Stable convergence behavior under summable perturbations of a class of projection methods for convex feasibility and optimization problems

Dan Butnariu; Ran Davidi; Gabor T. Herman; Ivan G. Kazantsev

doi:10.1109/JSTSP.2007.910263

Stable convergence behavior under summable perturbations of a class of projection methods for convex feasibility and optimization problems

Dan Butnariu, Ran Davidi, Gabor T. Herman, Ivan G. Kazantsev

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study the convergence behavior of a class of projection methods for solving convex feasibility and optimization problems. We prove that the algorithms in this class converge to solutions of the consistent convex feasibility problem, and that their convergence is stable under summable perturbations. Our class is a subset of the class of string-averaging projection methods, large enough to contain, among many other procedures, a version of the Cimmino algorithm, as well as the cyclic projection method. A variant of our approach is proposed to approximate the minimum of a convex functional subject to convex constraints. This variant is illustrated on a problem in image processing: namely, for optimization in tomography.

Original language	English
Pages (from-to)	540-547
Number of pages	8
Journal	IEEE Journal on Selected Topics in Signal Processing
Volume	1
Issue number	4
DOIs	https://doi.org/10.1109/JSTSP.2007.910263
State	Published - Dec 2007

Bibliographical note

Funding Information:
Manuscript received January 29, 2007; revised August 28, 2007. This work was supported by the National Institutes of Health through Grant HL70472. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Alex B. Gershman.

Funding Information:
D. Butnariu’s work on this paper was done during his 2006 visit to the Discrete Imaging and Graphics group of the Graduate Center of the City University of New York, and he gratefully acknowledges the inspiring scientific activities in which he was invited to take part, as well as the financial support from the grant from the National Institutes of Health that made his participation possible. The authors thank P. Combettes for his help with the implementation of the TV-minimizing algorithm based on [10] and Y. Censor for his criticisms of an earlier version.

Keywords

Cimmino algorithm
Convex feasibility
Cyclic projection method
Projection method
String-averaging
Tomographic optimization
Total variation

ASJC Scopus subject areas

Signal Processing
Electrical and Electronic Engineering

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10.1109/JSTSP.2007.910263

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abstract = "We study the convergence behavior of a class of projection methods for solving convex feasibility and optimization problems. We prove that the algorithms in this class converge to solutions of the consistent convex feasibility problem, and that their convergence is stable under summable perturbations. Our class is a subset of the class of string-averaging projection methods, large enough to contain, among many other procedures, a version of the Cimmino algorithm, as well as the cyclic projection method. A variant of our approach is proposed to approximate the minimum of a convex functional subject to convex constraints. This variant is illustrated on a problem in image processing: namely, for optimization in tomography.",

keywords = "Cimmino algorithm, Convex feasibility, Cyclic projection method, Projection method, String-averaging, Tomographic optimization, Total variation",

author = "Dan Butnariu and Ran Davidi and Herman, {Gabor T.} and Kazantsev, {Ivan G.}",

note = "Funding Information: Manuscript received January 29, 2007; revised August 28, 2007. This work was supported by the National Institutes of Health through Grant HL70472. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Alex B. Gershman. Funding Information: D. Butnariu{\textquoteright}s work on this paper was done during his 2006 visit to the Discrete Imaging and Graphics group of the Graduate Center of the City University of New York, and he gratefully acknowledges the inspiring scientific activities in which he was invited to take part, as well as the financial support from the grant from the National Institutes of Health that made his participation possible. The authors thank P. Combettes for his help with the implementation of the TV-minimizing algorithm based on [10] and Y. Censor for his criticisms of an earlier version.",

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AU - Kazantsev, Ivan G.

N1 - Funding Information: Manuscript received January 29, 2007; revised August 28, 2007. This work was supported by the National Institutes of Health through Grant HL70472. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Alex B. Gershman. Funding Information: D. Butnariu’s work on this paper was done during his 2006 visit to the Discrete Imaging and Graphics group of the Graduate Center of the City University of New York, and he gratefully acknowledges the inspiring scientific activities in which he was invited to take part, as well as the financial support from the grant from the National Institutes of Health that made his participation possible. The authors thank P. Combettes for his help with the implementation of the TV-minimizing algorithm based on [10] and Y. Censor for his criticisms of an earlier version.

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Stable convergence behavior under summable perturbations of a class of projection methods for convex feasibility and optimization problems

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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