh Getachew Abubakar, Jamilu King Mongkuts Univ Technol Thonburi KMUTT Fac Sci Dept Math Bangkok Thailand King Mongkuts Univ Technol Thonburi KMUTT Fac Sci Ctr Excellence Theoret & Computat Sci TaCS CoE NCAO Res CtrFixed Point Theory & Applicat Res Gr Bangkok Thailand Debre Berhan Univ Coll Computat & Nat Sci Dept Math Debre Berhan Ethiopia China Med Univ China Med Univ Hosp Dept Med Res Taichung Taiwan Usmanu Danfodiyo Univ Sokoto Dept Math Sokoto Nigeria

The multiple-sets split feasibility problem (MSSFP) requires finding a point closet to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. Motivated by the ball-relaxed projection algorithm proposed by Yu et al. for the split feasibility problem (SFP), in this paper, we introduce ball-relaxed projection algorithms for solving the MSSFP. Instead of the level sets or half-spaces, our algorithms require computing the orthogonal projections onto closed balls. We establish weak and strong convergence of the proposed algorithms to a solution of the MSSFP. Finally, we provide preliminary numerical experiments to show the efficiency and the implementation of our method.

关键词： Multiple-sets split feasibility problem strongly convex function Hilbert space relaxed cq algorithm ball self-adaptive technique

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