Algorithmic Projection Operators

1.

S. Agmon, The relaxation method for linear inequalities. Can. J. Math. 6, 382–392 (1954)

2.

R. Aharoni, A. Berman, Y. Censor, An interior point algorithm for the convex feasibility problem. Adv. Appl. Math. 4, 479–489 (1983)

3.

R. Aharoni, Y. Censor, Block-iterative projection methods for parallel computation of solutions to convex feasibility problems. Lin. Algebra Appl. 120, 165–175 (1989)

4.

D. Alevras, M.W. Padberg, Linear Optimization and Extensions. Problems and Solutions (Springer, Berlin, 2001)

5.

A. Aleyner, S. Reich, Block-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach spaces. J. Math. Anal. Appl. 343, 427–435 (2008)

6.

A. Aleyner, S. Reich, Random products of quasi-nonexpansive mappings in Hilbert space. J. Convex Anal. 16, 633–640 (2009)

7.

M. Altman, On the approximate solution of linear algebraic equations. Bulletin de l’Académie Polonaise des Sciences Cl. III 3 , 365–370 (1957)

8.

I. Amemiya, T. Ando, Convergence of random products of contractions in Hilbert space. Acta Sci. Math. (Szeged) 26, 239–244 (1965)

9.

R. Ansorge, Connections between the Cimmino-method and the Kaczmarz-method for solution of singular and regular systems of equations. Computing 33, 367–375 (1984)

10.

G. Appleby, D.C. Smolarski, A linear acceleration row action method for projecting onto subspaces. Electron. Trans. Numer. Anal. 20, 253–275 (2005)

11.

N. Aronszajn, Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

12.

A. Auslender, Optimisation, méthodes numériques (Masson, Paris, 1976)

13.

V.N. Babenko, Convergence of the Kaczmarz projection algorithm. Zh. Vychisl. Mat. Mat. Fiz. 24, 1571–1573 (1984) (in Russian)

14.

J.B. Baillon, R.E. Bruck, S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4, 1–9 (1978)

15.

S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrals. Fundamenta Mathematicae 3, 133–181 (1922)

16.

H.H. Bauschke, A norm convergence result on random products of relaxed projections in Hilbert space. Trans. Am. Math. Soc. 347, 1365–1373 (1995)

17.

H.H. Bauschke, Projection Algorithms and Monotone Operators, Ph.D. Thesis, Department of Mathematics, Simon Fraser, University, Burnaby, BC, Canada, 1996

18.

H.H. Bauschke, The approximation of fixed points of compositions of nonexpansive mapping in Hilbert space. J. Math. Anal. Appl. 202, 150–159 (1996)

19.

H.H. Bauschke, The composition of the projections onto closed convex sets in Hilbert space is asymptotically regular. Proc. Am. Math. Soc. 131, 141–146 (2002)

20.

H.H. Bauschke, J. Borwein, On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1, 185–212 (1993)

21.

H.H. Bauschke, J. Borwein, Dykstra’s alternating projection algorithm for two sets. J. Approx. Theor. 79, 418–443 (1994)

22.

H.H. Bauschke, J. Borwein, On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)

23.

H.H. Bauschke, J.M. Borwein, A.S. Lewis, The method of cyclic projections for closed convex sets in Hilbert space. Contemp. Math. 204, 1–38 (1997)

24.

H.H. Bauschke, P.L. Combettes, A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001)

25.

H.H. Bauschke, P.L. Combettes, S.G. Kruk, Extrapolation algorithm for affine-convex feasibility problems. Numer. Algorithms 41, 239–274 (2006)

26.

H.H. Bauschke, P.L. Combettes, D.R. Luke, Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization. J. Opt. Soc. Am. A 19, 1334–1345 (2002)

27.

H.H. Bauschke, P.L. Combettes, D.R. Luke, Hybrid projection-reflection method for phase retrieval. J. Opt. Soc. Am. A 20, 1025–1034 (2003)

28.

H.H. Bauschke, P.L. Combettes, D.R. Luke, Finding best approximation pairs relative to two closed convex sets in Hilbert spaces. J. Approx. Theor. 127, 178–192 (2004)

29.

H.H. Bauschke, P.L. Combettes, D.R. Luke, A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space. J. Approx. Theor. 141, 63–69 (2006)

30.

H.H. Bauschke, F. Deutsch, H. Hundal, S-H. Park, Accelerating the convergence of the method of alternating projections. Trans. Am. Math. Soc. 355, 3433–3461 (2003)

31.

H.H. Bauschke, S.G. Kruk, Reflection-projection method for convex feasibility problems with an obtuse cone. J. Optim. Theor. Appl. 120, 503–531 (2004)

32.

H.H. Bauschke, E. Matoušková, S. Reich, Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 56, 715–738 (2004)

33.

M.H. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming, Theory and Algorithms, 3rd edn. (Wiley, Hoboken, 2006)

34.

M. Benzi, C.D. Meyer, A direct projection method for sparse linear systems. SIAM J. Sci. Comput. 16, 1159–1176 (1995)

35.

A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (Academic, New York, 1979)

36.

V. Berinde, Iterative Approximation of Fixed Points (Springer, Berlin, 2007)

37.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, Bristol, 1998)

38.

D.P. Bertsekas, Nonlinear Programming (Athena Scientific, Belmont, 1995)

39.

D. Blatt, A.O. Hero, Energy-based sensor network source localization via projection onto convex sets (POCS). IEEE Trans. Signal Process. 54, 3614–3619 (2006)

40.

J.M. Borwein, A. Lewis, Convex Analysis and Nonlinear Optimization, Theory and Examples (Springer, New York, 2000)

41.

R. Bramley, A. Sameh, Row projection methods for large nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 168–193 (1992)

42.

L.M. Bregman, Finding the common point of convex sets by the method of successive projection (in Russian). Dokl. Akad. Nauk SSSR 162, 487–490 (1965); English translation in: Soviet Math. Dokl. 6, 688–692 (1965)

43.

L.E.J. Brouwer, Über Abbildung von Mannigfaltigkeiten. Math. Ann. 71, 97–115 (1912)

44.

F.E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space. Proc. Nat. Acad. Sci. USA 53, 1272–1276 (1965)

45.

F.E. Browder, Nonexpansive nonlinear operators in a Banach space. Proc. Nat. Acad. Sci. USA 54, 1041–1044 (1965)

46.

F.E. Browder, Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Rational Mech. Anal. 24, 82–90 (1967)

47.

F.E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Zeitschr. 100, 201–225 (1967)

48.

F.E. Browder, W.V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces. Bull. Am. Math. Soc. 72, 571–575 (1966)

49.

R.E. Bruck, Nonexpansive projections on subsets of Banach spaces. Pac. J. Math. 47, 341–355 (1973)

50.

R.E. Bruck, Random products of contractions in metric and Banach spaces. J. Math. Anal. Appl. 88, 319–332 (1982)

51.

