Projection methods: an annotated bibliography of books and reviews

Projections onto sets are used in a wide variety of methods in optimization theory but not every method that uses projections really belongs to the class of projection methods as we mean it here. Here, projection methods are iterative algorithms that use projections onto sets while relying on the general principle that when a family of (usually closed and convex) sets is present, then projections (or approximate projections) onto the given individual sets are easier to perform than projections onto other sets (intersections, image sets under some transformation, etc.) that are derived from the given family of individual sets. Projection methods employ projections (or approximate projections) onto convex sets in various ways. They may use different kinds of projections and, sometimes, even use different projections within the same algorithm. They serve to solve a variety of problems which are either of the feasibility or the optimization types. They have different algorithmic structures, of which some are particularly suitable for parallel computing, and they demonstrate nice convergence properties and/or good initial behavioural patterns. This class of algorithms has witnessed great progress in recent years and its member algorithms have been applied with success to many scientific, technological and mathematical problems. This annotated bibliography includes books and review papers on, or related to, projection methods that we know about, use and like. If you know of books or review papers that should be added to this list please contact us.

[1]  Constantin Popa,et al.  Projection Algorithms - Classical Results and Developments , 2012 .

[2]  J. Borwein,et al.  Techniques of variational analysis , 2005 .

[3]  A. Auslender.  Optimisation : méthodes numériques , 1976 .

[4]  W. Oettli,et al.  Mathematische Optimierung : Grundlagen und Verfahren , 1975 .

[5]  Y. Censor,et al.  Parallel Optimization:theory , 1997 .

[6]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[7]  I. Konnov.  Combined Relaxation Methods for Variational Inequalities , 2000 .

[8]  M. Raydan,et al.  Alternating Projection Methods , 2011 .

[9]  Heinz H. Bauschke,et al.  Projection Methods: Swiss Army Knives for Solving Feasibility and Best Approximation Problems with Halfspaces , 2013, 1301.4506.

[10]  C. Byrne.  Iterative Optimization in Inverse Problems , 2014 .

[11]  Y. Censor.  Iterative Methods for the Convex Feasibility Problem , 1984 .

[12]  Gabor T. Herman,et al.  Fundamentals of Computerized Tomography: Image Reconstruction from Projections , 2009, Advances in Pattern Recognition.

[13]  Gabor T. Herman,et al.  Image reconstruction from projections : the fundamentals of computerized tomography , 1980 .

[14]  Michele Benzi,et al.  GIANFRANCO CIMMINO'S CONTRIBUTIONS TO NUMERICAL MATHEMATICS , 2004 .

[15]  Alston S. Householder,et al.  The Theory of Matrices in Numerical Analysis , 1964 .

[16]  Nikolaĭ Stepanovich Kurpelʹ.  Projection-Iterative Methods for Solution of Operator Equations , 1976 .

[17]  Claude Brezinski,et al.  Projection methods for systems of equations , 1997 .

[18]  John W. Chinneck,et al.  Feasibility and Infeasibility in Optimization:: Algorithms and Computational Methods , 2007 .

[19]  S. Kaczmarz.  Approximate solution of systems of linear equations , 1993 .

[20]  Y. Censor,et al.  An iterative row-action method for interval convex programming , 1981 .

[21]  Boris Polyak,et al.  The method of projections for finding the common point of convex sets , 1967 .

[22]  P. Bahr,et al.  Sampling: Theory and Applications , 2020, Applied and Numerical Harmonic Analysis.

[23]  Y. Censor.  Row-Action Methods for Huge and Sparse Systems and Their Applications , 1981 .

[24]  Noël Gastinel,et al.  Linear numerical analysis , 1973 .

[25]  I. J. Schoenberg,et al.  The Relaxation Method for Linear Inequalities , 1954, Canadian Journal of Mathematics.

[26]  L. Bregman.  The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[27]  John M. Dye,et al.  Convergence of random products of compact contractions in Hilbert space , 1989 .

[28]  Yongyi Yang,et al.  Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics , 1998 .

[29]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[30]  E. H. Zarantonello.  Projections on Convex Sets in Hilbert Space and Spectral Theory: Part I. Projections on Convex Sets: Part II. Spectral Theory , 1971 .

[31]  P. L. Combettes,et al.  The Convex Feasibility Problem in Image Recovery , 1996 .

[32]  Heinz H. Bauschke,et al.  Fixed-Point Algorithms for Inverse Problems in Science and Engineering , 2011, Springer Optimization and Its Applications.

[33]  W. W. Breckner.  Blum, E. / Oettli, W., Mathematische Optimierung, Grundlagen und Verfahren, IX, 413 S., 5 Abb., Berlin‐Heidelberg‐New York. Springer‐Verlag. 1975. DM 148,‐. US $ 60.70 . , 1977 .

[34]  V. Berinde.  Iterative Approximation of Fixed Points , 2007 .

[35]  Ivan G. Kazantsev,et al.  A discrete spherical x-ray transform of orientation distribution functions using bounding cubes , 2009 .

[36]  Heinz H. Bauschke,et al.  The method of cyclic projections for closed convex sets in Hilbert space , 1997 .

[37]  A. Cegielski.  Iterative Methods for Fixed Point Problems in Hilbert Spaces , 2012 .

[38]  I. I. Eremin.  Theory of Linear Optimization , 2002 .

[39]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[40]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[41]  Y. Censor,et al.  Parallel Optimization: Theory, Algorithms, and Applications , 1997 .

[42]  Dan Butnariu,et al.  Asymptotic Behavior of Quasi-Nonexpansive Mappings , 2001 .

[43]  G. Herman,et al.  Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography. , 1970, Journal of theoretical biology.

[44]  Charles L. Byrne,et al.  Applied Iterative Methods , 2007 .

[45]  Y. Censor,et al.  Iterative Projection Methods in Biomedical Inverse Problems , 2008 .

[46]  Patrick L. Combettes,et al.  On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints , 2009, Computational Optimization and Applications.

[47]  Karin Schwab,et al.  Best Approximation In Inner Product Spaces , 2016 .

[48]  P. C. Parks.  S. Kaczmarz (1895–1939) , 1993 .

[49]  P. L. Combettes.  The foundations of set theoretic estimation , 1993 .

[50]  Frank Deutsch,et al.  Arbitrarily Slow Convergence of Sequences of Linear Operators: A Survey , 2011, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[51]  A. Galántai.  Projectors and Projection Methods , 2003 .

[52]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[53]  Lech Maligranda.  Stefan Kaczmarz (1895-1939) , 2014 .

[54]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.