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## On Projection Algorithms for Solving Convex Feasibility Problems (1996)

Citations: | 315 - 43 self |

### Citations

385 |
Minimization methods for nondifferentiable functions
- Shor
- 1985
(Show Context)
Citation Context ...hmic schemes (“cyclic” or “weighted” control). Basic results: Eremin [52], Polyak [86], Censor and Lent [28]. Comments: Quality of convergence is fairly well understood. References: Censor [22], Shor =-=[92]-=-. Areas of application: Solution of convex inequalities, minimization of convex nonsmooth functions. To improve, unify, and review algorithms for these branches, we must study a flexible algorithmic s... |

238 | Weak convergence of the sequence of successive approximations for nonexpansive mappings - Opial - 1967 |

225 |
Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography
- Gordon, Bender, et al.
- 1970
(Show Context)
Citation Context ... and every index i active at n. Then the sequence (x (n) ) converges linearly to some point in C with a rate independent of the starting point. EXAMPLE 6.22 (Kaczmarz [71], Gordon, Bender, and Herman =-=[59]-=-). Suppose X is finite dimensional and the projection algorithm is cyclic, unrelaxed, and has constant sets that are hyperplanes. Then the sequence (x (n) ) converges linearly to some point in C with ... |

206 | Convex Functions - Roberts, Varberg - 1973 |

168 |
Angenäherte auflösung von systemen linearer gleichungen
- Kaczmarz
- 1937
(Show Context)
Citation Context ... Discrete models. Properties: Each set Ci is a halfspace or a hyperplane. X is a Euclidean space (i.e., a finite-dimensional Hilbert space). Very flexible algorithmic schemes. Basic results: Kaczmarz =-=[71]-=-, Cimmino [29], Agmon [1], Motzkin and Schoenberg [83]. Comments: Behaviour in general Hilbert space and quality of convergence only partially understood. References: Censor [21, 23, 24], Censor and H... |

157 |
Linograms in image reconstruction from projections
- Edholm, Herman
(Show Context)
Citation Context ...y an infinite-dimensional Hilbert space. Fairly simple algorithmic schemes. Basic results: Gubin, Polyak, and Raik [60]. Comments: Quality of convergence is fairly well understood. References: Herman =-=[63]-=-, Youla and Webb [108], Stark [95]. Areas of application: Computerized tomography, signal processing. IV. Subgradient algorithms. Properties: Some sets Ci are of type 2. Fairly simple algorithmic sche... |

151 | Theory of extremal problems - Ioffe, Tihomirov - 1979 |

131 |
H.: Projections on convex sets in Hilbert space and spectral theory I
- Zarantonello
- 1971
(Show Context)
Citation Context ... d(·,C)is convex and continuous (hence weakly lower semicontinuous). A good reference on nonexpansive mappings is Goebel and Kirk’s recent book [58]. Many results on projections are in Zarantonello’s =-=[109]-=-. The algorithms’ quality of convergence will be discussed in terms of linear convergence: a sequence (xn) in X is said to converge linearly to its limit x (with rate β) if β ∈ [0, 1[ and there is som... |

125 |
The method of projections for finding the common point of convex sets
- Gubin, Polyak, et al.
- 1967
(Show Context)
Citation Context ...ron microscopy. III. Image reconstruction: Continuous models. Properties: X is usually an infinite-dimensional Hilbert space. Fairly simple algorithmic schemes. Basic results: Gubin, Polyak, and Raik =-=[60]-=-. Comments: Quality of convergence is fairly well understood. References: Herman [63], Youla and Webb [108], Stark [95]. Areas of application: Computerized tomography, signal processing. IV. Subgradie... |

124 |
On approximate solutions of systems of linear inequalities
- Hoffman
- 1952
(Show Context)
Citation Context ... of linearly regular N-tuples. FACT 5.23 (linear regularity and intersecting halfspaces). If each set Ci is a halfspace, then the N-tuple (C1,...,CN)is linearly regular. REMARK 5.24. In 1952, Hoffman =-=[66]-=- proved this fact, relying on some results by Agmon [1] for Euclidean spaces. It turns out that his proof also works for Hilbert spaces; a detailed proof will appear in the thesis of the first author.... |

123 | Geometric functional analysis and its applications - Holmes |

107 |
Convergence of convex sets and of solutions of variational inequalities
- Mosco
- 1969
(Show Context)
Citation Context ...ion algorithm. However, before we can do so, we first must understand the meaning of a focusing projection algorithm. A first prototype is formulated in terms of set convergence in the sense of Mosco =-=[82]-=- (see [10] for a good survey article on set convergence). It is essentially a reformulation of Tsukada’s [101] characterization of Mosco convergence. LEMMA 4.2. Suppose (Sn) is a sequence of closed co... |

