Finding common fixed points of a class of paracontractions

Crombez, Gilbert

doi:10.1023/b:amhu.0000028410.94541.fb

Cited by 15 publications

(17 citation statements)

References 12 publications

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“…We refer to a procedure defined by (7) as a perturbation-resilient amalgamated projection method. Amalgamated projection methods without perturbations (i.e., with b k 5 0, for all k 2 N in (7)) are string averaging methods as introduced in Censor et al (2001a), and further studied in Crombez (2002Crombez ( , 2004, Bauschke et al (2004), Bilbao-Castro et al (2003), and Censor and Tom (2003).…”

Section: Introductionmentioning

confidence: 99%

Perturbation‐resilient block‐iterative projection methods with application to image reconstruction from projections

Davidi¹,

Herman²,

Censor

2009

Int Trans Operational Res

109

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A block-iterative projection algorithm for solving the consistent convex feasibility problem in a finite-dimensional Euclidean space that is resilient to bounded and summable perturbations (in the sense that convergence to a feasible point is retained even if such perturbations are introduced in each iterative step of the algorithm) is proposed. This resilience can be used to steer the iterative process towards a feasible point that is superior in the sense of some functional on the points in the Euclidean space having a small value. The potential usefulness of this is illustrated in image reconstruction from projections, using both total variation and negative entropy as the functional.

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Section: Introductionmentioning

confidence: 99%

Perturbation‐resilient block‐iterative projection methods with application to image reconstruction from projections

Davidi¹,

Herman²,

Censor

2009

Int Trans Operational Res

109

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“…Compared to Theorem 1, we have a less restrictive assumption on (Fix T,C) but a more restrictive one on the parameters (r n , η n ). The proof of Theorem 2 is more or less implicit in the works by Crombez [20] and Polyak [14]; see Remark 2 and Remark 3.…”

Section: Finitely Convergent Cutter Methodsmentioning

confidence: 99%

“…For further information on cutters and subgradient projectors, we refer the reader to [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] and the references therein. Given r ≥ 0, we follow Crombez [20] and define the operator U r : X → X at x ∈ X by…”

Section: Introductionmentioning

confidence: 99%

On the Finite Convergence of a Projected Cutter Method

Bauschke

Wang

et al. 2014

J Optim Theory Appl

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The subgradient projection iteration is a classical method for solving a convex inequality. Motivated by works of Polyak and of Crombez, we present and analyze a more general method for finding a fixed point of a cutter, provided that the fixed point set has nonempty interior. Our assumptions on the parameters are more general than existing ones. Various limiting examples and comparisons are provided.

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“…By suitably averaging this set of operators, a fully parallel and a block-iterative algorithm can be obtained, both with a variable relaxation parameter. In Crombez [44] string-averaging is used to produce a asynchronous parallel algorithm that leads to a common fixed point of directed operators in Hilbert space. The assumption in this paper is that there exists a common fixed point that is an interior point.…”

Section: Algorithmmentioning

confidence: 99%

Sparse string-averaging and split common fixed points

Censor¹,

Segal²

2010

Nonlinear Analysis and Optimization I

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We review the common fixed point problem for the class of directed operators. This class is important because many commonly used nonlinear operators in convex optimization belong to it. We present our recent definition of sparseness of a family of operators and discuss a string-averaging algorithmic scheme that favorably handles the common fixed points problem when the family of operators is sparse. We also review some recent results on the multiple operators split common fixed point problem which requires to find a common fixed point of a family of operators in one space whose image under a linear transformation is a common fixed point of another family of operators in the image space.

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Finding common fixed points of a class of paracontractions

Cited by 15 publications

References 12 publications

Perturbation‐resilient block‐iterative projection methods with application to image reconstruction from projections

Perturbation‐resilient block‐iterative projection methods with application to image reconstruction from projections

On the Finite Convergence of a Projected Cutter Method

Sparse string-averaging and split common fixed points

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