R.E. Bruck, S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459–470 (1977)

52.

R.S. Burachik, J.O. Lopes, B.F. Svaiter, An outer approximation method for the variational inequality problem. SIAM J. Contr. Optim. 43, 2071–2088 (2005)

53.

D. Butnariu, Y. Censor, On the behavior of a block-iterative projection method for solving convex feasibility problems. Int. J. Comp. Math. 34, 79–94 (1990)

54.

D. Butnariu, Y. Censor, P. Gurfil, E. Hadar, On the behavior of subgradient projections methods for convex feasibility problems in Euclidean spaces. SIAM J. Opt. 19, 786–807 (2008)

55.

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)

56.

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)

57.

C.L. Byrne, Applied Iterative Methods (AK Peters, Wellesley, 2008)

58.

C.L. Byrne, Bounds on the largest singular value of a matrix and the convergence of simultaneous and block-iterative algorithms for sparse linear systems. Int. Trans. Oper. Res. 16, 465–479 (2009)

59.

T.D. Capricelli, P.L. Combettes, Parallel block-iterative reconstruction algorithms for binary tomography. Electron. Notes Discr. Math. 20, 263–280 (2005)

60.

G. Casssiani, G. Böhm, A. Vesnaver, R. Nicolich, A geostatistical framework for incorporating seismic tomography auxiliary data into hydraulic conductivity estimation. J. Hydrol. 206, 58–74 (1998)

61.

J. Cea, Optimisation: théorie et algorithmes (Dunod, Paris, 1971); Polish translation: Optymalizacja: Teoria i algorytmy (PWN, Warszawa, 1976)

62.

A. Cegielski, Relaxation Methods in Convex Optimization Problems (in Polish). Monographs, vol. 67, Institute of Mathematics, Higher College of Engineering, Zielona Góra, 1993

63.

A. Cegielski, in Projection Onto an Acute Cone and Convex Feasibility Problems, ed. by J. Henry i J.-P. Yvon. Lecture Notes in Control and Inform. Sci., vol. 197 (Springer, London, 1994), pp. 187–194

64.

A. Cegielski, A method of projection onto an acute cone with level control in convex minimization. Math. Program. 85, 469–490 (1999)

65.

A. Cegielski, Obtuse cones and Gram matrices with nonnegative inverse. Lin. Algebra Appl. 335, 167–181 (2001)

66.

A. Cegielski, A generalization of the Opial’s theorem. Contr. Cybern. 36, 601–610 (2007)

67.

A. Cegielski, Convergence of the projected surrogate constraints method for the linear split feasibility problems. J. Convex Anal. 14, 169–183 (2007)

68.

A. Cegielski, Projection methods for the linear split feasibility problems. Optimization 57, 491–504 (2008)

69.

A. Cegielski, Generalized relaxations of nonexpansive operators and convex feasibility problems. Contemp. Math. 513, 111–123 (2010)

70.

A. Cegielski, Y. Censor, in Opial-Type Theorems and the Common Fixed Point Problem, ed. by H.H. Bauschke, R.S. Burachik, P.L. Combettes, V. Elser, D.R. Luke, H. Wolkowicz. Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optimization and Its Applications, vol. 49 (Springer, New York, 2011), pp. 155–183

71.

A. Cegielski, Y. Censor, Extrapolation and local acceleration of an iterative process for common fixed point problems. J. Math. Anal. Appl. 394, 809–818 (2012)

72.

A. Cegielski, R. Dylewski, Selection strategies in projection methods for convex minimization problems. Discuss. Math. Differ. Incl. Contr. Optim. 22, 97–123 (2002)

73.

A. Cegielski, R. Dylewski, Residual selection in a projection method for convex minimization problems. Optimization 52, 211–220 (2003)

74.

A. Cegielski, R. Dylewski, Variable target value relaxed alternating projection method. Comput. Optim. Appl. 47, 455–476 (2010)

75.

A. Cegielski, A. Suchocka, Incomplete alternating projection method for large inconsistent linear systems. Lin. Algebra Appl. 428, 1313–1324 (2008)

76.

A. Cegielski, A. Suchocka, Relaxed alternating projection methods. SIAM J. Optim. 19, 1093–1106 (2008)

77.

A. Cegielski, R. Zalas, Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators (2012) Numer. Funct. Anal. Optim. (in print)

78.

Y. Censor, Row-action methods for huge and sparse systems and their applications. SIAM Rev. 23, 444–466 (1981)

79.

Y. Censor, Iterative methods for convex feasibility problems. Ann. Discrete Math. 20, 83–91 (1984)

80.

Y. Censor, An automatic relaxation method for solving interval linear inequalities. J. Math. Anal. Appl. 106, 19–25 (1985)

81.

Y. Censor, Parallel application of block-iterative methods in medical imaging and radiation therapy. Math. Program. 42, 307–325 (1988)

82.

Y. Censor, Binary steering in discrete tomography reconstruction with sequential and simultaneous iterative algorithms. Lin. Algebra Appl. 339, 111–124 (2001)

83.

Y. Censor, M.D. Altschuler, W.D. Powlis, On the use of Cimmino’s simultaneous projections method for computing a solution of the inverse problem in radiation therapy treatment planning. Inverse Probl. 4, 607–623 (1988)

84.

Y. Censor, A. Ben-Israel, Y. Xiao, J.M. Galvin, On linear infeasibility arising in intensity-modulated radiation therapy inverse planning. Lin. Algebra Appl. 428, 1406–1420 (2008)

85.

Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)

86.

Y. Censor, P.P.B. Eggermont, D. Gordon, Strong underrelaxation in Kaczmarz’s method for inconsistent systems. Numer. Math. 41, 83–92 (1983)

87.

Y. Censor, T. Elfving, New methods for linear inequalities. Lin. Algebra Appl. 42, 199–211 (1982)

88.

Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithm 8, 221–239 (1994)

89.

Y. Censor, T. Elfving, Block-iterative algorithms with diagonal scaled oblique projections for the linear feasibility problems. SIAM J. Matrix Anal. Appl. 24, 40–58 (2002)

90.

Y. Censor, T. Elfving, Iterative algorithms with seminorm-induced oblique projections. Abstr. Appl. Anal. 8, 387–406 (2003)

91.

Y. Censor, T. Elfving, G.T. Herman, in Averaging Strings of Sequential Iterations for Convex Feasibility Problems, ed. by D. Butnariu, Y. Censor, S. Reich. Inherently Parallel Algorithms in Feasibility and Optimization and their Applications (Elsevier, Amsterdam, 2001), pp. 101–113

92.

Y. Censor, T. Elfving, G.T. Herman, T. Nikazad, On diagonally relaxed orthogonal projection methods. SIAM J. Sci. Comput. 30, 473–504 (2008)

93.

Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)

94.