102 | The foundations of set theoretic estimation - Combettes - 1993 |

100 |
Fixed points of nonexpanding maps
- Halpern
- 1967
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Citation Context ... modification of [9, Thm. 1.1], one can show that (T nx0) converges in norm whenever S1,...,SN are closed affine subspaces. REMARK 2.22. We conclude this section by mentioning a method due to Halpern =-=[62]-=- which generates a sequence that converges in norm to the fixed point of T that is closest to the starting point. For extensions of Halpern’s result, the interested reader is referred to Lions’s [77],... |

93 |
Image Recovery: Theory and Application
- Stark
- 1987
(Show Context)
Citation Context ...space. Fairly simple algorithmic schemes. Basic results: Gubin, Polyak, and Raik [60]. Comments: Quality of convergence is fairly well understood. References: Herman [63], Youla and Webb [108], Stark =-=[95]-=-. Areas of application: Computerized tomography, signal processing. IV. Subgradient algorithms. Properties: Some sets Ci are of type 2. Fairly simple algorithmic schemes (“cyclic” or “weighted” contro... |

87 |
The Relaxation Method for Linear Inequalities
- Agmon
- 1954
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Citation Context ...es: Each set Ci is a halfspace or a hyperplane. X is a Euclidean space (i.e., a finite-dimensional Hilbert space). Very flexible algorithmic schemes. Basic results: Kaczmarz [71], Cimmino [29], Agmon =-=[1]-=-, Motzkin and Schoenberg [83]. Comments: Behaviour in general Hilbert space and quality of convergence only partially understood. References: Censor [21, 23, 24], Censor and Herman [27], Viergever [10... |

84 |
The method of successive projection for finding a common point of convex sets
- Bregman
- 1965
(Show Context)
Citation Context ...test set control is an old and successful concept. In 1954, Agmon [1] and Motzkin and Schoenberg [83] studied projection algorithms for solving linear inequalities using remotest set control. Bregman =-=[16]-=- considered the situation when there is an arbitrary collection of intersecting closed convex sets. We will recapture Agmon’s main result [1, Thm. 3] and some generalizations in §6. 5. Guaranteeing no... |

82 |
Approximation of fixed points of nonexpansive mappings
- Wittmann
- 1992
(Show Context)
Citation Context ...ates a sequence that converges in norm to the fixed point of T that is closest to the starting point. For extensions of Halpern’s result, the interested reader is referred to Lions’s [77], Wittmann’s =-=[104]-=-, and the first author’s [11]. 3. The algorithm: Basic properties and convergence results. Setting. Suppose D is a closed convex nonempty set and C1,...,CN are finitely many closed convex subsets of D... |

80 |
On the convergence of von Neumanns alternating projection algorithm for two sets, Set-Valued Analysis
- Bauschke, Borwein
(Show Context)
Citation Context ... as von Neumann’s alternating projection algorithm. Since projections are idempotent, one can view the sequence generated by the random projection algorithm as an alternating projection algorithm. In =-=[13]-=-, we discussed this algorithm in some detail and provided sufficient conditions for norm (or even linear) convergence. In 1965, Amemiya and Ando [5] proved weak convergence for Case (ii)—this is still... |

77 | The convex feasibility problem in image recovery - Combettes - 1996 |

73 |
Row-action methods for huge and sparse systems and their applications
- Censor
- 1981
(Show Context)
Citation Context ...sic results: Kaczmarz [71], Cimmino [29], Agmon [1], Motzkin and Schoenberg [83]. Comments: Behaviour in general Hilbert space and quality of convergence only partially understood. References: Censor =-=[21, 23, 24]-=-, Censor and Herman [27], Viergever [102], Sezan [91]. Areas of application: Medical imaging and radiation therapy treatment planning (computerized tomography), electron microscopy. III. Image reconst... |

72 |
Nonexpansive projections and resolvents of accretive operators
- Bruck, Reich
- 1977
(Show Context)
Citation Context ...use) the fact that the class of attracting mappings properly contains all three of the following classes: the class of strictly nonexpansive mappings, Bruck and Reich’s strongly nonexpansive mappings =-=[19]-=-, and a very nice class of nonexpansive mappings introduced by De Pierro and Iusem [41, Def. 1]. The mapping x ↦−→ 1−ln(1+e x ) is a first example of a mapping that is strictly nonexpansive but not av... |

72 | Inconsistent signal feasibility problems: Least-squares solutions in a product space - Combettes - 1994 |

69 |
The method of cyclic projections for closed convex sets in Hilbert space
- Bauschke, Borwein, et al.
- 1997
(Show Context)
Citation Context ... one obtains the method of cyclic projections; the conclusion of the last example becomes Bregman’s [16, Thm. 1]. The case when the sets Ci do not necessarily intersect is discussed in some detail in =-=[14]-=-.s402 H. H. BAUSCHKE AND J. M. BORWEIN EXAMPLE 6.13 (Crombez’s [38, Thm. 3]). Suppose the projection algorithm is weighted and has constant sets. Suppose further the relaxation parameters and weights ... |

