Projection methods: an annotated bibliography of books and reviews

Censor, Yair; Cegielski, Andrzej

doi:10.1080/02331934.2014.957701

Cited by 56 publications

(45 citation statements)

References 28 publications

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“…In this case the algorithmic operator and the basic algorithm (1) it entails can be any of the wide variety of feasibility-seeking algorithms, see, e.g., [3, 4, 8, 11, 12, 19]. …”

Section: The Superiorization Methodologymentioning

confidence: 99%

“…This class of algorithms has witnessed great progress in recent years and its member algorithms have been applied with success to many scientific, technological and mathematical problems. See, e.g., the 1996 review [3], the recent annotated bibliography of books and reviews [11] and its references, the excellent book [8], or [12]. …”

Section: Linear Superiorizationmentioning

confidence: 99%

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Can linear superiorization be useful for linear optimization problems?

Censor¹

2017

Inverse Problems

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Linear superiorization considers linear programming problems but instead of attempting to solve them with linear optimization methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward reduced (not necessarily minimal) target function values. The two questions that we set out to explore experimentally are (i) Does linear superiorization provide a feasible point whose linear target function value is lower than that obtained by running the same feasibility-seeking algorithm without superiorization under identical conditions? and (ii) How does linear superiorization fare in comparison with the Simplex method for solving linear programming problems? Based on our computational experiments presented here, the answers to these two questions are: “yes” and “very well”, respectively.

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“…In this case the algorithmic operator and the basic algorithm (1) it entails can be any of the wide variety of feasibility-seeking algorithms, see, e.g., [3, 4, 8, 11, 12, 19]. …”

Section: The Superiorization Methodologymentioning

confidence: 99%

Section: Linear Superiorizationmentioning

confidence: 99%

Can linear superiorization be useful for linear optimization problems?

Censor¹

2017

Inverse Problems

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“…This class of algorithms has witnessed great progress in recent years and its member algorithms have been applied with success to many scientific, technological and mathematical problems. See, e.g., the 1996 review [18], the recent annotated bibliography of books and reviews [19] and its references, the excellent book [27], or [20].…”

Section: Projection Methods For Feasibility-seekingmentioning

confidence: 99%

“…To solve the split feasibility formulation of Problem 3 we propose to use the CQ-algorithm [22] for the sets C and Q given by (19) and (20), respectively. It has the advantage that it does not require to calculate the inverse A −1 1 of A 1 in order to "go back" from R m 1 to R n within the iterative process.…”

Section: Algorithmic Approachmentioning

confidence: 99%

Sparsity constrained split feasibility for dose-volume constraints in inverse planning of intensity-modulated photon or proton therapy

et al. 2017

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A split feasibility formulation for the inverse problem of intensity-modulated radiation therapy (IMRT) treatment planning with dose-volume constraints (DVCs) included in the planning algorithm is presented. It involves a new type of sparsity constraint that enables the inclusion of a percentage-violation constraint in the model problem and its handling by continuous (as opposed to integer) methods. We propose an iterative algorithmic framework for solving such a problem by applying the feasibility-seeking CQ-algorithm of Byrne combined with the automatic relaxation method (ARM) that uses cyclic projections. Detailed implementation instructions are furnished. Functionality of the algorithm was demonstrated through the creation of an intensity-modulated proton therapy plan for a simple 2D C-shaped geometry and also for a realistic base- of-skull chordoma treatment site. Monte Carlo simulations of proton pencil beams of varying energy were conducted to obtain dose distributions for the 2D test case. A research release of the Pinnacle3 proton treatment planning system was used to extract pencil beam doses for a clinical base-of-skull chordoma case. In both cases the beamlet doses were calculated to satisfy dose-volume constraints according to our new algorithm. Examination of the dose-volume histograms following inverse planning with our algorithm demonstrated that it performed as intended. The application of our proposed algorithm to dose-volume constraint inverse planning was successfully demonstrated. Comparison with optimized dose distributions from the research release of the Pinnacle3 treatment planning system showed the algorithm could achieve equivalent or superior results.

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“…Otherwise, if the distance of the point from the set is greater than the depth of the enlargement, then the valiant projector brings the point closer to the set Z in the orthogonal projection direction of the point onto the set. From the term 1 − β d Z (x) 2 we see that the farther the point is from the set, the operator will progress towards the set in larger steps. Like in [15], the overall step-size also incorporates an additional user-chosen relaxation parameter τ .…”

Section: The Valiant Projectormentioning

confidence: 99%

Convergence Analysis of Processes with Valiant Projection Operators in Hilbert Space

Censor

Mansour

2017

J Optim Theory Appl

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Convex feasibility problems require to find a point in the intersection of a finite family of convex sets. We propose to solve such problems by performing set-enlargements and applying a new kind of projection operators called valiant projectors. A valiant projector onto a convex set implements a special relaxation strategy, proposed by Goffin in 1971, that dictates the move toward the projection according to the distance from the set. Contrary to past realizations of this strategy, our valiant projection operator implements the strategy in a continuous fashion. We study properties of valiant projectors and prove convergence of our new valiant projections method. These results include as a special case and extend the 1985 automatic relaxation method of Censor.

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Projection methods: an annotated bibliography of books and reviews

Abstract: Contents 1 Introduction 2 Early beginnings: von Neumann, Kaczmarz and Cimmino 3 Books on projection methods 4 Review papers on projection methods 5 The end is open

Cited by 56 publications

References 28 publications

Can linear superiorization be useful for linear optimization problems?

Can linear superiorization be useful for linear optimization problems?

Sparsity constrained split feasibility for dose-volume constraints in inverse planning of intensity-modulated photon or proton therapy

Convergence Analysis of Processes with Valiant Projection Operators in Hilbert Space

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