On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints

Censor, Yair; Chen, Wei; Combettes, Patrick L.; Davidi, Ran; Herman, Gabor T.

doi:10.1007/s10589-011-9401-7

Cited by 142 publications

(158 citation statements)

References 74 publications

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“…Moreover, these algorithms were designed to efficiently solve inconsistent feasibility problems (when the intersection of the convex set is empty). Thorough comparisons between projection methods have been performed in [50,51].…”

Section: From Gradient Descent To Proximal Algorithmsmentioning

confidence: 99%

Epigraphical projection and proximal tools for solving constrained convex optimization problems

Chierchia

Pustelnik

Pesquet

et al. 2014

SIViP

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We propose a proximal approach to deal with convex optimization problems involving nonlinear constraints. A large family of such constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower level set of a sum of convex functions evaluated over different, but possibly overlapping, blocks of the signal. For this class of constraints, the associated projection operator generally does not have a closed form. We circumvent this difficulty by splitting the lower level set into as many epigraphs as functions involved in the sum. A closed half-space constraint is also enforced, in order to limit the sum of the introduced epigraphical variables to the upper bound of the original lower level set.In this paper, we focus on a family of constraints involving linear transforms of ℓ1,p balls. Our main theoretical contribution is to provide closed form expressions of the epigraphical projections associated with the Euclidean norm (p = 2) and the sup norm (p = +∞). The proposed approach is validated in the context of image restoration with missing samples, by making use of TV-like constraints. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems.

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Section: From Gradient Descent To Proximal Algorithmsmentioning

confidence: 99%

Epigraphical projection and proximal tools for solving constrained convex optimization problems

Chierchia

Pustelnik

Pesquet

et al. 2014

SIViP

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“…Figure 5 (part a) shows the relative errors of different strategies with 5% noise level. The strategy (6) with γ k = γ I k gives an acceptable result compared with other strategies. Using the results shown in Figure 5 (part b), adding more noise to the right-hand side of (1) gives fast semiconvergence for constant relaxation parameter λ = 30 and the CGLS method, whereas our strategies show stable behavior.…”

Section: Numerical Resultsmentioning

confidence: 93%

“…We take x 0 = 0 in all numerical tests. We also compare different strategies, (5), (6), and λ = 1 (without relaxation parameter), where Algorithm 1 is used. Furthermore, the results of the CGLS method and error minimizing relaxation strategy (EMR) [31, (3.16) We next give some notes on the fast semiconvergence phenomenon and implementation of Algorithm 1.…”

Section: Experimental Issuesmentioning

confidence: 99%

Choosing the relaxation parameter in sequential block-iterativemethods for linear systems

Nikazad¹,

Heidarzade²

2017

Turk J Math

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“…In the proposed method, the regularization term is the square of the FV function as shown in (6). The first term in (6) consists of components |v i − w i | which are comparable to |w i − w i−1 | forming the FV function.…”

Section: Epigraph Set Of Filtered Variation Functionmentioning

confidence: 99%

Projections onto the epigraph set of the filtered variation function based deconvolution algorithm

Tofighi

Çetin

2016

2016 IEEE Global Conference on Signal and Information Processing (GlobalSIP)

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Abstract-A new deconvolution algorithm based on orthogonal projections onto the hyperplanes and the epigraph set of a convex cost function is presented. In this algorithm, the convex sets corresponding to the cost function are defined by increasing the dimension of the minimization problem by one. The Filtered Variation (FV) function is used as the convex cost function in this algorithm. Since the FV cost function is a convex function in R N , then the corresponding epigraph set is also a convex set in the lifted set in R N +1 . At each step of the iterative deconvolution algorithm, starting with an arbitrary initial estimate in R N +1 , first the projections onto the hyperplanes are performed to obtain the first deconvolution estimate. Then an orthogonal projection is performed onto the epigraph set of the FV cost function, in order to regularize and denoise the deconvolution estimate, in a sequential manner. The algorithm converges to the deblurred image.

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On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints

Cited by 142 publications

References 74 publications

Epigraphical projection and proximal tools for solving constrained convex optimization problems

Epigraphical projection and proximal tools for solving constrained convex optimization problems

Choosing the relaxation parameter in sequential block-iterativemethods for linear systems

Projections onto the epigraph set of the filtered variation function based deconvolution algorithm

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