67 |
Mathematical theory of image restoration by the method of convex projections
- Youla
- 1987
(Show Context)
Citation Context ...f. By Corollary 5.14 (cf. Remark 5.15), the N-tuple (C1,...,CN) is boundedly linearly regular. Now apply Theorem 5.7. REMARK 6.7. An extended version of Youla and Webb’s well-written paper is Youla’s =-=[106]-=-. Analogously, we can prove the following. EXAMPLE 6.8 (Gubin, Polyak, and Raik’s [60, Thm. 1.(a)]). Suppose the projection algorithm is cyclic and has constant sets. Suppose further there is some ɛ>0... |

65 |
Convergence theorems for sequences of nonlinear operators in Banach spaces
- Browder
- 1967
(Show Context)
Citation Context ...r. We will analyze algorithms in a very broad and adaptive framework that is essentially due to Fl˚am and Zowe [53]. (Related frameworks with somewhat different ambitions were investigated by Browder =-=[17]-=- and Schott [89].) The algorithmic scheme is as follows. Given the current iterate x (n) , the next iterate x (n+1) is obtained by (∗) x (n+1) := A (n) x (n) � N� := λ (n) � i (1 − α (n) i )Id + α (n)... |

63 | The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space
- Bauschke
- 1996
(Show Context)
Citation Context ... in norm to the fixed point of T that is closest to the starting point. For extensions of Halpern’s result, the interested reader is referred to Lions’s [77], Wittmann’s [104], and the first author’s =-=[11]-=-. 3. The algorithm: Basic properties and convergence results. Setting. Suppose D is a closed convex nonempty set and C1,...,CN are finitely many closed convex subsets of D with nonempty intersection: ... |

62 |
Decomposition through formalization in a product space
- Pierra
- 1984
(Show Context)
Citation Context ...(x (n) ,C2)−→ 0, but d(x (n) ,C1∩C2)= �x (n) �−→1.Therefore, (C1,C2)is not boundedly regular and the proof is complete. A short excursion into a useful product space. We build—in the spirit of Pierra =-=[85]-=-— the product space and define the diagonal and the product X := � N i=1 1 (X, N 〈·, ·〉) ∆ :={(x1,...,xN)∈X:x1 =x2 =···=xN ∈X} B := � N i=1 Ci. This allows us to identify the set C with ∆ ∩ B. Then (s... |

62 |
Minimization of unsmooth functionals
- Polyak
- 1969
(Show Context)
Citation Context ...mography, signal processing. IV. Subgradient algorithms. Properties: Some sets Ci are of type 2. Fairly simple algorithmic schemes (“cyclic” or “weighted” control). Basic results: Eremin [52], Polyak =-=[86]-=-, Censor and Lent [28]. Comments: Quality of convergence is fairly well understood. References: Censor [22], Shor [92]. Areas of application: Solution of convex inequalities, minimization of convex no... |

56 |
The Relaxation Method for Linear Inequalities
- Motzkin, Schoenberg
- 1954
(Show Context)
Citation Context ...ce or a hyperplane. X is a Euclidean space (i.e., a finite-dimensional Hilbert space). Very flexible algorithmic schemes. Basic results: Kaczmarz [71], Cimmino [29], Agmon [1], Motzkin and Schoenberg =-=[83]-=-. Comments: Behaviour in general Hilbert space and quality of convergence only partially understood. References: Censor [21, 23, 24], Censor and Herman [27], Viergever [102], Sezan [91]. Areas of appl... |

54 | Practical and mathematical aspects of the problem of reconstructing objects from radiographs - SMITH, SOLOMON, et al. - 1977 |

49 |
The product of projection operators
- Halperin
- 1962
(Show Context)
Citation Context ...ir applications. I. Best approximation theory. Properties: Each set Ci is a closed subspace. The algorithmic scheme is simple (“cyclic” control). Basic results: von Neumann [103, Thm. 13.7], Halperin =-=[61]-=-. Comments: The generated sequence converges in norm to the point inC that is closest to the starting point. Quality of convergence is well understood. References: Deutsch [44]. ∗Received by the edito... |

47 |
Block-iterative projection methods for parallel computation of solutions to convex feasibility problems. Linear Algebra and its Applications 120
- Aharoni, Censor
- 1989
(Show Context)
Citation Context ... of computerized tomography. It is worthwhile to point out that the scheme (∗) can be thought of as a combination of the schemes investigated by Aharoni, Berman, and Censor [2] and Aharoni and Censor =-=[3]-=-. In Euclidean spaces, norm convergence results were obtained by Fl˚am and Zowe for (∗) and by Aharoni and Censor [3] for the restricted version. However, neither behaviour in general Hilbert spaces n... |