Y. Censor, A. Gibali, Projections onto super-half-spaces for monotone variational inequality problems in finite-dimensional spaces. J. Nonlinear Convex Anal. 9, 461–475 (2008)

95.

Y. Censor, A. Gibali, S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)

96.

Y. Censor, A. Gibali, S. Reich, Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61, 1119–1132 (2012)

97.

Y. Censor, A. Gibali, S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Meth. Software 26, 827–845 (2011)

98.

Y. Censor, D. Gordon, R. Gordon, Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems. Parallel Comput. 27, 777–808 (2001)

99.

Y. Censor, D. Gordon, R. Gordon, BICAV: A block-iterative, parallel algorithm for sparse systems with pixel-related weighting. IEEE Trans. Med. Imag. 20, 1050–1060 (2001)

100.

Y. Censor, G.T. Herman, On some optimization techniques in image reconstruction from projections. Appl. Numer. Math. 3, 365–391 (1987)

101.

Y. Censor, A.N. Iusem, S.A. Zenios, An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Math. Program. 81, 373–400 (1998)

102.

Y. Censor, A. Lent, Cyclic subgradient projections. Math. Program. 24, 233–235 (1982)

103.

Y. Censor, A. Motova, A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 327, 1244–1256 (2007)

104.

Y. Censor, A. Segal, The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)

105.

Y. Censor, A. Segal, On the string averaging method for sparse common fixed point problems. Int. Trans. Oper. Res. 16, 481–494 (2009)

106.

Y. Censor, A. Segal, Sparse string-averaging and split common fixed points. Contemp. Math. 513, 125–142 (2010)

107.

Y. Censor, E. Tom, Convergence of string-averaging projection schemes for inconsistent convex feasibility problems. Optim. Meth. Software 18, 543–554 (2003)

108.

Y. Censor, S.A. Zenios, Parallel Optimization, Theory, Algorithms and Applications (Oxford University Press, New York, 1997)

109.

A.E. Çetin, H. Özaktaş, H.M. Ozaktas, Resolution enhancement of low resolution wavefields with POCS algorithm. Electron. Lett. 9, 1808–1810 (2003)

110.

W. Chen, D. Craft, T.M. Madden, K. Zhang, H.M. Kooy, G.T. Herman, A fast optimization algorithm for multicriteria intensity modulated proton therapy planning. Med. Phys. 7, 4938–4945 (2010)

111.

W. Chen, G.T. Herman, Efficient controls for finitely convergent sequential algorithms. ACM Trans. Math. Software 37, 1–23 (2010)

112.

W. Cheney, A.A. Goldstein, Proximity maps for convex sets. Proc. Am. Math Soc. 10, 448–450 (1959)

113.

C.E. Chidume, Quasi-nonexpansive mappings and uniform asymptotic regularity. Kobe J. Math. 3, 29–35 (1986)

114.

Ch. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations (Springer, London, 2009)

115.

H. Choi, R.G. Baraniuk, Multiple wavelet basis image denoising using Besov ball projections. IEEE Signal Process. Lett. 11, 717–720 (2004)

116.

G. Cimmino, Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari. La Ricerca Scientifica, II 9, 326–333 (1938)

117.

P.L. Combettes, Inconsistent signal feasibility problems: Least-square solutions in a product space. IEEE Trans. Signal Process. 42, 2955–2966 (1994)

118.

P.L. Combettes, in The Convex Feasibility Problem in Image Recovery, ed. by P. Hawkes. Advances in Imaging and Electron Physics, vol. 95 (Academic, New York, 1996), pp. 155–270

119.

P.L. Combettes, Hilbertian convex feasibility problem: Convergence of projection methods. Appl. Math. Optim. 35, 311–330 (1997)

120.

P.L. Combettes, Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections. IEEE Trans. Image Process. 6, 493–506 (1997)

121.

P.L. Combettes, in Quasi-Fejérian Analysis of Some Optimization Algorithm, ed. by D. Butnariu, Y. Censor, S. Reich. Inherently Parallel Algorithms in Feasibility and Optimization and their Applications (Elsevier, Amsterdam, 2001), pp. 115–152

122.

P.L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53, 475–504 (2004)

123.

P.L. Combettes, P. Bondon, Hard-constrained inconsistent signal feasibility problems. IEEE Trans. Signal Process. 47, 2460–2468 (1999)

124.

P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

125.

P.L. Combettes, H. Puh, Iterations of parallel convex projections in Hilbert spaces. Numer. Funct. Anal. Optim. 15, 225–243 (1994)

126.

G. Crombez, A geometrical look at iterative methods for operators with fixed points. Numer. Funct. Anal. Optim. 26, 157–175 (2005)

127.

G. Crombez, A hierarchical presentation of operators with fixed points on Hilbert spaces. Numer. Funct. Anal. Optim. 27, 259–277 (2006)

128.

Y.-H. Dai, Fast algorithms for projection on an ellipsoid. SIAM J. Optim. 16, 986–1006 (2006)

129.

A. Dax, A note of the convergence of linear stationary iterative process. Lin. Algebra Appl. 129, 131–142 (1990)

130.

A. Dax, Linear search acceleration of iterative methods. Lin. Algebra Appl. 130, 43–63 (1990)

131.

A. Dax, On hybrid acceleration of a linear stationary iterative process. Lin. Algebra Appl. 130, 99–110 (1990)

132.

A. Dax, The convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations. SIAM Rev. 32, 611–635 (1990)

133.

L. Debnath, P. Mikusiński, Hilbert Spaces with Applications, 2nd edn. (Academic, San Diego, 1999)

134.

A.R. De Pierro, A.N. Iusem, A simultaneous projections method for linear inequalities. Lin. Algebra Appl. 64, 243–253 (1985)

135.

A.R. De Pierro, A.N. Iusem, A parallel projection method of finding a common point of a family of convex sets. Pesquisa Operacional 5, 1–20 (1985)

136.

A.R. De Pierro, A.N. Iusem, A finitely convergent “row-action” method for the convex feasibility problem. Appl. Math. Optim. 17, 225–235 (1988)

137.

A.R. De Pierro, A.N. Iusem, On the asymptotic behavior of some alternate smoothing series expansion iterative methods. Lin. Algebra Appl. 130, 3–24 (1990)

138.

F. Deutsch, in Applications of von Neumann’s Alternating Projections Algorithm, ed. by P. Kenderov. Mathematical Methods in Operations Research (Sophia, Bulgaria, 1983), pp. 44–51

139.

F. Deutsch, in The Method of Alternating Orthogonal Projections, ed. by S.P. Singh. Approximation Theory, Spline Functions and Applications (Kluwer Academic, The Netherlands, 1992), pp. 105–121

140.

F. Deutsch, Best Approximation in Inner Product Spaces (Springer, New York, 2001)

141.

