A Cyclic and Simultaneous Iterative Method for Solving the Multiple-Sets Split Feasibility Problem

Wen, Meng; Peng, Jigen; Tang, Yuchao

doi:10.1007/s10957-014-0701-9

Cited by 17 publications

(34 citation statements)

References 19 publications

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“…The MSSFP was introduced by Censor et al [12] to resolve the intensity modulated radiation therapy treatment planning [4]. Many numerical algorithms have been developed to solve the MSSFP; see [12][13][14][15][16][17] and references therein. More precisely, Censor et al in [12] proposed a projection gradient method, where the step size depends on the Lipschitz constant of the gradient operator.…”

Section: Mathematics Subject Classification 90c25 • 90c30 • 47j25 1 Introductionmentioning

confidence: 99%

“…Moreover, considering the MSSFP in Hilbert spaces, Xu in [13] proposed a fixed-point method with the step size relying on the Lipschitz constant of the gradient operator. Adopting the self-adaptive step-size rule, Wen et al in [17] introduced the cyclic/simultaneous iteration methods for the MSSFP and proposed the relaxed cyclic/simultaneous iteration methods for the general MSSFP where the involved sets are given by the level sets of a series of convex functions. Global weak convergence results of these theorems were established therein.…”

Section: Mathematics Subject Classification 90c25 • 90c30 • 47j25 1 Introductionmentioning

confidence: 99%

“…Global weak convergence results of these theorems were established therein. However, there are some gaps in the proofs of [17, In this paper, we consider a family of projection gradient methods for solving the MSSFP in Hilbert spaces, which include the cyclic/simultaneous iteration methods introduced in [17] as special cases. For the general case where the involved sets are given by level sets of a series of convex functions, the projections onto the level sets are not easily implemented in general.…”

Section: Mathematics Subject Classification 90c25 • 90c30 • 47j25 1 Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Family of Projection Gradient Methods for Solving the Multiple-Sets Split Feasibility Problem

Wang

et al. 2019

J Optim Theory Appl

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In the present paper, we explore a family of projection gradient methods for solving the multiple-sets split feasibility problem, which include the cyclic/simultaneous iteration methods introduced in Wen et al. (J Optim Theory Appl 166:844-860, 2015) as special cases. For the general case, where the involved sets are given by level sets of convex functions, the calculation of the projection onto the level sets is complicated in general, and thus, the resulting projection gradient method cannot be implemented easily. To avoid this difficulty, we introduce a family of relaxed projection gradient methods, in which the projections onto the approximated halfspaces are adopted in place of the ones onto the level sets. They cover the relaxed cyclic/simultaneous iteration methods introduced in Wen et al. (J Optim Theory Appl 166:844-860, 2015) as special cases. Global weak convergence theorems are established for these methods. In particular, as direct applications of the established theorems, our results fill some gaps and deal with the imperfections that appeared in Wen et al. (J Optim Theory Appl 166:844-860, 2015) and hence improve and extend the corresponding results therein.

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Section: Mathematics Subject Classification 90c25 • 90c30 • 47j25 1 Introductionmentioning

confidence: 99%

Section: Mathematics Subject Classification 90c25 • 90c30 • 47j25 1 Introductionmentioning

confidence: 99%

Section: Mathematics Subject Classification 90c25 • 90c30 • 47j25 1 Introductionmentioning

confidence: 99%

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A Family of Projection Gradient Methods for Solving the Multiple-Sets Split Feasibility Problem

Wang

et al. 2019

J Optim Theory Appl

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“…The number of k is the iteration number when the objection function g(x) of (3) satisfies g(x) ≤ δ for some given small δ. The numerical results obtained by the relaxed iterative algorithm of Zhao and Yang [17], Wen et al [18] and iterative sequence (18), where k is the same as in Table 2.…”

Section: Two Mssfp Problemsmentioning

confidence: 99%

Iterative methods for solving the multiple-sets split feasibility problem with splitting self-adaptive step size

Tang¹,

Zhu²,

Yu³

2015

Fixed Point Theory Appl

Self Cite

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We introduce an iterative algorithm for solving the multiple-sets split feasibility problem with splitting self-adaptive step size. This step size is calculated directly from the iteration process without need to know the spectral norm of linear operators. We also generalize the chosen step size to a relaxed iterative algorithm. Theoretical convergence is proved in an infinite dimensional Hilbert space. Some numerical experiments are presented to verify the effectiveness of our proposed methods. MSC: 90C25; 90C30; 47J25

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“…Many iterative algorithms have been developed to solve the MSCFP and the MSFP. See, for example, [ 7 – 14 ] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Solving the multiple-set split equality common fixed-point problem of firmly quasi-nonexpansive operators

Zhao

Zong

2018

J Inequal Appl

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In this paper, we propose parallel and cyclic iterative algorithms for solving the multiple-set split equality common fixed-point problem of firmly quasi-nonexpansive operators. We also combine the process of cyclic and parallel iterative methods and propose two mixed iterative algorithms. Our several algorithms do not need any prior information about the operator norms. Under mild assumptions, we prove weak convergence of the proposed iterative sequences in Hilbert spaces. As applications, we obtain several iterative algorithms to solve the multiple-set split equality problem.

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A Cyclic and Simultaneous Iterative Method for Solving the Multiple-Sets Split Feasibility Problem

Cited by 17 publications

References 19 publications

A Family of Projection Gradient Methods for Solving the Multiple-Sets Split Feasibility Problem

A Family of Projection Gradient Methods for Solving the Multiple-Sets Split Feasibility Problem

Iterative methods for solving the multiple-sets split feasibility problem with splitting self-adaptive step size

Solving the multiple-set split equality common fixed-point problem of firmly quasi-nonexpansive operators

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