47 |
On certain inequalities and characteristic value problems for analytic functions and for functions of two variables
- Friedrichs
- 1937
(Show Context)
Citation Context ...etely characterize regularity of an N-tuple of closed subspaces. We begin with the case when N = 2. Recall that the angle γ = γ(C1,C2)∈ [0,π/2] between two subspaces C1,C2 is given by (see Friedrichs =-=[54]-=- or Deutsch [42, 43]) cos γ = sup{〈c1,c2〉:c1 ∈C1∩(C1 ∩ C2) ⊥ ,c2 ∈C2∩(C1 ∩ C2) ⊥ , �c1�=�c2�=1}. PROPOSITION 5.16. If C1,C2 are two closed subspaces and γ is the angle between them, then the following... |

47 | Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections - Combettes - 1997 |

46 |
An example concerning fixed points
- Genel, Lindenstrauss
- 1975
(Show Context)
Citation Context ...erges in norm to some point in C. Proof. This follows from Lemma 3.2.(iv) and Theorem 2.16. REMARK 3.4. If the interior of C is empty, then the convergence might only be weak: Genel and Lindenstrauss =-=[56]-=- present an example of a firmly nonexpansive self mapping T of some closed convex nonempty set in ℓ2 such that the sequence of iterates (T n x0) converges weakly but not in norm for some starting poin... |

45 |
Iterative algorithms for large partitioned linear systems, with applications to image reconstruction
- Eggermont, Herman, et al.
- 1981
(Show Context)
Citation Context ... REMARK 6.26. Herman et al. [65] used block control variants of Example 6.20 for image reconstruction. Their algorithms are based on a (more matrix-theoretic) framework by Eggermont, Herman, and Lent =-=[49]-=-. Weighted control. EXAMPLE 6.27 (Trummer’s [98, Thm. 8]). Suppose X is finite dimensional and the projection algorithm is weighted, unrelaxed, and has constant sets that are hyperplanes Ci = {x ∈ X :... |

39 | Dotson: Approximating fixed points of nonexpansive mappings - Senter, G - 1974 |

39 | Hilbertian convex feasibility problem: convergence of projection methods - Combettes - 1997 |

36 |
Rate of convergence of the method of alternating projections,” in Parametric Optimization and Approximation
- Deutsch
- 1983
(Show Context)
Citation Context ...ze regularity of an N-tuple of closed subspaces. We begin with the case when N = 2. Recall that the angle γ = γ(C1,C2)∈ [0,π/2] between two subspaces C1,C2 is given by (see Friedrichs [54] or Deutsch =-=[42, 43]-=-) cos γ = sup{〈c1,c2〉:c1 ∈C1∩(C1 ∩ C2) ⊥ ,c2 ∈C2∩(C1 ∩ C2) ⊥ , �c1�=�c2�=1}. PROPOSITION 5.16. If C1,C2 are two closed subspaces and γ is the angle between them, then the following conditions are equi... |

35 |
Iteration processes for nonexpansive mappings
- Goebel, Kirk
- 1983
(Show Context)
Citation Context ...Fejér monotone sequences; see [79, Proof of Thm. 1] and [90, Proof of Thm. 2]. However, tremendous progress has been made and today the iteration is studied in normed or even more general spaces (see =-=[15, 57]-=- for further information). EXAMPLE 2.20 (Example 2.14 continued). The sequence (T nx0) converges weakly to some fixed point of T for every x0. Proof. (T nx0) is asymptotically regular (Example 2.14) a... |

34 |
On the use of Cimmino's simultaneous projections method for computing a solution of the inverse problem in radiation therapy treatment planning, Inverse Problems 4
- Censor, Altschuler, et al.
- 1988
(Show Context)
Citation Context ...hm or of intermittent control. Following Censor [21], we call an algorithm almost cyclic if it is intermittent and singular. We say the algorithm considers only blocks and speak of block control (cf. =-=[25]-=-) and a block algorithm if the following two conditions hold. 1. There is a decomposition J1 ∩···∩JM ={1,...,N}with Jm �= ∅ and Jm ∩ Jm ′ =∅ for all m, m ′ ∈{1,...,M}and m �= m ′ . 2. There is a posit... |

33 |
Krasnoselski-Mann iterations in normed spaces
- Borwein, Reich, et al.
- 1992
(Show Context)
Citation Context ...Fejér monotone sequences; see [79, Proof of Thm. 1] and [90, Proof of Thm. 2]. However, tremendous progress has been made and today the iteration is studied in normed or even more general spaces (see =-=[15, 57]-=- for further information). EXAMPLE 2.20 (Example 2.14 continued). The sequence (T nx0) converges weakly to some fixed point of T for every x0. Proof. (T nx0) is asymptotically regular (Example 2.14) a... |

29 | Fixed Point Theory - Istratescu - 1981 |

28 | Topology and Normed Spaces - JAMESON - 1974 |

25 |
On some optimization techniques in image reconstruction from projections
- Censor, Herman
- 1987
(Show Context)
Citation Context ...ino [29], Agmon [1], Motzkin and Schoenberg [83]. Comments: Behaviour in general Hilbert space and quality of convergence only partially understood. References: Censor [21, 23, 24], Censor and Herman =-=[27]-=-, Viergever [102], Sezan [91]. Areas of application: Medical imaging and radiation therapy treatment planning (computerized tomography), electron microscopy. III. Image reconstruction: Continuous mode... |

