Variable metric forward-backward splitting with applications to monotone inclusions in duality

Optimization

Volumn 63, Issue 9, 2014, Pages 1289-1318

  Combettes, Patrick L ^a   Vũ, Bǎng C ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

cocoercive operator; composite operator; demiregularity; duality; forward backward splitting algorithm; monotone inclusion; monotone operator; primal dual algorithm; quasi Fej r sequence; variable metric

Indexed keywords

EID: 84903576420     PISSN: 02331934     EISSN: 10294945     Source Type: Journal    
DOI: 10.1080/02331934.2012.733883     Document Type: Article

References (42)

1. Attouch, H, Briceño-Arias, LM and Combettes, PL. 2010. A parallel splitting method for coupled monotone inclusions. SIAM J. Control Optim., 48: 3246 - 3270.
2. Attouch, H and Théra, M. 1996. A general duality principle for the sum of two operators. J. Convex Anal., 3: 1 - 24.
3. Bauschke, HH, Borwein, JM and Combettes, PL. 2003. Bregman monotone optimization algorithms. SIAM J. Control Optim., 42: 596 - 636.
4. Bauschke, HH and Combettes, PL. 2011. Convex Analysis and Monotone Operator Theory in Hilbert Spaces, New York: Springer.
5. Bonnans, JF, Ch. Gilbert, J, Lemaréchal, C and Sagastizábal, CA. 1995. A family of variable metric proximal methods. Math. Program., 68: 15 - 47.
6. Boţ, RI. 2010. Conjugate Duality in Convex Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 637, New York: Springer.
7. Briceño-Arias, LM and Combettes, PL. 2013. " Monotone operator methods for Nash equilibria in non-potential games ". In Computational and Analytical Mathematics, Edited by: Bailey, D, Bauschke, HH, Borwein, P, Garvan, F, Théra, M, Vanderwerff, J and Wolkowicz, H. New York: Springer.
8. Burke, JV and Qian, M. 1999. A variable metric proximal point algorithm for monotone operators. SIAM J. Control Optim., 37: 353 - 375.
9. Burke, JV and Qian, M. 2000. On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating. Math. Program., 88: 157 - 181.
10. Chen, X and Fukushima, M. 1999. Proximal quasi-Newton methods for nondifferentiable convex optimization. Math. Program., 85: 313 - 334.
11. Chen, GH-G and Rockafellar, RT. 1997. Convergence rates in forward-backward splitting. SIAM J. Optim., 7: 421 - 444.
12. Combettes, PL. 2004. Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization, 53: 475 - 504.
13. Combettes, PL, Dũng, D and Vũ, BC. 2010. Dualization of signal recovery problems. Set-Valued Var. Anal., 18: 373 - 404.
14. Combettes, PL, Dũng, D and Vũ, BC. 2011. Proximity for sums of composite functions. J. Math. Anal. Appl., 380: 680 - 688.
15. Combettes, PL and Pesquet, J-C. 2011. " Proximal splitting methods in signal processing ". In Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Edited by: Bauschke, HH, Burachik, RS, Combettes, PL, Elser, V, Luke, DR and Wolkowicz, H. 185 - 212. New York: Springer.
16. Combettes, PL and Pesquet, J-C. 2012. Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued Var. Anal., 20: 307 - 330.
17. Combettes, PL and Vũ, BC. 2013. Variable metric quasi-Fejér monotonicity. Nonlinear Anal., 78: 17 - 31.
18. Combettes, PL and Wajs, VR. 2005. Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul., 4: 1168 - 1200.
19. Davidon, WC. 1991. Variable metric method for minimization, Argonne National Laboratory research and development report 5990, 1959. reprinted in SIAM J. Optim., 1: 1 - 17.
20. De Vito, E, Umanità, V and Villa, S. 2011. A consistent algorithm to solve Lasso, elastic-net and Tikhonov regularization. J. Complexity, 27: 188 - 200.
21. Duchi, J and Singer, Y. 2009. Efficient online and batch learning using forward backward splitting. J. Mach. Learn. Res., 10: 2899 - 2934.
22. Eckstein, J and Ferris, MC. 1999. Smooth methods of multipliers for complementarity problems. Math. Program., 86: 65 - 90.
23. Facchinei, F and Pang, J-S. 2003. Finite-Dimensional Variational Inequalities and Complementarity Problems, New York: Springer-Verlag.
24. Glowinski, R and Le Tallec, P. 1989. Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, Philadelphia, PA: SIAM.
25. Hiriart-Urruty, J-B and Lemaréchal, C. 1993. Convex Analysis and Minimization Algorithms, New York: Springer-Verlag.
26. Kato, T. 1980. Perturbation Theory for Linear Operators, 2nd, New York: Springer-Verlag.
27. Lemaréchal, C and Sagastizábal, C. 1997. Variable metric bundle methods: From conceptual to implementable forms. Math. Program., 76: 393 - 410.
28. Lotito, PA, Parente, LA and Solodov, MV. 2009. A class of variable metric decomposition methods for monotone variational inclusions. J. Convex Anal., 16: 857 - 880.
29. Mercier, B. 1979. Topics in Finite Element Solution of Elliptic Problems, Lectures on Mathematics, no. 63, Bombay, India: Tata Institute of Fundamental Research.
30. Mercier, B. 1980. Inéquations Variationnelles de la Mécanique, Orsay, France: Université de Paris-XI. (Publications Mathématiques d'Orsay, no. 80.01)
31. Mosco, U. 1972. Dual variational inequalities. J. Math. Anal. Appl., 40: 202 - 206.
32. Parente, LA, Lotito, PA and Solodov, MV. 2008. A class of inexact variable metric proximal point algorithms. SIAM J. Optim., 19: 240 - 260.
33. Pearson, JD. 1969. Variable metric methods of minimisation. Comput. J., 12: 171 - 178.
34. Pennanen, T. 2000. Dualization of generalized equations of maximal monotone type. SIAM J. Optim., 10: 809 - 835.
35. Qi, L and Chen, X. 1997. A preconditioning proximal Newton method for nondifferentiable convex optimization. Math. Program., 76: 411 - 429.
36. Qian, M. The variable metric proximal point algorithm: Theory and application, Ph.D. thesis, University of Washington, Seattle, WA, 1992
37. Robinson, SM. 1999. Composition duality and maximal monotonicity. Math. Program., 85: 1 - 13.
38. Rockafellar, RT. 1967. Duality and stability in extremum problems involving convex functions. Pacific J. Math., 21: 167 - 187.
39. Tseng, P. 1990. Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming. Math. Program., 48: 249 - 263.
40. Tseng, P. 1991. Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control Optim., 29: 119 - 138.
41. Vũ, BC. A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comput. Math., published on-line 2011. http://www.springerlink.com/content/m177247u22644173/
42. Zhu, DL and Marcotte, P. 1996. Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J. Optim., 6: 714 - 726.