F. Deutsch, in Accelerating the Convergence of the Method of Alternating Projections via a Line Search: A Brief Survey, ed. by D. Butnariu, Y. Censor, S. Reich. Inherently Parallel Algorithms in Feasibility and Optimization and their Application, Studies in Computational Mathematics, vol. 8 (Elsevier Science, Amsterdam, 2001), pp. 203–217

142.

F. Deutsch, H. Hundal, The rate of convergence for the cyclic projections algorithm, I. Angles between convex sets. J. Approx. Theor. 142, 36–55 (2006)

143.

F. Deutsch, H. Hundal, The rate of convergence for the cyclic projections algorithm, II. Norms of nonlinear operators. J. Approx. Theor. 142, 56–82 (2006)

144.

F. Deutsch, I. Yamada, Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19, 33–56 (1998)

145.

J.B. Diaz, F.T. Metcalf, On the set of subsequential limit points of successive approximations. Trans. Am. Math. Soc. 135, 459–485 (1969)

146.

L.T. Dos Santos, A parallel subgradient projections method for the convex feasibility problem. J. Comput. Appl. Math. 18, 307–320 (1987)

147.

W.G. Dotson Jr., On the Mann iterative process. Trans. Am. Math. Soc. 149, 65–73 (1970)

148.

W.G. Dotson, Fixed points of quasi-nonexpansive mappings. J. Austral. Math. Soc. 13, 167–170 (1972)

149.

J. Douglas, H.H. Rachford, On the numerical solution of heat conduction problems in two or three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)

150.

R. Dudek, Iterative method for solving the linear feasibility problem. J. Optim Theor. Appl. 132, 401–410 (2007)

151.

J. Dye, M.A. Khamsi, S. Reich, Random products of contractions in Banach spaces. Trans. Am. Math. Soc. 325, 87–99 (1991)

152.

J.M. Dye, S. Reich, On the unrestricted iteration of projections in Hilbert space. J. Math. Anal. Appl. 156, 101–119 (1991)

153.

J. Dye, S. Reich, Unrestricted iterations of nonexpansive mappings in Hilbert space. Nonlinear Anal. 18, 199–207 (1992)

154.

R. Dylewski, Selection of Linearizations in Projection Methods for Convex Optimization Problems (in Polish), Ph.D. thesis, University of Zielona Góra, Institute of Mathematics, 2003

155.

R. Dylewski, Projection method with residual selection for linear feasibility problems. Discuss. Math. Differ. Incl. Contr. Optim. 27, 43–50 (2007)

156.

M.G. Eberle, M.C. Maciel, Finding the closest Toeplitz matrix. Computat. Appl. Math. 22, 1–18 (2003)

157.

J. Eckstein, D.P. Bertsekas, On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)

158.

I. Ekeland, R. Témam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976)

159.

T. Elfving, A projection method for semidefinite linear systems and its applications. Lin. Algebra Appl. 391, 57–73 (2004)

160.

T. Elfving, T. Nikazad, Stopping rules for Landweber-type iteration. Inverse Probl. 23, 1417–1432 (2007)

161.

V. Elser, I. Rankenburg, P. Thibault, Searching with iterated maps. Proc. Natl. Acad. Sci. USA 104, 418–423 (2007)

162.

L. Elsner, I. Koltracht, P. Lancaster, Convergence properties of ART and SOR algorithms. Numer. Math. 59, 91–106 (1991)

163.

L. Elsner, I. Koltracht, M. Neumann, On the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data. Lin. Algebra Appl. 130, 65–82 (1990)

164.

L. Elsner, I. Koltracht, M. Neumann, Convergence of sequential and asynchronous nonlinear paracontractions. Numer. Math. 62, 305–319 (1992)

165.

F. Facchinei, J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I, Volume II (Springer, New York, 2003)

166.

M. Fiedler, V. Pták, On matrices with non-positive off-diagonal elements and positive principal minors. Czech. Math. J. 12, 382–400 (1962)

167.

S. Fitzpatrick, R.R. Phelps, Differentiability of the metric projection in Hilbert space. Trans. Am. Math. Soc. 270, 483–501 (1982)

168.

S.D. Flåm, J. Zowe, Relaxed outer projections, weighted averages and convex feasibility. BIT 30, 289–300 (1990)

169.

R. Fletcher, Practical Methods of Optimization (Wiley, Chichester, 1987)

170.

K. Friedricks, On certain inequalities and characteristic value problems for analytic functions and for functions of two variables. Trans. Am. Math. Soc. 41, 321–364 (1937)

171.

M. Fukushima, A relaxed projection method for variational inequalities. Math. Program. 35, 58–70 (1986)

172.

E.M. Gafni, D.P. Bertsekas, Two metric projection methods for constrained optimization. SIAM J. Contr. Optim. 22, 936–964 (1984)

173.

A. Galántai, Projectors and Projection Methods (Kluwer Academic, Boston, 2004)

174.

A. Galántai, On the rate of convergence of the alternating projection method in finite dimensional spaces. J. Math. Anal. Appl. 310, 30–44 (2005)

175.

U. García-Palomares, Parallel projected aggregation methods for solving the convex feasibility problem. SIAM J. Optim. 3, 882–900 (1993)

176.

U. García-Palomares, A superlinearly convergent projection algorithm for solving the convex inequality problem. Oper. Res. Lett. 22, 97–103 (1998)

177.

W.B. Gearhart, M. Koshy, Acceleration schemes for the method of alternating projections. J. Comput. Appl. Math. 26, 235–249 (1989)

178.

C. Geiger, Ch. Kanzow, Numerische Verfahren zur Lösung unrestingierter Optimierungsaufgaben (Springer, Berlin, 1999)

179.

C. Geiger, Ch. Kanzow, Theorie und Numerik restringierter Optimierungsaufgaben (Springer, Berlin, 2002)

180.

J.R. Giles, Convex Analysis with Application in Differentiation of Convex Functions (Pitman Advanced Publishing Program, Boston, 1982)

181.

P.E. Gill, W. Murray, M.H. Wright, Numerical Linear Algebra and Optimization (Addison-Wesley, Redwood City, 1991)

182.

W. Glunt, T.L. Hayden, R. Reams, The nearest ‘doubly stochastic’ matrix to a real matrix with the same first moment. Numer. Lin. Algebra Appl. 5, 475–482 (1998)

183.

K. Goebel, Concise Course on Fixed Points Theorems (Yokohama Publishing, Yokohama, 2002); Polish translation: Twierdzenia o punktach stałych (Wydawnictwo UMCS, Lublin, 2005)

184.

K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory (Cambridge University Press, Cambridge, 1990); Polish translation: Zagadnienia metrycznej teorii punktów stałych (Wydawnictwo UMCS, Lublin, 1999)

185.

K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (Marcel Dekker, New York, 1984)

186.

J.L. Goffin, The relaxation method for solving systems of linear inequalities. Math. Oper. Res. 5, 388–414 (1980)

187.

J.L. Goffin, On the finite convergence of the relaxation method for solving systems of inequalities. Operations Research Center, Report ORC 71–36, University of California, Berkeley, 1971

188.

