A monotone+ skew splitting model for composite monotone inclusions in duality

SIAM Journal on Optimization

Volumn 21, Issue 4, 2011, Pages 1230-1250

  Briceno Arias, Luis M ^a   Combettes, Patrick L ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

Composite operator; Convex optimization; Decomposition; Duality; Fenchel Rockafellar duality; Forward backward forward algorithm; Minimization algorithm; Monotone inclusion; Monotone operator; Operator splitting

Indexed keywords

COMPOSITE OPERATORS; DUALITY; FENCHEL-ROCKAFELLAR DUALITY; MINIMIZATION ALGORITHM; MONOTONE OPERATORS; OPERATOR-SPLITTING;

ALGORITHMS; CONVEX OPTIMIZATION; DECOMPOSITION; LINEAR TRANSFORMATIONS;

PROBLEM SOLVING;

EID: 84855933316     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/10081602X     Document Type: Article

References (43)

1. G. Alduncin, Composition duality principles for mixed variational inequalities, Math. Comput. Modelling, 41 (2005), pp. 639-654. (Pubitemid 41097562)
2. H. Attouch, L. M. Briceño-Arias, and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim., 48 (2010), pp. 3246-3270.
3. H. Attouch and M. Théra, A general duality principle for the sum of two operators, J. Convex Anal., 3 (1996), pp. 1-24. (Pubitemid 126408057)
4. H. Attouch and M. Théra, A duality proof of the Hille-Yosida theorem, in Progress in Partial Differential Equations: The Metz Surveys, 4, Pitman. Res. Notes Math. Ser. 345, Longman, Harlow, 1996, pp. 18-35.
5. H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejérmonotone methods in Hilbert spaces, Math. Oper. Res., 26 (2001), pp. 248-264. (Pubitemid 34050614)
6. H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011.
7. H. H. Bauschke, P. L. Combettes, and S. Reich, The asymptotic behavior of the composition of two resolvents, Nonlinear Anal., 60 (2005), pp. 283-301.
8. G. Chen and M. Teboulle, A proximal-based decomposition method for convex minimization problems, Math. Programming, 64 (1994), pp. 81-101.
9. P. L. Combettes, Quasi-Fejérian analysis of some optimization algorithms, in Inherently Parallel Algorithms for Feasibility and Optimization, D. Butnariu, Y. Censor, and S. Reich, eds., Elsevier, New York, 2001, pp. 115-152.
10. P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), pp. 475-504. (Pubitemid 40190083)
11. P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), pp. 727-748.
12. P. L. Combettes, Dinh Dũng, and B. C. Vũ, Dualization of signal recovery problems, Set-Valued Var. Anal., 18 (2010), pp. 373-404.
13. P. L. Combettes and J.-C. Pesquet, Proximal splitting methods in signal processing, in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Ser. Optim. Appl. 49, H. H. Bauschke et al., eds., Springer, New York, 2011, pp. 185-212.
14. P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), pp. 1168-1200. (Pubitemid 44932287)
15. J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), pp. 293-318.
16. J. Eckstein and M. C. Ferris, Smooth methods of multipliers for complementarity problems, Math. Programming, 86 (1999), pp. 65-90.
17. J. Eckstein and B. F. Svaiter, A family of projective splitting methods for the sum of two maximal monotone operators, Math. Programming, 111 (2008), pp. 173-199. (Pubitemid 46993959)
18. I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels, Collection Études Mathématiques, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974. Also published as Convex Analysis and Variational Problems (corrected reprint of the 1976 English edition), Classics Appl. Math. 28, SIAM, Philadelphia, 1999.
19. F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003.
20. M. Fukushima, The primal Douglas-Rachford splitting algorithm for a class of monotone mappings with applications to the traffic equilibrium problem, Math. Programming, 72 (1996), pp. 1-15. (Pubitemid 126372924)
21. M. Fortin and R. Glowinski, eds., Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems, North-Holland, Amsterdam, 1983.
22. D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems, M. Fortin and R. Glowinski, eds., North-Holland, Amsterdam, 1983, pp. 299-331.
23. R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, Stud. Appl. Math. 9, SIAM, Philadelphia, 1989.
24. P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), pp. 964-979.
25. L. McLinden, An extension of Fenchel's duality theorem to saddle functions and dual minimax problems, Pacific J. Math., 50 (1974), pp. 135-158.
26. B. Mercier, Lectures on Topics in Finite Element Solution of Elliptic Problems, Tata Inst. Fund. Res. Lectures on Math. and Phys. 63, Tata Institute of Fundamental Research, Bombay, 1979.
27. B. Mercier, Inéquations Variationnelles de la Mécanique, Publications Mathématiques d'Orsay 80.01, Université de Paris-XI, Orsay, France, 1980.
28. U. Mosco, Dual variational inequalities, J. Math. Anal. Appl., 40 (1972), pp. 202-206.
29. T. Pennanen, Dualization of generalized equations of maximal monotone type, SIAM J. Optim., 10 (2000), pp. 809-835.
30. T. Pennanen, A splitting method for composite mappings, Numer. Funct. Anal. Optim., 23 (2002), pp. 875-890.
31. S. M. Robinson, Composition duality and maximal monotonicity, Math. Programming, 85 (1999), pp. 1-13.
32. S. M. Robinson, Generalized duality in variational analysis, in Advances in Convex Analysis and Global Optimization, N. Hadjisavvas and P. M. Pardalos, eds., Kluwer, Dordrecht, The Netherlands, 2001, pp. 205-219.
33. R. T. Rockafellar, Duality and stability in extremum problems involving convex functions, Pacific J. Math., 21 (1967), pp. 167-187.
34. R. T. Rockafellar, Conjugate Duality and Optimization, CBMS-NSF Reg. Conf. Ser. Appl. Math. 16, SIAM, Philadelphia, 1974.
35. R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
36. R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), pp. 97-116.
37. R. T. Rockafellar, Duality and optimality in multistage stochastic programming, Ann. Oper. Res., 85 (1999), pp. 1-19.
38. B. F. Svaiter, On weak convergence of the Douglas-Rachford method, SIAM J. Control Optim., 49 (2011), pp. 280-287.
39. P. Tseng, Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming, Math. Programming, 48 (1990), pp. 249-263.
40. P. Tseng, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM J. Control Optim., 29 (1991), pp. 119-138.
41. P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), pp. 431-446. (Pubitemid 30594547)
42. C. Zǎlinescu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, NJ, 2002.
43. E. Zeidler, Nonlinear Functional Analysis and Its Applications, Vols. I-V, Springer-Verlag, New York, 1985-1993.