Inexact and accelerated proximal point algorithms

Journal of Convex Analysis

Volumn 19, Issue 4, 2012, Pages 1167-1192

  Salzo, Saverio ^a   Villa, Silvia ^b  

^a UNIVERSITY OF GENOA   (Italy)

^b ISTITUTO ITALIANO DI TECNOLOGIA   (Italy)

Author keywords

Accelerated proximal point algorithms; Approximation criteria; Global convergence rates

Indexed keywords

EID: 84872004272     PISSN: 09446532     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Conference Paper

References (34)

1. A. Auslender: Numerical methods for nondifferentiable convex optimization, Math. Weirprogram. Study 30 ( 1987) 102-126
2. A. Beel, M. Teboulle: A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Sci. 2( 1) ( 2009) 183-202.
3. S. Becker, J. Bobin, E. Candes: NESTA: A fast and accurate first-order method for sparse recovery, SIAM J. Imaging Sci. 4( 1) ( 2011) 1-39.
4. 1-unified variational framework for image restoration, in: Computer Vision - ECCV 2004 (Prague, 2004), Part IV, T. Pajdla, J. Matas (eds.), Lecture Notes in Computer Science 3024, Springer, Berlin ( 2004) 1-13.
5. J. R. Birge, L. Qi, Z. Wei: Convergence analysis of some methods for minimizing a nonsmooth convex function, J. Optim. Theory Appl. 97(2) (1998) 357-383. (Pubitemid 128357528)
6. R. S. Burachik, A. N. Iusem, B. F. Svaiter: Enlargement of monotone operators with applications to variational inequalities, Set-Valued Anal. 5( 2) ( 1997) 159-180.
7. R. S. Burachik, B. F. Svaiter: A relative error tolerance for a family of generalized proximal point methods, Math. Oper. Res. 26( 4) ( 2001) 816-831.
8. A. Chambolle: An algorithm for total variation minimization and applications, J. Math. Imaging Vis. 20( 1- 2) ( 2004) 89-97.
9. P. L. Combettes, J.-C. Pesquet: Proximal splitting methods in signal processing, in: Fixed-Point Algorithms for Inverse Problems in Science and Engineering ( Banff, 2009), H. H. Bauschke, R. Burachik, P. L. Combettes, V. Elser, D. R. Luke, H. Wolkowicz (eds.), Springer, New York ( 2010) 185-212.
10. P. L. Combettes, V. R. Wajs: Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul. 4( 4) ( 2005) 1168-1200.
11. R. Cominetti: Coupling the proximal point algorithm with approximation methods, J. Optim. Theory Appl. 95( 3) ( 1997) 581-600.
12. R. Correa, C. Lemaréchal: Convergence of some algorithms for convex minimization, Math. Program., Ser. B 62( 2) ( 1993) 261-275.
13. M. Fornasier (ed.): Theoretical Foundations and Numerical Methods for Sparse Recovery (Vienna, 2009), Radon Series on Computational and Applied Mathematics 9, De Gruyter, Berlin ( 2010).
14. O. Güler: On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim. 29( 2) ( 1991) 403-419.
15. O. Güler: New proximal point algorithms for convex minimization, SIAM J. Optim. 2( 4) ( 1992) 649-664.
16. J.-B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimization Algorithms II, Grundlehren der Mathematischen Wissenschaften 306, Springer, Berlin ( 1993).
17. L. Jacob, G. Obozinski, J.-P. Vert: Group lasso with overlap and graph lasso, in: ICML 2009 ( Montréal, 2009), L. Bottou et al. (ed.), ACM, New York ( 2009) 433-440.
18. B. Lemaire. About the convergence of the proximal method, in: Advances in Optimization (Lambrecht, 1991), W. Oettli et al. (ed.), Lecture Notes in Econom. and Math. Systems 382, Springer, Berlin ( 1992) 39-51.
19. B. Martinet: Régularisation d'inéquations variationnelles par approximations successives, Rev. Franç. Inform. Rech. Opér. 4 ( R-3) ( 1970) 154-158.
20. R. D. C. Monteiro, B. F. Svaiter: Convergence rate of inexact proximal point methods with relative error criteria for convex optimization, preprint ( 2010).
21. J.-J. Moreau: Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. Fr. 93 ( 1965) 273-299.
22. S. Mosci, L. Rosasco, M. Santoro, A. Verri, S. Villa: Solving structured sparsity regularization with proximal methods, in: Machine Learning and Knowledge Discovery in Databases ( Barcelona, 2010), Part II, J. Balcázar, F. Bonchi, A. Gionis, M. Sebag (eds.), Lecture Notes in Computer Science 6322, Springer, Berlin ( 2010) 418-433.
23. A. S. Nemirovsky, D. B. Yudin: Problem Complexity and Method Efficiency in Optimization, Wiley, New York ( 1983).
24. Y. Nesterov: Introductory Lectures on Convex Optimization. A Basic Course, Applied Optimization 87, Kluwer, Boston ( 2004).
25. Y. E. Nesterov: On an approach to the construction of optimal methods for minimizing smooth convex functions, Ehkon. Mat. Metody 24( 3) ( 1988) 509-517.
26. R. T. Rockafellar: Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res. 1( 2) ( 1976) 97-116.
27. R. T. Rockafellar: Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14( 5) ( 1976) 877-898.
28. M. V. Solodov, B. F. Svaiter: A comparison of rates of convergence of two inexact proximal point algorithms, in: Nonlinear Optimization and Related Topics (Erice, 1998), G. Di Pillo et al. (ed.), Appl. Optim. 36, Kluwer, Dordrecht ( 2000) 415-427.
29. M. V. Solodov, B. F. Svaiter: Error bounds for proximal point subproblems and associated inexact proximal point algorithms, Math. Program., Ser. B 88( 2) ( 2000) 371-389.
30. M. V. Solodov, B. F. Svaiter: An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions, Math. Oper. Res. 25( 2) ( 2000) 214-230.
31. M. V. Solodov, B. F. Svaiter: A unified framework for some inexact proximal point algorithms, Numer. Funct. Anal. Optim. 22( 7- 8) ( 2001) 1013-1035.
32. P. Tseng: On accelerated proximal gradient methods for convex-concave optimization, Technical Report (2008), SIAM J. Optim., submitted.
33. C. Zǎ;linescu: Convex Analysis in General Vector Spaces, World Scientific, River Edge ( 2002).
34. A. Zaslavski: Convergence of a proximal point method in the presence of computational errors in Hilbert spaces, SIAM J. Optim. 20( 5) ( 2010) 2413-2421.