D. Göhde, Zum Prinzip der kontraktiven Abbildung. Math. Nachr. 30, 251–258 (1965)

189.

R. Gordon, R. Bender, G.T. Herman, Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography. J. Theoret. Biol. 29, 471–481 (1970)

190.

D. Gordon, R. Gordon, Component-averaged row projections: a robust, block-parallel scheme for sparse linear systems. SIAM J. Sci. Comput. 27, 1092–1117 (2005)

191.

A. Granas, J. Dugundji, Fixed Point Theory (Springer, New York, 2003)

192.

K.M. Grigoriadis, A.E. Frazho, R.E. Skelton, Application of alternating convex projection methods for computation of positive Toeplitz matrices. IEEE Trans. Signal Process. 42, 1873–1875 (1994)

193.

K.M. Grigoriadis, R.E. Skelton, Low-order control design for LMI problems using alternating projection methods. Automatica 32, 1117–1125 (1996)

194.

K.M. Grigoriadis, R.E. Skelton, Alternating convex projection methods for discrete-time covariance control design. J. Optim. Theor. Appl. 88, 399–432 (1996)

195.

J. Gu, H. Stark, Y. Yang, Wide-band smart antenna design using vector space projection methods. IEEE Trans. Antenn. Propag. 52, 3228–3236 (2004)

196.

L.G. Gurin, B.T. Polyak, E.V. Raik, The method of projection for finding the common point in convex sets. Zh. Vychisl. Mat. Mat. Fiz. 7, 1211–1228 (1967) (in Russian); English translation in: USSR Comput. Math. Phys. 7, 1–24 (1967)

197.

R. Haller, R. Szwarc, Kaczmarz algorithm in Hilbert space. Studia Math. 169, 123–132 (2005)

198.

I. Halperin, The product of projection operators. Acta Sci. Math. (Szeged) 23, 96–99 (1962)

199.

H.W. Hamacher, K.-H. Küfer, Inverse radiation therapy planning – a multiple objective optimization approach. Discrete Appl. Math. 118, 145–161 (2002)

200.

S.-P. Han, A successive projection method. Math. Program. (Ser. A) 40, 1–14 (1988)

201.

M. Hanke, W. Niethammer, On the acceleration of Kaczmarz’s method for inconsistent linear systems. Lin. Algebra Appl. 130, 83–98 (1990)

202.

Y. Haugazeau, Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes (Thèse, Université de Paris, Paris, 1968)

203.

H. He, S. Liu, H. Zhou, An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings. Nonlinear Anal. 72, 3124–3135 (2010)

204.

G.T. Herman, A relaxation method for reconstructing objects from noisy X-rays. Math. Program. 8, 1–19 (1975)

205.

G.T. Herman, Fundamentals of Computerized Tomography: Image Reconstruction from Projections, 2nd edn. (Springer, London, 2009)

206.

G.T. Herman, W. Chen, A fast algorithm for solving a linear feasibility problem with application to intensity-modulated radiation therapy. Lin. Algebra Appl. 428, 1207–1217 (2008)

207.

N.J. Higham, Computing a nearest symmetric positive semidefinite matrix. Lin. Algebra Appl. 103, 103–118 (1988)

208.

N.J. Higham, Computing the nearest correlation matrix – a problem from finance. IMA J. Numer. Anal. 22, 329–343 (2002)

209.

J.-B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Minimization Algorithms, Vol I, Vol II (Springer, Berlin, 1993)

210.

J.-B. Hiriart-Urruty, C. Lemaréchal, Fundamentals of Convex Analysis (Springer, Berlin, 2001)

211.

S.A. Hirstoaga, Iterative selection methods for common fixed point problems. J. Math. Anal. Appl. 324, 1020–1035 (2006)

212.

J. Höffner, P. Decker, E.L. Schmidt, W. Herbig, J. Rittler, P. Weiß, Development of a fast optimization preview in radiation treatment planning. Strahlentherapie und Onkologie 172, 384–394 (1996)

213.

H.S. Hundal, An alternating projection that does not converge in norm. Nonlinear Anal. 57, 35–61 (2004)

214.

J.K. Hunter, B. Nachtergaele, Applied Analysis (World Scientific, Singapore, 2000)

215.

A.N. Iusem, A.R. De Pierro, Convergence results for an accelerated nonlinear Cimmino algorithm. Numer. Math. 49, 367–378 (1986)

216.

A.N. Iusem, A.R. De Pierro, A simultaneous iterative method for computing projections on polyhedra. SIAM J. Contr. Optim. 25, 231–243 (1987)

217.

A.N. Iusem, A.R. De Pierro, On the convergence properties of Hildreth’s quadratic programming algorithm. Math. Program. (Ser. A) 47, 37–51 (1990)

218.

A.N. Iusem, B.F. Svaiter, A row-action method for convex programming. Math. Program. 64, 149–171 (1994)

219.

B.K. Jennison, J.P. Allebach, D.W. Sweeney, Iterative approaches to computer-generated holography. Opt. Eng. 28, 629–637 (1989)

220.

M. Jiang, G. Wang, Development of iterative algorithms for image reconstruction. J. X-Ray Sci. Tech. 10, 77–86 (2002)

221.

M. Jiang, G. Wang, Convergence studies on iterative algorithms for image reconstruction. IEEE Trans. Med. Imag. 22, 569–579 (2003)

222.

B. Johansson, T. Elfving, V. Kozlov, Y. Censor, P.-E. Forssén, G. Granlund, The application of an oblique-projected Landweber method to a model of supervised learning. Math. Comput. Model. 43, 892–909 (2006)

223.

S. Kaczmarz, Angenäherte Auflösung von Systemen linearer Gleichungen. Bulletin International de l’Académie Polonaise des Sciences et des Lettres A35, 355–357 (1937); English translation: S. Kaczmarz, Approximate solution of systems of linear equations. Int. J. Contr. 57, 1269–1271 (1993)

224.

A.C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988)

225.

I.G. Kazantsev, S. Schmidt, H.F. Poulsen, A discrete spherical x-ray transform of orientation distribution functions using bounding cubes. Inverse Probl. 25, 105009 (2009)

226.

D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications (Academic, New York, 1980)

227.

W.A. Kirk, A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72, 1004–1006 (1965)

228.

Yu.N. Kiseliov, Algorithms of projection of a point onto an ellipsoid. Lithuanian Math. J. 34, 141–159 (1994)

229.

K.C. Kiwiel, Block-iterative surrogate projection methods for convex feasibility problems. Lin. Algebra Appl. 215, 225–259 (1995)

230.

K.C. Kiwiel, The efficiency of subgradient projection methods for convex optimization. I. General level methods. SIAM J. Contr. Optim. 34, 660–676 (1996)

231.

