zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A customized proximal point algorithm for convex minimization with linear constraints. (English) Zbl 1287.90048
Summary: This paper demonstrates a customized application of the classical proximal point algorithm (PPA) to the convex minimization problem with linear constraints. We show that if the proximal parameter in metric form is chosen appropriately, the application of PPA could be effective to exploit the simplicity of the objective function. The resulting subproblems could be easier than those of the augmented Lagrangian method (ALM), a benchmark method for the model under our consideration. The efficiency of the customized application of PPA is demonstrated by some image processing problems.

90C25Convex programming
Full Text: DOI
[1] Afonso, M., Bioucas-Dias, J., Figueiredo, M.: An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Trans. Image Process. 20, 681--695 (2010) · doi:10.1109/TIP.2010.2076294
[2] Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183--202 (2009) · Zbl 1175.94009 · doi:10.1137/080716542
[3] Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Method. Academic Press, New York (1982) · Zbl 0572.90067
[4] Blum, E., Oettli, W.: Mathematische Optimierung, Econometrics and Operations Research XX. Springer, Berlin (1975) · Zbl 0315.90062
[5] Burke, J.V., Qian, M.J.: A variable metric proximal point algorithm for monotone operators. SIAM J. Control Optim. 37, 353--375 (1998) · Zbl 0918.90112 · doi:10.1137/S0363012992235547
[6] Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89--97 (2004) · doi:10.1023/B:JMIV.0000011320.81911.38
[7] Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM Rev. 43, 129--159 (2001) · Zbl 0979.94010 · doi:10.1137/S003614450037906X
[8] Daubechies, I., Defrise, M., Mol, C.D.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57, 1413--1457 (2004) · Zbl 1077.65055 · doi:10.1002/cpa.20042
[9] Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program., Ser. A 55, 293--318 (1992) · Zbl 0765.90073 · doi:10.1007/BF01581204
[10] Fadili, J., Starck, J.L., Elad, M., Donoho, D.: Mcalab: reproducible research in signal and image decomposition and inpainting. Comput. Sci. Eng. 12(1), 44--63 (2010) · doi:10.1109/MCSE.2010.14
[11] Figueiredo, M.A.T., Nowak, R.D., Wright, S.J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1(4), 586--597 (2007) · doi:10.1109/JSTSP.2007.910281
[12] Glowinski, R., Marrocco, A.: Approximation par éléments finis d’ordre un et résolution par pénalisation-dualité d’une classe de problèmes non linéaires. RAIRO. Rech. Opér. R2, 41--76 (1975)
[13] Gol’shtein, E.G., Tret’yakov, N.V.: Modified Lagrangian in convex programming and their generalizations. Math. Program. Stud. 10, 86--97 (1979) · doi:10.1007/BFb0120845
[14] Hansen, P., Nagy, J., O’Leary, D.: Deblurring Images: Matrices, Spectra, and Filtering. SIAM, Philadelphia (2006) · Zbl 1112.68127
[15] Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303--320 (1969) · Zbl 0174.20705 · doi:10.1007/BF00927673
[16] He, B.S., Yuan, X.M.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5, 119--149 (2012) · Zbl 1250.90066 · doi:10.1137/100814494
[17] Malgouyres, F., Guichard, F.: Edge direction preserving image zooming: a mathematical and numerical analysis. SIAM J. Numer. Anal. 39(1), 1--37 (2001) · Zbl 1001.68174 · doi:10.1137/S0036142999362286
[18] Martinet, B.: Regularisation, d’inéquations variationelles par approximations succesives. Rev. Francaise Inf. Rech. Oper. 4, 154--159 (1970)
[19] Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (1999) · Zbl 0930.65067
[20] Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283--298. Academic Press, New York (1969) · Zbl 0194.47701
[21] Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97--116 (1976) · Zbl 0402.90076 · doi:10.1287/moor.1.2.97
[22] Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259--268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[23] Starck, J.L., Murtagh, F., Fadili, J.M.: Sparse Image and Signal Processing, Wavelets, Curvelets, Morphological Diversity. Cambridge University Press, Cambridge (2010) · Zbl 1196.94008
[24] Weiss, P., Aubert, G., Blanc-Féraud, L.: Some applications of l constraints in image processing. INRIA Research Report (2006)
[25] Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for l 1-minimization with applications to compressed sensing. SIAM J. Imaging Sci. 1, 143--168 (2008) · Zbl 1203.90153 · doi:10.1137/070703983
[26] Zhang, X.Q., Burger, M., Osher, S.: A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46, 20--46 (2011) · Zbl 1227.65052 · doi:10.1007/s10915-010-9408-8