DC-Programming versus ℓ₀-Superiorization for Discrete Tomography

[ApS15] MOSEK ApS, The mosek optimization toolbox for matlab manual. version 7.1 (revision 28)., 2015.Search in Google Scholar

[Bat07] K. J. Batenburg, A Network Flow Algorithm for Reconstructing Binary Images from Discrete X-rays, J. Math. Imaging Vis. 27 (2007), no. 2, 175-191.10.1007/s10851-006-9798-2Search in Google Scholar

[BDHK07] D. Butnariu, R. Davidi, G. T. Herman, and I. G. Kazantsev, Stable convergence behavior under summable perturbations of a class of projection methods for convex feasibility and optimization problems, IEEE Journal of Selected Topics in Signal Processing 1 (2007), 540-547.10.1109/JSTSP.2007.910263Search in Google Scholar

[BGH70] R. Bender, R. Gordon, and G. T. Herman, Algebraic reconstruction techniques (art) for three-dimensional electron microscopy and x-ray photography, Journal of Theoretical Biology 29 (1970), 471-81.10.1016/0022-5193(70)90109-8Search in Google Scholar

[BKS08] A. M. Bagirov, B. Karas¨oen, and M. Sezer, Discrete gradient method: drivative-free method for nonsmooth optimization, Journal of Optimization Theory and Applications 137 (2008), no. 2, 317334.10.1007/s10957-007-9335-5Search in Google Scholar

[Byr99] C. L. Byrne, Iterative projection onto convex sets using multiple Bregman distances, Inverse Problems 15 (1999), no. 5, 1295-1313. MR 171536610.1088/0266-5611/15/5/313Search in Google Scholar

[Byr08] C. L. Byrne, Iterative methods for fixed point problems in hilbert spaces, Applied Iterative Methods, AK Peters, Wellsely, MA, USA, 2008.Search in Google Scholar

[CAP88] Y. Censor, M. D. Altschuler, and 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 Problems 4 (1988), no. 3, 607-623. MR 96563910.1088/0266-5611/4/3/006Search in Google Scholar

[CBMAT06] Y. Censor, T. Bortfeld, B. Martin, and A A. Trofimov, unified approach for inversion problems in intensity-modulated radiation therapy, Physics in Medicine and Biology 51 (2006), 2353-2365.10.1088/0031-9155/51/10/001Search in Google Scholar

[CDH10] Y. Censor, R. Davidi, and G. T. Herman, Perturbation resilience and superiorization of iterative algorithms, Inverse Problems 26 (2010), no. 6, 065008, 12. MR 264716210.1088/0266-5611/26/6/065008Search in Google Scholar

[CDH+14] Y. Censor, R. Davidi, G. T. Herman, R.W. Schulte, and L. Tetruashvili, Projected subgradient minimization versus superiorization, Journal of Optimization Theory and Applications 160 (2014), 730-747.10.1007/s10957-013-0408-3Search in Google Scholar

[CE94] Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms 8 (1994), no. 2-4, 221-239. MR 130922210.1007/BF02142692Search in Google Scholar

[Ceg12] A. Cegielski, Iterative methods for fixed point problems in hilbert spaces, Lecture Notes in mathematics 2057, Springer-Verlag, Berlin, Heidelberg, Germany, 2012.10.1007/978-3-642-30901-4Search in Google Scholar

[CEH01] Y. Censor, T. Elfving, and G. T. Herman, Averaging strings of sequential iterations for convex feasibility problems, Inherently parallel algorithms in feasibility and optimization and their applications (Haifa, 2000), Stud. Comput. Math., vol. 8, North-Holland, Amsterdam, 2001, pp. 101-113. MR 185321910.1016/S1570-579X(01)80009-4Search in Google Scholar

[Cen15] Y. Censor, Weak and Strong Superiorization: Between Feasibility- Seeking and Minimization, An. S¸ t. Univ. Ovidius Constant¸a 23 (2015), 41-54.10.1515/auom-2015-0046Search in Google Scholar

[Cenml] Y. Censor, Superiorization and perturbation resilience of algorithms: A bibliography compiled and continuously updated, http://math.haifa.ac.il/yair/bib-superiorization-censor.html.Search in Google Scholar

[CHE34] Y. Censor, G. T. Herman, and M. Jiang (Guest Editors), Superiorization: Theory and applications, Special Issue of the journal Inverse Problems, Volume 33, Number 4, 2017, http://iopscience.iop.org/issue/0266-5611/33/4.10.1088/1361-6420/aa5debSearch in Google Scholar

[CHS16] Y. Censor, H. Heaton, and R. Schulte, Superiorization with componentwise perturbations and its application to image reconstruction, November 4, 2016.Search in Google Scholar

[CL82] Y. Censor and A. Lent, Short Communication: Cyclic subgradient projections, Math. Programming 24 (1982), no. 1, 233-235. MR 155295510.1007/BF01585107Search in Google Scholar

[CS08] R. Chartrand and V. Staneva, Restricted Isometry Properties and Nonconvex Compressive Sensing, Inverse Problems 24 (2008), no. 3, 035020.10.1088/0266-5611/24/3/035020Search in Google Scholar