K.C. Kiwiel, The efficiency of subgradient projection methods for convex optimization, II. Implementations and extensions. SIAM J. Contr. Optim. 34, 677–697 (1996)

232.

K.C. Kiwiel, Monotone Gram matrices and deepest surrogate inequalities in accelerated relaxation methods for convex feasibility problems. Lin. Algebra Appl. 252, 27–33 (1997)

233.

K.C. Kiwiel, B. Łopuch, Surrogate projection methods for finding fixed points of firmly nonexpansive mappings. SIAM J. Opt. 7, 1084–1102 (1997)

234.

A. Kiełbasiński, H. Schwetlick, Numerical Linear Algebra (in German) (Verlag Harri Deutsch, Thun, 1988); Polish translation: Numeryczna algebra liniowa (WNT, Warszawa, 1992)

235.

E. Kopecká, S. Reich, A note on the von Neumann alternating projections algorithm. J. Nonlinear Convex Anal. 5, 379–386 (2004)

236.

E. Kopecká, S. Reich, Another note on the von Neumann alternating projections algorithm. J. Nonlinear Convex Anal. 11, 455–460 (2010)

237.

G.M. Korpelevich, The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody 12, 747–756 (1976)

238.

M.A. Krasnosel’skiĭ, Two remarks on the method of successive approximations (in Russian). Uspehi Mat. Nauk 10, 123–127 (1955)

239.

S. Kwapień, J. Mycielski, On the Kaczmarz algorithm of approximation in infinite-dimensional spaces. Studia Math. 148, 5–86 (2001)

240.

L. Landweber, An iteration formula for Fredholm integral equations of the first kind. Am. J. Math. 73, 615–624 (1951)

241.

S. Lee, P.S. Cho, R.J. Marks, S.Oh, Conformal radiotherapy computation by the method of alternating projections onto convex sets. Phys. Med. Biol. 42, 1065–1086 (1997)

242.

S.-H. Lee, K.-R. Kwon, Mesh watermarking based projection onto two convex sets. Multimedia Syst. 13, 323–330 (2008)

243.

A. Lent, in A Convergent Algorithm for Maximum Entropy Image Restoration with a Medical X-ray Application, ed. by R. Shaw. Image Analysis and Evaluation (SPSE, Washington DC), pp. 249–257

244.

A. Lent, Y. Censor, Extensions of Hildreth’s row-action method for quadratic programming. SIAM J. Contr. Optim. 18, 444–454 (1980)

245.

A.W.-C. Liew, H. Yan, N.-F. Law, POCS-based blocking artifacts suppression using a smoothness constraint set with explicit region modeling. IEEE Trans. Circ. Syst. Video Tech. 15, 795–800 (2005)

246.

P.L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)

247.

C. Liu, An acceleration scheme for row projection methods. J. Comput. Appl. Math. 57, 363–391 (1995)

248.

Y.M. Lu, M. Karzand, M. Vetterli, Demosaicking by alternating projections: theory and fast one-step implementation. IEEE Trans. Image Process. 19, 2085–2098 (2010)

249.

P.-E. Maingé, Inertial iterative process for fixed points of certain quasi-nonexpansive mappings. Set-Valued Anal. 15, 67–79 (2007)

250.

P.-E. Maingé, Extension of the hybrid steepest descent method to a class of variational inequalities and fixed point problems with nonself-mappings. Numer. Funct. Anal. Optim. 29, 820–834 (2008)

251.

P.-E. Maingé, New approach to solving a system of variational inequalities and hierarchical problems. J. Optim. Theor. Appl. 138, 459–477 (2008)

252.

W.R. Mann, Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)

253.

Şt. Măruşter, The solution by iteration of nonlinear equations in Hilbert spaces. Proc. Am. Math. Soc. 63, 69–73 (1977)

254.

Şt. Măruşter, Quasi-nonexpansivity and the convex feasibility problem. An. Ştiinţ. Univ. Al. I. Cuza Iaşi Inform. (N.S.) 15, 47–56 (2005)

255.

Şt. Măruşter, C. Popîrlan, On the Mann-type iteration and the convex feasibility problem. J. Comput. Appl. Math. 212, 390–396 (2008)

256.

Şt. Măruşter, C. Popîrlan, On the regularity condition in a convex feasibility problem. Nonlinear Anal. 70, 1923–1928 (2009)

257.

E. Masad, S. Reich, A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007)

258.

E. Matoušková, S. Reich, The Hundal example revisited. J. Nonlinear Convex Anal. 4, 411–427 (2003)

259.

S.F. McCormick, The methods of Kaczmarz and row orthogonalization for solving linear equations and least squares problems in Hilbert space. Indiana Univ. Math. J. 26, 1137–1150 (1977)

260.

Yu.I. Merzlyakov, On a relaxation method of solving systems of linear inequalities (in Russian). Zh. Vychisl. Mat. Mat. Fiz. 2, 482–487 (1962)

261.

D. Michalski, Y. Xiao, Y. Censor, J.M. Galvin, The dose-volume constraint satisfaction problem for inverse treatment planning with field segments. Phys. Med. Biol. 49, 601–616 (2004)

262.

W. Mlak, Introduction to Hilbert Spaces (in Polish) (PWN, Warsaw, 1982)

263.

W. Mlak, Hilbert Spaces and Operator Theory (Kluwer Academic, Boston, 1991)

264.

J.-J. Moreau, Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris 255, 238–240 (1962)

265.

J. Moreno, B. Datta, M. Raydan, A symmetry preserving alternating projection method for matrix model updating. Mech. Syst. Signal Process. 23, 1784–1791 (2009)

266.

T.S. Motzkin, I.J. Schoenberg, The relaxation method for linear inequalities. Can. J. Math. 6, 393–404 (1954)

267.

J. Musielak, Introduction to Functional Analysis (in Polish) (PWN, Warszawa, 1989)

268.

J. Mycielski, S. Świerczkowski, Uniform approximation with linear combinations of reproducing kernels. Studia Math. 121, 105–114 (1996)

269.

N. Nadezhkina, W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006)

270.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, Chichester, 1986)

271.

J. von Neumann, in Functional Operators – Vol. II. The Geometry of Orthogonal Spaces. Annals of Mathematics Studies, vol. 22 (Princeton University Press, Princeton, 1950) (Reprint of mimeographed lecture notes first distributed in 1933)

272.

O. Nevanlinna, S. Reich, Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces. Israel J. Math. 32, 44–58 (1979)

273.

N. Ogura, I. Yamada, Non-strictly convex minimization over the fixed point set of an asymptotically shrinking nonexpansive mapping. Numer. Funct. Anal. Optim. 23, 113–137 (2002)

274.

N. Ogura, I. Yamada, Nonstrictly convex minimization over the bounded fixed point set of a nonexpansive mapping. Numer. Funct. Anal. Optim. 24, 129–135 (2003)

275.

