Extrapolation algorithm for affine-convex feasibility problems

Bauschke, Heinz H.; Combettes, Patrick L.; Kruk, Serge

doi:10.1007/s11075-005-9010-6

Cited by 77 publications

(78 citation statements)

References 43 publications

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“…For the case where m is large and each of the sets X i has a simple form, incremental methods that make successive projections on the component sets X i have a long history (see e.g., Gubin et al [25], and recent papers such as Bauschke [6], Bauschke et al [2,3], and Cegielski and Suchocka [19], and their bibliographies). We may consider the following generalized version of the classical feasibility problem,…”

Section: Iterated Projection Algorithmsmentioning

confidence: 99%

“…3 Here {α k } is a positive scalar sequence, and we assume that each f i : n → is a convex function and X is a closed convex set. The motivation for this method is that with a favorable structure of the components, the proximal iteration (3) may be obtained in closed form or be relatively simple, in which case it may be preferable to a gradient or subgradient iteration.…”

Section: Introductionmentioning

confidence: 99%

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Incremental proximal methods for large scale convex optimization

2011

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We consider the minimization of a sum m i=1 f i (x) consisting of a large number of convex component functions f i . For this problem, incremental methods consisting of gradient or subgradient iterations applied to single components have proved very effective. We propose new incremental methods, consisting of proximal iterations applied to single components, as well as combinations of gradient, subgradient, and proximal iterations. We provide a convergence and rate of convergence analysis of a variety of such methods, including some that involve randomization in the selection of components. We also discuss applications in a few contexts, including signal processing and inference/machine learning.

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Section: Iterated Projection Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Incremental proximal methods for large scale convex optimization

2011

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“…The field of projection methods is vast and we can only mention here a few recent works that can give the reader some good starting points. Such a list includes, among many others, the paper of Lakshminarayanan and Lent [58] on the SIRT method, the works of Crombez [43,46], the connection with variational inequalities, see, e.g., Noor [62], Yamada's [68] which is motivated by real-world problems of signal processing, and the many contributions of Bauschke and Combettes, see, e.g., Bauschke, Combettes and Kruk [7] and references therein. Consult Bauschke and Borwein [4] and Censor and Zenios [33,Chapter 5] for a tutorial review and a book chapter, respectively.…”

Section: Projection Methods: Advantages and Earlier Workmentioning

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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“…xiii-xiv] and the references therein), it is natural to consider also the CFP with infinitely many sets appearing in the formulation of the problem, and this is done in the present paper. A few other works considering the CFP with infinitely many sets exist, for instance, [10,12,44] and [23,25], but some do not consider the SSP.…”

Section: The Number Of Involved Setsmentioning

confidence: 99%

Zero-convex functions, perturbation resilience, and subgradient projections for feasibility-seeking methods

Censor

Reem

2014

Math. Program.

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The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, we show that the sequential subgradient projection method is perturbation resilient. By this we mean that under appropriate conditions the sequence generated by the method converges weakly, and sometimes also strongly, to a point in the intersection of the given subsets of the feasibility problem, despite certain perturbations which are allowed in each iterative step. Unlike previous works on solving the convex feasibility problem, the involved functions, which induce the feasibility problem's subsets, need not be convex. Instead, we allow them to belong to a wider and richer class of functions satisfying a weaker condition, that we call "zero-convexity". This class, which is introduced and discussed here, holds a promise to solve optimization problems in various areas, especially in non-smooth and nonconvex optimization. The relevance of this study to approximate minimization and to the recent superiorization methodology for constrained optimization is explained.

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Extrapolation algorithm for affine-convex feasibility problems

Cited by 77 publications

References 43 publications

Incremental proximal methods for large scale convex optimization

Incremental proximal methods for large scale convex optimization

Sparse string-averaging and split common fixed points

Zero-convex functions, perturbation resilience, and subgradient projections for feasibility-seeking methods

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