23 |
Cyclic subgradient projections
- Censor, Lent
- 1982
(Show Context)
Citation Context ...ssing. IV. Subgradient algorithms. Properties: Some sets Ci are of type 2. Fairly simple algorithmic schemes (“cyclic” or “weighted” control). Basic results: Eremin [52], Polyak [86], Censor and Lent =-=[28]-=-. Comments: Quality of convergence is fairly well understood. References: Censor [22], Shor [92]. Areas of application: Solution of convex inequalities, minimization of convex nonsmooth functions. To ... |

22 | Surrogate projection methods for finding fixed points of firmly nonexpansive mappings - Kiwiel, CLopuch - 1997 |

20 |
Approximation de points fixes de contractions, Comptes Rendus de l’Académie des Sciences 284
- Lions
- 1977
(Show Context)
Citation Context ... [62] which generates a sequence that converges in norm to the fixed point of T that is closest to the starting point. For extensions of Halpern’s result, the interested reader is referred to Lions’s =-=[77]-=-, Wittmann’s [104], and the first author’s [11]. 3. The algorithm: Basic properties and convergence results. Setting. Suppose D is a closed convex nonempty set and C1,...,CN are finitely many closed c... |

20 | Construction d’un point fixe commun à une famille de contractions fermes - Combettes - 1995 |

19 |
Convergence of random products of contractions in Hilbert space
- Amemiya, Ando
- 1965
(Show Context)
Citation Context ...have received much attention in radiation therapy treatment planning; see [25] and the subsection on polyhedra in §6. • Equivalent to the phrase “almost cyclic” is Amemiya and Ando’s “quasi-periodic” =-=[5]-=- or Browder’s “admissible (for finitely many sets)” [17]. THEOREM 3.20 (weak topology results). (i) Suppose the algorithm is focusing and intermittent. If lim n:n active for i µ (n) i > 0 for every in... |

19 |
An overview of convex projections theory and its application to image recovery problems, Ultramicroscopy 40
- Sezan
- 1992
(Show Context)
Citation Context ...and Schoenberg [83]. Comments: Behaviour in general Hilbert space and quality of convergence only partially understood. References: Censor [21, 23, 24], Censor and Herman [27], Viergever [102], Sezan =-=[91]-=-. Areas of application: Medical imaging and radiation therapy treatment planning (computerized tomography), electron microscopy. III. Image reconstruction: Continuous models. Properties: X is usually ... |

17 |
Convergence of sequential and asynchronous nonlinear paracontractions
- Elsner, Koltracht, et al.
- 1992
(Show Context)
Citation Context ...not require nonexpansivity in the definition of attracting mappings; see, for example, Bruck’s “strictly quasi-nonexpansive mappings” [18], Elsner, Koltracht, and Neumann’s “paracontracting mappings” =-=[51]-=-, Eremin’s “F -weakly Fejér maps” [52], and Istratescu’s “T -mappings” [69, Chap. 6]. For our purposes, however, the above definitions are already general enough. As we will see, the class of strongly... |

17 | Relaxation methods for image reconstruction - Herman, Lent, et al. - 1978 |

16 | Parallel application of block-iterative methods in medical imaging and radiation therapy - Censor - 1988 |

15 |
A Generalization of Polyak’s Convergence Result for Subgradient Optimization
- Allen, Helgason, et al.
- 1987
(Show Context)
Citation Context ... hence fi(x (nk) ) −→ 0. Therefore, fi(x) ≤ 0 and x ∈ Ci. The property that ∂fi is uniformly bounded on bounded sets is a standard assumption for theorems on subgradient algorithms; see, for example, =-=[52, 86, 28, 4, 40]-=-. We now characterize this property. PROPOSITION 7.8 (uniform boundedness of subdifferentials on bounded sets). Suppose f : X −→ R is a convex function. Then the following conditions are equivalent. (... |

15 | Une remarque sur le comportement asymptotique des semigroupes non lineaires - Baillon, Brézis - 1976 |

15 | A norm convergence result on random products of relaxed projections in Hilbert space - Bauschke - 1995 |

14 | On the behavior of a block-iterative projection method for solving convex feasibility problems - Butnariu, Censor - 1990 |

14 |
Relaxed outer projections, weighted averages, and convex feasibility
- Fl˚am, Zowe
- 1990
(Show Context)
Citation Context ...ble to draw conclusions on the quality of convergence. This is our objective in this paper. We will analyze algorithms in a very broad and adaptive framework that is essentially due to Fl˚am and Zowe =-=[53]-=-. (Related frameworks with somewhat different ambitions were investigated by Browder [17] and Schott [89].) The algorithmic scheme is as follows. Given the current iterate x (n) , the next iterate x (... |

