On the convergence of CQ algorithm with variable steps for the split equality problem

The split equality problem (SEP) has extraordinary utility and broad applicability in many areas of applied mathematics. Recently, Byrne and Moudafi (2013) proposed a CQ algorithm for solving it. In this paper, we propose a modification for the CQ algorithm, which computes the stepsize adaptively and performs an additional projection step onto two half-spaces in each iteration. We further propose a relaxation scheme for the self-adaptive projection algorithm by using projections onto half-spaces instead of those onto the original convex sets, which is much more practical. Weak convergence results for both algorithms are analyzed.

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An Improved Alternating CQ Algorithm for Solving Split Equality Problems

Mathematics ◽

10.3390/math9243313 ◽

2021 ◽

Vol 9 (24) ◽

pp. 3313

Author(s):

Yan-Juan He ◽

Li-Jun Zhu ◽

Nan-Nan Tan

Keyword(s):

Signal Processing ◽

Weak Convergence ◽

Medical Image ◽

Phase Retrieval ◽

Scientific Field ◽

Iterative Sequence ◽

Split Equality Problem ◽

Cq Algorithm ◽

Equality Problem ◽

Medical Image Reconstruction

The CQ algorithm is widely used in the scientific field and has a significant impact on phase retrieval, medical image reconstruction, signal processing, etc. Moudafi proposed an alternating CQ algorithm to solve the split equality problem, but he only obtained the result of weak convergence. The work of this paper is to improve his algorithm so that the generated iterative sequence can converge strongly.

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Convergence rate analysis of an iterative algorithm for solving the multiple-sets split equality problem

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2219-z ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

Shijie Sun ◽

Meiling Feng ◽

Luoyi Shi

Keyword(s):

Convergence Rate ◽

Iterative Algorithm ◽

Sufficient Conditions ◽

Linear Convergence ◽

Regularity Property ◽

Step Size ◽

Bounded Linear ◽

Split Equality Problem ◽

Linear Convergence Rate ◽

Equality Problem

Abstract This paper considers an iterative algorithm of solving the multiple-sets split equality problem (MSSEP) whose step size is independent of the norm of the related operators, and investigates its sublinear and linear convergence rate. In particular, we present a notion of bounded Hölder regularity property for the MSSEP, which is a generalization of the well-known concept of bounded linear regularity property, and give several sufficient conditions to ensure it. Then we use this property to conclude the sublinear and linear convergence rate of the algorithm. In the end, some numerical experiments are provided to verify the validity of our consequences.

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Two-step methods and relaxed two-step methods for solving the split equality problem

Computational and Applied Mathematics ◽

10.1007/s40314-021-01465-y ◽

2021 ◽

Vol 40 (3) ◽

Author(s):

Dianlu Tian ◽

Lining Jiang

Keyword(s):

Split Equality Problem ◽

Equality Problem

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Iterative algorithm for the split equality problem in Hilbert spaces

Journal of Applied Analysis ◽

10.1515/jaa-2016-0008 ◽

2016 ◽

Vol 22 (1) ◽

Cited By ~ 2

Author(s):

Godwin Chidi Ugwunnadi

Keyword(s):

Strong Convergence ◽

Iterative Algorithm ◽

Hilbert Spaces ◽

Split Equality Problem ◽

Equality Problem

AbstractIn this paper, we studied the split equality problems (SEP) with a new proposed iterative algorithm and established the strong convergence of the proposed algorithm to solution of the split equality problems (SEP).

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Strong convergence of iterative algorithms for the split equality problem

Journal of Inequalities and Applications ◽

10.1186/1029-242x-2014-478 ◽

2014 ◽

Vol 2014 (1) ◽

pp. 478 ◽

Cited By ~ 7

Author(s):

Luo Shi ◽

Rudong Chen ◽

Yujing Wu

Keyword(s):

Strong Convergence ◽

Iterative Algorithms ◽

Split Equality Problem ◽

Equality Problem

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A Relaxed Self-Adaptive Projection Algorithm for Solving the Multiple-Sets Split Equality Problem

Journal of Function Spaces ◽

10.1155/2020/6183214 ◽

2020 ◽

Vol 2020 ◽

pp. 1-12

Author(s):

Haitao Che ◽

Haibin Chen

Keyword(s):

Fixed Point ◽

Equation System ◽

Projection Algorithm ◽

Step Size ◽

Split Equality Problem ◽

Fixed Point Equation ◽

Point Equation ◽

Equality Problem ◽

Adaptive Step ◽

Self Adaptive

In this article, we introduce a relaxed self-adaptive projection algorithm for solving the multiple-sets split equality problem. Firstly, we transfer the original problem to the constrained multiple-sets split equality problem and a fixed point equation system is established. Then, we show the equivalence of the constrained multiple-sets split equality problem and the fixed point equation system. Secondly, we present a relaxed self-adaptive projection algorithm for the fixed point equation system. The advantage of the self-adaptive step size is that it could be obtained directly from the iterative procedure. Furthermore, we prove the convergence of the proposed algorithm. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithm.

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