S. Oh, R.J. Marks, L.E. Atlas, Kernel synthesis for generalized time-frequency distributions using the method of alternating projections onto convex sets. IEEE Trans. Signal Process. 42, 1653–1661 (1994)

276.

J.G. O’Hara, P. Pillay, H.-K. Xu, Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces. Nonlinear Anal. 54, 1417–1426 (2003)

277.

S.O. Oko, Surrogate methods for linear inequalities. J. Optim. Theor. Appl. 72, 247–268 (1992)

278.

Z. Opial, Nonexpansive and Monotone Mappings in Banach Spaces. Lecture Notes 67-1, Center for Dynamical Systems, Brown University, Providence, RI, 1967

279.

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

280.

S.C. Park, M.K. Park, M.G. Kang, Super-resolution image reconstruction: A technical overview. IEEE Signal Process. Mag. 20, 21–36 (2003)

281.

J. Park, D.C. Park, R.J. Marks, M. El-Sharkawi, Recovery of image blocks using the method of alternating projections. IEEE Trans. Image Process. 14, 461–474 (2005)

282.

S.N. Penfold, R.W. Schulte, Y. Censor, A.B. Rosenfeld, Total variation superiorization schemes in proton computed tomography image reconstruction. Med. Phys. 37, 5887–5895 (2010)

283.

W.V. Petryshyn, T.E. Williamson Jr., Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings. J. Math. Anal. Appl. 43, 459–497 (1973)

284.

G. Pierra, Decomposition through formalization in a product space. Math. Program. 28, 96–115 (1984)

285.

C. Popa, Least-squared solution of overdetermined inconsistent linear systems using Kaczmarz’s relaxation. Int. J. Comp. Math. 55, 79–89 (1995)

286.

C. Popa, Extensions of block-projections methods with relaxation parameters to inconsistent and rank-deficient least-squares problems. BIT 38, 151–176 (1998)

287.

C. Popa, R. Zdunek, Kaczmarz extended algorithm for tomographic image reconstruction from limited data. Math. Comput. Simulat. 65, 579–598 (2004)

288.

S. Prasad, Generalized array pattern synthesis by the method of alternating orthogonal projections. IEEE Trans. Antenn. Propag. 28, 328–332 (1980)

289.

J.L. Prince, A.S. Willsky, A geometric projection-space reconstruction algorithm. Lin. Algebra Appl. 130, 151–191 (1990)

290.

E. Pustylnik, S. Reich, A.J. Zaslavski, Convergence of infinite products of nonexpansive operators in Hilbert space. J. Nonlinear Convex Anal. 11, 461–474 (2010)

291.

E. Pustylnik, S. Reich, A.J. Zaslavski, Convergence of non-cyclic infinite products of operators. J. Math. Anal. Appl. 380, 759–767 (2011)

292.

B. Qu, N. Xiu, A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21, 1655–1665 (2005)

293.

B. Qu, N. Xiu, A new halfspace-relaxation projection method for the split feasibility problem. Lin. Algebra Appl. 428, 1218–1229 (2008)

294.

S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)

295.

S. Reich, A limit theorem for projections. Lin. Multilinear Algebra 13, 281–290 (1983)

296.

S. Reich and I. Shafrir, The asymptotic behavior of firmly nonexpansive mappings. Proc. Am. Math. Soc. 101, 246–250 (1987)

297.

S. Reich, A.J. Zaslavski, Attracting mappings in Banach and hyperbolic spaces. J. Math. Anal. Appl. 253, 250–268 (2001)

298.

R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)

299.

R.T. Rockafellar, Monotone operators and the proximal point algorithm. SIAM J. Contr. Optim. 14, 877–898 (1976)

300.

W. Rudin, Functional Analysis, 2nd edn. (McGraw-Hill, New York, 1991); Polish translation: Analiza funkcjonalna (PWN, Warszawa, 2002)

301.

A.A. Samsonov, E.G. Kholmovski, D.L. Parker, C.R. Johnson, POCSENSE: POCS-based reconstruction for sensitivity encoded magnetic resonance imaging. Magn. Reson. Med. 52, 1397–1406 (2004)

302.

J. Schauder, Der Fixpunktsatz in Funktionalräumen. Studia Math. 2, 171–180 (1930)

303.

D. Schott, A general iterative scheme with applications to convex optimization and related fields. Optimization 22, 885–902 (1991)

304.

H.D. Scolnik, N. Echebest, M.T. Guardarucci, M.C. Vacchino, A class of optimized row projection methods for solving large nonsymmetric linear systems. Appl. Numer. Math. 41, 499–513 (2002)

305.

H.D. Scolnik, N. Echebest, M.T. Guardarucci, M.C. Vacchino, Acceleration scheme for parallel projected aggregation methods for solving large linear systems. Ann. Oper. Res. 117, 95–115 (2002)

306.

H.D. Scolnik, N. Echebest, M.T. Guardarucci, M.C. Vacchino, Incomplete oblique projections for solving large inconsistent linear systems. Math. Program. 111, 273–300 (2008)

307.

A. Segal, Directed Operators for Common Fixed Point Problems and Convex Programming Problems, Ph.D. Thesis, University of Haifa, Haifa, Israel, 2008

308.

H.F. Senter, W.G. Dotson Jr., Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44, 375–380 (1974)

309.

A. Serbes, L. Durak, Optimum signal and image recovery by the method of alternating projections in fractional Fourier domains. Comm. Nonlinear Sci. Numer. Simulat. 15, 675–689 (2010)

310.

N.T. Shaked, J. Rosen, Multiple-viewpoint projection holograms synthesized by spatially incoherent correlation with broadband functions. J. Opt. Soc. Am. A 25, 2129–2138 (2008)

311.

G. Sharma, Set theoretic estimation for problems in subtractive color. Color Res. Appl. 25, 333–348 (2000)

312.

K.K. Sharma, S.D. Joshi, Extrapolation of signals using the method of alternating projections in fractional Fourier domains. Signal Image Video Process. 2, 177–182 (2008)

313.

K.T. Smith, D.C. Solman, S.L. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs. Bull. Am. Math. Soc. 83, 1227–1270 (1977)

314.

R.A. Soni, K.A. Gallivan, W.K. Jenkins, Low-complexity data reusing methods in adaptive filtering. IEEE Trans. Signal Process. 52, 394–405 (2004)

315.

H. Stark, P. Oskoui, High resolution image recovery from image-plane arrays, using convex projections. J. Opt. Soc. Am. A 6, 1715–1726 (1989)

316.

H. Stark, Y. Yang, Vector Space Projections. A Numerical Approach to Signal and Image Processing, Neural Nets and Optics (Wiley, New York, 1998)

317.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, 3rd edn. (Springer, New York, 2002)

318.

C. Sudsukh, Strong convergence theorems for fixed point problems, equilibrium problems and applications. Int. J. Math. Anal. (Ruse) 3, 1867–1880 (2009)

319.

S. Świerczkowski, A model of following. J. Math. Anal. Appl. 222, 547–561 (1998)

320.