13 |
Unrestricted iterations of nonexpansive mappings in Hilbert space, Nonlinear Analysis 18
- Dye, Reich
- 1992
(Show Context)
Citation Context ... • As we commented in Remarks 4.7, the problem becomes much harder without a compactness assumption. Nevertheless, some interesting results were obtained by Bruck [18], Youla [107], and Dye and Reich =-=[48]-=-. An immediate consequence of Example 6.1 is the following. EXAMPLE 6.3 (Bruck’s [18, Cor. 1.2]). Suppose the projection algorithm is singular and has constant sets where (at least) one is boundedly c... |

13 | Block-iterative surrogate projection methods for convex feasibility problems - Kiwiel - 1995 |

13 |
On the convergence of products of firmly nonexpansive mappings
- Tseng
- 1992
(Show Context)
Citation Context ...Weak topology results on intermittent focusing algorithms are given. We actually study a more general form of the iteration (∗) without extra work; as a by-product, we obtain a recent result by Tseng =-=[100]-=- and make connections with work by Browder [17] and Baillon [7]. At the start of §4, we exclusively consider algorithms such as (∗), which we name projection algorithms. Prototypes of focusing and lin... |

13 |
Convergence of best approximations in a smooth Banach space
- Tsukada
- 1984
(Show Context)
Citation Context ...gorithm. A first prototype is formulated in terms of set convergence in the sense of Mosco [82] (see [10] for a good survey article on set convergence). It is essentially a reformulation of Tsukada’s =-=[101]-=- characterization of Mosco convergence. LEMMA 4.2. Suppose (Sn) is a sequence of closed convex sets and there is some closed convex nonempty set S with S ⊆ Sn for all n. Then the following conditions ... |

11 | An interior points algorithm for the convex feasibility problem - Aharoni, Berman, et al. - 1983 |

11 |
Reconstructing pictures from projections: on the convergence of the ART algorithm with relaxation
- Trummer
- 1981
(Show Context)
Citation Context ... the last example, which was discovered by Kaczmarz as early as 1937. • Kaczmarz’s method is well understood even in the infeasible case; we refer the interested reader to Tanabe’s [96] and Trummer’s =-=[97, 99]-=-. • The iteration described in Example 6.21 is also known as “ART” (algebraic reconstruction technique). EXAMPLE 6.24 (Trummer’s [97, first part of Thm. 1]). Suppose X is finite dimensional and the pr... |

9 |
A.N.: A simultaneous projections method for linear inequalities
- Pierro, Iusem
- 1985
(Show Context)
Citation Context ...ar and independent of the starting point. For a slightly more restrictive scheme, De Pierro and Iusem could also identify the limit of (x (n) ) in the infeasible case as a least squares solution; see =-=[39]-=-. Consideration of remotest sets control. EXAMPLE 6.42. Suppose the projection algorithm considers remotest sets and has constant sets that are halfspaces. Suppose further that (i (n) ) is a sequence ... |

9 |
Fejér mappings and convex programming
- Eremin
- 1969
(Show Context)
Citation Context ...mputerized tomography, signal processing. IV. Subgradient algorithms. Properties: Some sets Ci are of type 2. Fairly simple algorithmic schemes (“cyclic” or “weighted” control). Basic results: Eremin =-=[52]-=-, Polyak [86], Censor and Lent [28]. Comments: Quality of convergence is fairly well understood. References: Censor [22], Shor [92]. Areas of application: Solution of convex inequalities, minimization... |

9 | The Solution by Iteration of Nonlinear Equations in Hilbert Spaces - Maruster - 1977 |

9 | Iterative methods for the convex feasibility problem - Censor - 1984 |

8 |
Random products of contractions in metric and banach spaces
- Bruck
- 1982
(Show Context)
Citation Context ...ttracting, or κ-attracting mappings. REMARKS 2.2. Some authors do not require nonexpansivity in the definition of attracting mappings; see, for example, Bruck’s “strictly quasi-nonexpansive mappings” =-=[18]-=-, Elsner, Koltracht, and Neumann’s “paracontracting mappings” [51], Eremin’s “F -weakly Fejér maps” [52], and Istratescu’s “T -mappings” [69, Chap. 6]. For our purposes, however, the above definitions... |

8 |
Image recovery by convex combinations of projections
- Crombez
- 1991
(Show Context)
Citation Context ...he relaxation parameters and weights satisfy 0 < α (n) i ≡ αi < 2 and 0 <λ (n) i ≡ λi for every index i and all n. Then the sequence (x (n) ) converges weakly to some point in C. REMARK 6.14. Crombez =-=[38]-=- assumed in addition that one of the sets is the entire space (which has the identity as projection). Consideration of remotest sets control. EXAMPLE 6.15 (Gubin, Polyak, and Raik’s [60, Thm. 1.(a)] f... |

7 | Convergence of the cyclical relaxation method for linear inequalities - Mandel - 1984 |

