Alternating Iterative Algorithms for Split Equality Problem of Strictly Pseudononspreading Mapping

In this article, we study the extended split equality problem and extended split equality fixed point problem, which are extensions of the convex feasibility problem. For solving the extended split equality problem, we present two self-adaptive stepsize algorithms with internal perturbation projection and obtain the weak and the strong convergence of the algorithms, respectively. Furthermore, based on the operators being quasinonexpansive, we offer an iterative algorithm to solve the extended split equality fixed point problem. We introduce a way of selecting the stepsize which does not need any prior information about operator norms in the three algorithms. We apply our iterative algorithms to some convex and nonlinear problems. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithms.

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Gradient Methods with Selection Technique for the Multiple-Sets Split Equality Problem

Mathematics ◽

10.3390/math7100928 ◽

2019 ◽

Vol 7 (10) ◽

pp. 928 ◽

Cited By ~ 1

Author(s):

Dianlu Tian ◽

Lining Jiang ◽

Luoyi Shi

Keyword(s):

Inverse Problem ◽

Gradient Methods ◽

Iterative Algorithms ◽

Split Feasibility Problem ◽

Fixed Point Problem ◽

Feasibility Problem ◽

Point Problem ◽

Split Equality Problem ◽

Selection Technique ◽

Equality Problem

The inverse problem is one of the four major problems in computational mathematics. There is an inverse problem in medical image reconstruction and radiotherapy that is called the multiple-sets split equality problem. The multiple-sets split equality problem is a unified form of the split feasibility problem, split equality problem, and split common fixed point problem. In this paper, we present two iterative algorithms for solving it. The suggested algorithms are based on the gradient method with a selection technique. Based on this technique, we only need to calculate one projection in each iteration.

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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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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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