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Maximum entropy and feasibility methods for convex and nonconvex inverse problems. (English) Zbl 1241.45009

The author gives a survey of maximum entropy and feasibility methods for convex and nonconvex inverse problems. In particular he discusses two approaches to solve nonconvex and convex feasiblity problem via an algebraic iterative method and entropy optimization methods. More importance is given to the Fenchel conjugate.

MSC:

45Q05 Inverse problems for integral equations
49N45 Inverse problems in optimal control
90C25 Convex programming
49N15 Duality theory (optimization)
90C90 Applications of mathematical programming
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[1] Avellaneda M, Proceedings of the International Congress of Mathematicians pp 545– (1998)
[2] DOI: 10.1093/imanum/8.1.141 · Zbl 0638.65055
[3] DOI: 10.1007/BF01027691 · Zbl 0801.47042
[4] DOI: 10.1137/S0036144593251710 · Zbl 0865.47039
[5] DOI: 10.1007/978-1-4419-9467-7 · Zbl 1218.47001
[6] DOI: 10.1364/JOSAA.19.001334
[7] Borwein JM, Featured SIAM Rev. 48 pp 585– (2006)
[8] Borwein J, Experimental Mathematics in Action (2007)
[9] Borwein JM, SIAMOpt 4 pp 464– (2003)
[10] DOI: 10.1007/s10107-007-0134-4 · Zbl 1158.90007
[11] DOI: 10.1137/0805004 · Zbl 0820.49016
[12] J.M. Borwein, P. Howlett, and J. Piantadosi,Copulas with maximum entropy, Optim. Lett. (2011), 27 pp. DOI: 10.1007/s11590-010-0254-2 · Zbl 1280.90093
[13] DOI: 10.1137/0329017 · Zbl 0797.49030
[14] DOI: 10.1007/BF01581072 · Zbl 0778.90049
[15] DOI: 10.1137/0804008 · Zbl 0808.46022
[16] Borwein JM, Convex Analysis and Nonlinear Optimization, 2. ed. (2005)
[17] DOI: 10.1007/s002110050090 · Zbl 0840.65147
[18] Borwein JM, SIAG/OPT News and Views 5 pp 9– (1994)
[19] DOI: 10.1080/00036819508840375 · Zbl 0833.65055
[20] Borwein JM, Handbook of Mathematical Methods in Imaging pp 229– (2010)
[21] DOI: 10.1007/978-1-4419-9569-8_5 · Zbl 1357.49115
[22] DOI: 10.1007/s001860050004 · Zbl 0960.90090
[23] J.M. Borwein and B. Sims,The Douglas–Rachford algorithm in the absence of convexity, inFixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optimization and its Applications, Vol. 49, Chap. 6, Springer-Verlag, New York, 2011, pp. 93–109 · Zbl 1259.90098
[24] Borwein JM, Convex Functions (2010)
[25] Borwein JM, Techniques of Variational Analysis (2005)
[26] DOI: 10.1137/060658163 · Zbl 1151.37061
[27] Bose CJ, Maximum entropy estimates for risk-neutral probability measures with nonstrictly-convex data, preprint (2011)
[28] Boyd S, Convex Optimization 317 (2004)
[29] Burg JP, Paper presented at 37th Meeting of the Society of Exploration Geophysicists, Oklahoma City, OK, 1967
[30] Clarke FH, Graduate Texts in Mathematics 178 (1998)
[31] DOI: 10.1016/0167-6377(82)90043-8
[32] Davidson K, Real Analysis and Applications, Undergraduate Texts in Mathematics (2008)
[33] DOI: 10.1137/0802010 · Zbl 0805.49022
[34] DOI: 10.1109/78.80970
[35] DOI: 10.1137/1.9781611971088
[36] DOI: 10.1073/pnas.0606359104 · Zbl 1160.90495
[37] Fabian M, Banach Space Theory (2010)
[38] DOI: 10.4153/CJM-1949-007-x · Zbl 0038.20902
[39] Fenchel W, Convex Cones, Sets and Functions (1953)
[40] DOI: 10.2307/2296568
[41] DOI: 10.1137/0146029 · Zbl 0608.42002
[42] Gravel SS, Divide and concur: A general approach constraint satisfaction (2008)
[43] DOI: 10.1016/j.hm.2003.07.001 · Zbl 1102.01021
[44] Havel J, Gamma: Exploring Euler’s Constant (2003)
[45] H.S. Hundal,An alternating projection that does not converge in norm, Nonlinear Anal. 57 (2004), pp. 35–61 · Zbl 1070.46013
[46] DOI: 10.1109/TASSP.1984.1164296
[47] DOI: 10.1109/23.467872
[48] M.A. Limber, T.A. Manteuffel, S.F. McCormick, and D.S. Sholl,Optimal resolution in maximum entropy image reconstruction from projections with multigrid acceleration, Proceedings of the 6th Annual Copper Mountain Conference on Multigrid Methods, 1993
[49] DOI: 10.1088/0266-5611/21/1/004 · Zbl 1146.78008
[50] D. MacHale,Comic Sections: The Book of Mathematical Jokes, Humour, Wit and Wisdom, Boole Press, Dublin, 1993
[51] DOI: 10.1016/0146-664X(79)90034-0
[52] Penot J-P, Calculus without Derivatives · Zbl 1264.49014
[53] I. Peterson, The Sudoku Solution,Science News, Math Trek, December 23, 2008
[54] Priestly MB, Non-linear and Nonstationary Time Series Analysis (1988)
[55] Pukelsheim F, Optimal Design of Experiments (1993)
[56] Rockafellar RT, Convex Analysis (1970) · Zbl 0932.90001
[57] Simons S, From Hahn–Banach to Monotonicity, Lecture Notes in Mathematics 1693 (2008)
[58] J. von Neumann,Functional operators, Vol. II: The geometry of orthogonal spaces, inAnnals of Mathematical Studies, Vol. 22, Princeton University Press, Princeton, NJ, 1950 (MR0032011 (11,240f). This is a reprint of mimeographed lecture notes first circulated in 1933)
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