7 |
On a relaxation method of solving systems of linear inequalities
- Merzlyakov
- 1963
(Show Context)
Citation Context ...most violated constraints. Suppose further 0 <α (n) 1 ≡ α1 < 2. Then the sequence (x (n) ) converges linearly to some point in C with a rate independent of the starting point. REMARK 7.39. Merzlyakov =-=[80]-=- actually considered a more general version, where the ω (n) 1,i need not necessarily sum up to 1. EXAMPLE 7.40 (Yang and Murty’s [105]). In the polyhedral framework, suppose X is finite dimensional, ... |

7 |
A general iterative scheme with applications to convex optimization and related fields
- Schott
- 1991
(Show Context)
Citation Context ...ze algorithms in a very broad and adaptive framework that is essentially due to Fl˚am and Zowe [53]. (Related frameworks with somewhat different ambitions were investigated by Browder [17] and Schott =-=[89]-=-.) The algorithmic scheme is as follows. Given the current iterate x (n) , the next iterate x (n+1) is obtained by (∗) x (n+1) := A (n) x (n) � N� := λ (n) � i (1 − α (n) i )Id + α (n) � (n) i P i � x... |

7 |
New iterative methods for linear inequalities
- Yang, Murty
(Show Context)
Citation Context ...ceptionally simple proof of an important result by De Pierro and Iusem [40]. It is very satisfactory that analogous results are obtained for algorithms suggested by Dos Santos [47] and Yang and Murty =-=[105]-=-. For the reader’s convenience, an index is included. We conclude this section with a collection of frequently-used facts, definitions, and notation. The “stage” throughout this paper is a real Hilber... |

6 |
parallel subgradient method for the convex feasibility problem
- Santos, “A
- 1987
(Show Context)
Citation Context ...ing, thus yielding a conceptionally simple proof of an important result by De Pierro and Iusem [40]. It is very satisfactory that analogous results are obtained for algorithms suggested by Dos Santos =-=[47]-=- and Yang and Murty [105]. For the reader’s convenience, an index is included. We conclude this section with a collection of frequently-used facts, definitions, and notation. The “stage” throughout th... |

6 |
Parallel-projected aggregation methods for solving the convex feasibility problem
- Garćıa-Palomares
- 1993
(Show Context)
Citation Context ...but are not. The manuscript is merely a snapshot of what the authors knew in mid-1993; time, of course, has not stood still. The manuscripts sent to us recently by Combettes [30–37], García-Palomares =-=[55]-=-, and Kiwiel [72–75] deal with exciting new generalizations and deserve much attention. A synthesis of a selection of these results may be found in the first author’s Ph.D. thesis (Projection Algorith... |

6 |
Image reconstruction by the method of convex projections : Part 1
- Youla, Webb
- 1982
(Show Context)
Citation Context ...onal Hilbert space. Fairly simple algorithmic schemes. Basic results: Gubin, Polyak, and Raik [60]. Comments: Quality of convergence is fairly well understood. References: Herman [63], Youla and Webb =-=[108]-=-, Stark [95]. Areas of application: Computerized tomography, signal processing. IV. Subgradient algorithms. Properties: Some sets Ci are of type 2. Fairly simple algorithmic schemes (“cyclic” or “weig... |

5 | New methods for linear inequalities - Censor, Elfving - 1982 |

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Comportement asymptotique des contractions et semi-groupes de contractions - equations de schroedinger non lineaires et divers, Thèse
- Baillon
- 1978
(Show Context)
Citation Context ...en. We actually study a more general form of the iteration (∗) without extra work; as a by-product, we obtain a recent result by Tseng [100] and make connections with work by Browder [17] and Baillon =-=[7]-=-. At the start of §4, we exclusively consider algorithms such as (∗), which we name projection algorithms. Prototypes of focusing and linearly focusing (a stronger, more quantitative version) projecti... |

4 | set theoretic image recovery by extrapolated iterations of parallel subgradient projections - “Convex - 1997 |

4 |
cas de convergence des itérées d’une contraction d’un espace hilbertien
- Moreau, Un
- 1978
(Show Context)
Citation Context ... rate of convergence of (xn) follows easily from the estimate given in (v). REMARKS 2.17. As far as we know, the notion of Fejér monotonicity was coined by Motzkin and Schoenberg [83] in 1954. Moreau =-=[81]-=- inspired (iii); see also [69, Thm. 6.5.3]. (iv) rests on an idea by Baillon and Brezis [8, Lemme 3] and partially extends [46, Thm. 3.4.(c)]. Finally, (v) and (vi) appeared implicitly in Gubin, Polya... |

4 | Convergence of sequences of sets - Baronti, Papini - 1986 |

4 | On the method of cyclic projections for convex sets in Hilbert space.” Research report - Bauschke, Borwein, et al. - 1994 |