[CZ97] Y. Censor and S. A. Zenios, Parallel optimization: Theory, algorithms, and applications, Oxford University Press, New York, NY, USA, 1997.Search in Google Scholar

[CZ13] Y. Censor and A. J. Zaslavski, Convergence and perturbation resilience of dynamic string-averaging projection methods, Computational Optimization and Applications 54 (2013), no. 1, 65-76.10.1007/s10589-012-9491-xSearch in Google Scholar

[CZ15] , Strict Fej´er monotonicity by superiorization of feasibilityseeking projection methods, J. Optim. Theory Appl. 165 (2015), no. 1, 172-187. MR 332742010.1007/s10957-014-0591-xSearch in Google Scholar

[FM11] G. M. Fung and O. L. Mangasarian, Equivalence of Minimal `0- and `p- Norm Solutions of Linear Equalities, Inequalities and Linear Programs for Sufficiently Small p, J. Optim. Theory. Appl. 151 (2011), no. 1, 1-10.10.1007/s10957-011-9871-xSearch in Google Scholar

[FR13] S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Springer, 2013.10.1007/978-0-8176-4948-7Search in Google Scholar

[GR84] K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. MR 744194Search in Google Scholar

[Her14] G. T. Herman, Superiorization for image analysis, in:, 2014.10.1007/978-3-319-07148-0_1Search in Google Scholar

[HK99] G. T. Herman and A. Kuba, Discrete Tomography: Foundations, Algorithms, and Applications, Applied and Numerical Harmonic Analysis, Birkhauser, December 1999 1999.10.1007/978-1-4612-1568-4Search in Google Scholar

[HSH12] P. C. Hansen and M. Saxild-Hansen, AIR Tools - A MATLAB Package of Algebraic Iterative Reconstruction Methods, J. Comput. Appl. Math. 236 (2012), no. 8, 2167-2178.10.1016/j.cam.2011.09.039Search in Google Scholar

[HT99] R. Horst and N. Thoai, DC programming: Overview, J. Optim. Theory Appl. 103 (1999), no. 1, 1-43.10.1023/A:1021765131316Search in Google Scholar

[Kac37] S. Kaczmarz, Angen¨oherte aufl¨osung von systemen linearer gleichungen, Bulletin de l’Acad´emie Polonaise des Sciencesat Lettres A35 (1937), 355-357.Search in Google Scholar

[Man96] O. L. Mangasarian, Machine Learning via Polyhedral Concave Minimization, Applied Mathematics and Parallel Computing: Festschrift for Klaus Ritter (H. Fischer, B. Riedm¨uller, and S. Sch¨affler, eds.), Physica-Verlag HD, Heidelberg, 1996, pp. 175-188.10.1007/978-3-642-99789-1_13Search in Google Scholar

[Nat95] B. K. Natarajan, Sparse Approximate Solutions to Linear Systems, SIAM J. Computing 24 (1995), no. 5, 227-234.10.1137/S0097539792240406Search in Google Scholar

[PDEB86] T. Pham Dinh and S. El Bernoussi, Algorithms for Solving a Class of Nonconvex Optimization Problems. Methods of Subgradients, Fermat Days 85: Mathematics for Optimization (J.-B. Hiriart-Urruty, ed.), North-Holland Mathematics Studies, vol. 129, North-Holland, 1986, pp. 249-271.10.1016/S0304-0208(08)72402-2Search in Google Scholar

[PDTH97] T. Pham Dinh and A. L. Thi Hoai, Convex Analysis Approach to D.C. Programming: Theory, Algorithms and Applications, Acta Math. Viet. 22 (1997), no. 1, 289-355.Search in Google Scholar

[PDTH06] , A D.C. Optimization Algorithm for Solving the Trust-Region Subproblem, SIAM J. Optim. 8 (2006), no. 2, 476-50510.1137/S1052623494274313Search in Google Scholar

[PZC⁺17] S. Penfold, R. Zalas, M. Casiraghi, M. Brooke, Y. Censor, and R. Schulte, Sparsity constrained split feasibility for dose-volume constraints in inverse planning of intensity-modulated photon or proton therapy, Physics in Medicine and Biology 62 (2017), 3599-3618.10.1088/1361-6560/aa602b598904128379849Search in Google Scholar

[Roc70] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, Princeton University Press, 1970.Search in Google Scholar

[RP17] D. Reem and A. De Pierro, A new convergence analysis and perturbation resilience of some accelerated proximal forward-backward algorithms with errors, Inverse Problems 33 (2017), no. 2, 28pp.10.1088/1361-6420/33/4/044001Search in Google Scholar

[THPD05] A. L. Thi Hoai and T. Pham Dinh, The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems, Annals of Operations Research 133 (2005), no. 1, 23-46.10.1007/s10479-004-5022-1Search in Google Scholar

[Tol78] J. F. Toland, Duality in Nonconvex Optimization, J Math Anal Appl. 66 (1978), no. 2, 399-415.10.1016/0022-247X(78)90243-3Search in Google Scholar

[YLHX15] P. Yin, Y. Lou, Q. He, and J. Xin, Minimization of ℓ_1-2for Compressed Sensing, SIAM J. Sci. Computing 37 (2015), no. 1, A536-A563.10.1137/140952363Search in Google Scholar