W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)

321.

K. Tanabe, Projection method for solving a singular system of linear equations and its applications. Numer. Math. 17, 203–214 (1971)

322.

K. Tanabe, Characterization of linear stationary iterative processes for solving a singular system of linear equations. Numer. Math. 22, 349–359 (1974)

323.

G. Tetzlaff, K. Arnold, A. Raabe, A. Ziemann, Observations of area averaged near-surface wind- and temperature-fields in real terrain using acoustic travel time tomography. Meteorologische Zeitschrift 11, 273–283 (2002)

324.

Ch. Thieke, T. Bortfeld, A. Niemierko, S. Nill, From physical dose constraints to equivalent uniform dose constraints in inverse radiotherapy planning. Med. Phys. 30, 2332–2339 (2003)

325.

J. van Tiel, Convex Analysis, An Introductory Text (Wiley, Chichester, 1984)

326.

M.J. Todd, Some Remarks on the Relaxation Method for Linear Inequalities, Technical Report, vol. 419, Cornell University, Cornell, Ithaca, 1979

327.

Ph.L. Toint, Global convergence of a class of trust region methods for nonconvex minimization in Hilbert space. IMA J. Numer. Anal. 8, 231–252 (1988)

328.

W. Treimer, U. Feye-Treimer, Two dimensional reconstruction of small angle scattering patterns from rocking curves. Physica B 241–243, 1228–1230 (1998)

329.

J.A. Tropp, I.S. Dhillon, R.W. Heath, T. Strohmer, Designing structured tight frames via an alternating projection method. IEEE Trans. Inform. Theor. 51, 188–209 (2005)

330.

M.R. Trummer, SMART – an algorithm for reconstructing pictures from projections. J. Appl. Math. Phys. 34, 746–753 (1983)

331.

P. Tseng, On the convergence of the products of firmly nonexpansive mappings. SIAM J. Optim. 2, 425–434 (1992)

332.

A. Van der Sluis, H.A. Van der Vorst, in Numerical Solution of Large Sparse Linear Algebraic Systems Arising from Tomographic Problems, ed. by G. Nolet. Seismic Tomography (Reidel, Dordrecht, 1987)

333.

V.V. Vasin, A.L. Ageev, Ill-Posed Problems with A Priori Information (VSP, Utrecht, 1995)

334.

S. Webb, Intensity Modulated Radiation Therapy (Institute of Physics Publishing, Bristol, 2001)

335.

S. Webb, The Physics of Conformal Radiotherapy (Institute of Physics Publishing, Bristol, 2001)

336.

R. Webster, Convexity (Oxford University Press, Oxford, 1994)

337.

R. Wegmann, Conformal mapping by the method of alternating projections. Numer. Math. 56, 291–307 (1989)

338.

R. Wittmann, Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992)

339.

P. Wolfe, Finding the nearest point in a polytope. Math. Program. 11, 128–149 (1976)

340.

B.J. van Wyk, M.A. van Wyk, A POCS-based graph matching algorithm. IEEE Trans. Pattern Anal. Mach. Intell. 26, 1526–1530 (2004)

341.

Y. Xiao, Y. Censor, D. Michalski, J.M. Galvin, The least-intensity feasible solution for aperture-based inverse planning in radiation therapy. Ann. Oper. Res. 119, 183–203 (2003)

342.

H.-K. Xu, Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

343.

H.-K. Xu, An iterative approach to quadratic optimization. J. Optim. Theor. Appl. 116, 659–678 (2003)

344.

H.-K. Xu, A variable Krasnosel’skiĭ-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)

345.

I. Yamada, in The Hybrid Steepest Descent Method for the Variational Inequality Problem Over the Intersection of Fixed Point Sets of Nonexpansive Mappings, ed. by D. Butnariu, Y. Censor, S. Reich. Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, Studies in Computational Mathematics, vol. 8 (Elsevier Science, Amsterdam, 2001), pp. 473–504

346.

I. Yamada, N. Ogura, Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619–655 (2004)

347.

I. Yamada, N. Ogura, Adaptive projected subgradient method for asymptotic minimization of sequence of nonnegative convex functions. Numer. Funct. Anal. Optim. 25, 593–617 (2004)

348.

I. Yamada, N. Ogura, N. Shirakawa, in A Numerically Robust Hybrid Steepest Descent Method for the Convexly Constrained Generalized Inverse Problems, ed. by Z. Nashed, O. Scherzer. Inverse Problems, Image Analysis and Medical Imaging, American Mathematical Society, Contemp. Math., vol. 313 (2002), pp. 269–305

349.

I. Yamada, N. Ogura, Y. Yamashita, K. Sakaniwa, Quadratic optimization of fixed points of nonexpansive mappings in Hilbert space. Numer. Funct. Anal. Optim. 19, 165–190 (1998)

350.

Q.Z. Yang, The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20, 1261–1266 (2004)

351.

K. Yang, K.G. Murty, New iterative methods for linear inequalities. JOTA 72, 163–185 (1992)

352.

Q. Yang, J. Zhao, Generalized KM theorems and their applications. Inverse Probl. 22, 833–844 (2006)

353.

Y. Yao, Y.-C. Liou, Weak and strong convergence of Krasnoselski–Mann iteration for hierarchical fixed point problems. Inverse Probl. 24, 015015, 8 (2008)

354.

D. Youla, Generalized image restoration by the method of alternating orthogonal projections. IEEE Trans. Circ. Syst. 25, 694–702 (1978)

355.

M. Yukawa, I. Yamada, Pairwise optimal weight realization – Acceleration technique for set-theoretic adaptive parallel subgradient projection algorithm. IEEE Trans. Signal Process. 54, 4557–4571 (2006)

356.

M. Zaknoon, Algorithmic Developments for the Convex Feasibility Problem, Ph.D. Thesis, University of Haifa, Haifa, Israel, 2003

357.

E.H. Zarantonello, in Projections on Convex Sets in Hilbert Space and Spectral Theory, ed. by E.H. Zarantonello. Contributions to Nonlinear Functional Analysis (Academic, New York, 1971), pp. 237–424

358.

E. Zeidler, Nonlinear Functional Analysis and Its Applications, III – Variational Methods and Optimization (Springer, New York, 1985)

359.

J. Zhang, A.K. Katsaggelos, in Image Recovery Using the EM Algorithm, ed. by V.K. Madisetti, D.B. Williams. Digital Signal Processing Handbook (CRC Press LLC, Boca Raton, 1999)

360.

D.F. Zhao, The principles and practice of iterative alternating projection algorithm: Solution for non-LTE stellar atmospheric model with the method of linearized separation. Chin. Astron. Astrophys. 25, 305–316 (2001)

361.

J. Zhao, Q. Yang, Several solution methods for the split feasibility problem. Inverse Probl. 21, 1791–1799 (2005)

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4. Algorithmic Projection Operators

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