4 | On variable block algebraic reconstruction techniques, in - Censor - 1991 |

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Connections between the Cimmino-method and the Kaczmarz-method for the solution of singular and regular systems of equations
- ANSORGE
- 1984
(Show Context)
Citation Context ...y used with great success; see Censor’s survey article [23]. We present a variation of Cimmino’s method that includes a method suggested by Ansorge. EXAMPLE 6.32 (a generalization of Ansorge’s method =-=[6]-=-). Suppose X is finite dimensional and the projection algorithm has constant sets where CN = X. Suppose further that and α (n) N ≡ 1, λ(n) N ≡ λN > 0, α (n) 1 ≡···≡α(n) N−1 ≡ 2, λ (n) ⎧ ⎪⎨ i := ⎪⎩ (1 ... |

3 | convex feasibility problem: Convergence of projection methods - Hilbertian - 1997 |

3 | Extrapolated projection method for the euclidean convex feasibility problem - Combettes, Puh - 1993 |

3 |
Introduction to discrete reconstruction methods in medical imaging
- VIERGEVER
- 1988
(Show Context)
Citation Context ...[1], Motzkin and Schoenberg [83]. Comments: Behaviour in general Hilbert space and quality of convergence only partially understood. References: Censor [21, 23, 24], Censor and Herman [27], Viergever =-=[102]-=-, Sezan [91]. Areas of application: Medical imaging and radiation therapy treatment planning (computerized tomography), electron microscopy. III. Image reconstruction: Continuous models. Properties: X... |

2 | On the asymptotic behaviour of nonexpansive mappings and semigroups in Banach spaces - BAILLON, BRUCK, et al. - 1978 |

2 |
Convergence of sequences of sets, in Methods of Functional Analysis in Approximation Theory
- BARONTI, PAPINI
- 1986
(Show Context)
Citation Context ...thm. However, before we can do so, we first must understand the meaning of a focusing projection algorithm. A first prototype is formulated in terms of set convergence in the sense of Mosco [82] (see =-=[10]-=- for a good survey article on set convergence). It is essentially a reformulation of Tsukada’s [101] characterization of Mosco convergence. LEMMA 4.2. Suppose (Sn) is a sequence of closed convex sets ... |

2 | signal feasibility problems: Least-squares solutions in a product space - “Inconsistent - 1994 |

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Projection method for solving a linear system of linear equations and its applications
- TANABE
- 1971
(Show Context)
Citation Context ...esults is certainly the last example, which was discovered by Kaczmarz as early as 1937. • Kaczmarz’s method is well understood even in the infeasible case; we refer the interested reader to Tanabe’s =-=[96]-=- and Trummer’s [97, 99]. • The iteration described in Example 6.21 is also known as “ART” (algebraic reconstruction technique). EXAMPLE 6.24 (Trummer’s [97, first part of Thm. 1]). Suppose X is finite... |

1 |
Calcolo approssimate per le soluzioni dei sistemi di equazioni lineari. La Ricerca scientifica ed il Progresso tecnico nell' Economia nazionale
- CIMMINO
- 1938
(Show Context)
Citation Context ...ls. Properties: Each set Ci is a halfspace or a hyperplane. X is a Euclidean space (i.e., a finite-dimensional Hilbert space). Very flexible algorithmic schemes. Basic results: Kaczmarz [71], Cimmino =-=[29]-=-, Agmon [1], Motzkin and Schoenberg [83]. Comments: Behaviour in general Hilbert space and quality of convergence only partially understood. References: Censor [21, 23, 24], Censor and Herman [27], Vi... |

1 | of parallel convex projections in Hilbert space - Iterations - 1994 |

1 | ROUHANI, Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space - DJAFARI - 1990 |

1 |
Multilevel image reconstruction, in Multiresolution Image Processing and Analysis
- HERMAN, LEVKOWITZ, et al.
- 1984
(Show Context)
Citation Context ..., we can similarly recapture Trummer’s [97, second part of Thm. 1], where he describes an iteration that yields a nonnegative solution (assuming there exists at least one). REMARK 6.26. Herman et al. =-=[65]-=- used block control variants of Example 6.20 for image reconstruction. Their algorithms are based on a (more matrix-theoretic) framework by Eggermont, Herman, and Lent [49]. Weighted control. EXAMPLE ... |

1 | ADSTR OM. An embedding theorem for spaces of convex sets - R - 1952 |

1 |
personal communication
- SIMONIČ
(Show Context)
Citation Context ...) (C1,C2)is linearly regular. (v) (C1,C2)is boundedly linearly regular. (vi) (C1,C2)is regular. (vii) (C1,C2)is boundedly regular. Proof. “(i)⇐⇒(ii)” is due to Deutsch [42, Lem. 2.5.(4)] and Simonič (=-=[93]-=-, a proof can be found in [13, Lem. 4.10]). “(ii)⇐⇒(iii)” is well known (see Jameson’s [70, Cor. 35.6]). “(ii)�⇒(iv)⇐⇒(v)⇐⇒(vi)�⇒(vii)”: Combine Fact 5.12.(ii) and Proposition 5.9. “(vii)�⇒(i)”: Let u... |