Attouch-Théra duality revisited: Paramonotonicity and operator splitting

Journal of Approximation Theory

Volumn 164, Issue 8, 2012, Pages 1065-1084

  Bauschke, Heinz H ^a   Boţ, Radu I ^b   Hare, Warren L ^a   Moursi, Walaa M ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Attouch Th ra duality; Douglas Rachford splitting; Eckstein Ferris Pennanen Robinson duality; Fenchel duality; Fenchel Rockafellar duality; Firmly nonexpansive mapping; Fixed point; Hilbert space; Maximal monotone operator; Nonexpansive mapping; Paramonotonicity; Resolvent; Subdifferential operator; Total duality

Indexed keywords

EID: 84862884952     PISSN: 00219045     EISSN: 10960430     Source Type: Journal    
DOI: 10.1016/j.jat.2012.05.008     Document Type: Article

References (52)

1. Attouch H., Théra M. A general duality principle for the sum of two operators. Journal of Convex Analysis 1996, 3:1-24.
2. H.H. Bauschke, New demiclosedness principles for (firmly) nonexpansive operators, March 2011 (submitted for publication). arxiv:1103.0991.
3. Bauschke H.H., Borwein J.M., Wang X. Fitzpatrick functions and continuous linear monotone operators. SIAM Journal on Optimization 2007, 18:789-809.
4. Bauschke H.H., Combettes P.L. A weak-to-strong convergence principle for Fejér monotone methods in Hilbert spaces. Mathematics of Operations Research 2001, 26:248-264.
5. Bauschke H.H., Combettes P.L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces 2011, Springer.
6. Bauschke H.H., Combettes P.L., Luke D.R. Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization. Journal of the Optical Society of America A 2002, 19:1334-1345.
7. Bauschke H.H., Combettes P.L., Luke D.R. Finding best approximation pairs relative to two closed convex sets in Hilbert spaces. Journal of Approximation Theory 2004, 127:178-192.
8. Bauschke H.H., Combettes P.L., Luke D.R. A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space. Journal of Approximation Theory 2006, 141:63-69.
9. Bauschke H.H., Wang X., Yao L. Monotone linear relations: maximality and Fitzpatrick functions. Journal of Convex Analysis 2009, 16:673-686.
10. Bauschke H.H., Wang X., Yao L. Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter. Journal of Mathematical Analysis and Applications 2010, 370:224-241.
11. Borwein J.M., Vanderwerff J.D. Convex Functions 2010, Cambridge University Press.
12. Boţ R.I, Grad S.M., Wanka G. New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces. Nonlinear Analysis: Theory, Methods & Applications 2008, 69:323-336.
13. Boţ R.I, Grad S.M., Wanka G. On strong and total Lagrange duality for convex optimization problems. Journal of Mathematical Analysis and Applications 2008, 337:1315-1325.
14. Brézis H. Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert 1973, North-Holland/Elsevier.
15. Briceño-Arias L.M., Combettes P.L. A monotone + skew splitting model for composite monotone inclusions in duality. SIAM Journal on Optimization 2011, 21:1230-1250.
16. Burachik R.S., Iusem A.N. A generalized proximal point algorithm for the variational inequality problem in a Hilbert space. SIAM Journal on Optimization 1998, 8:197-216.
17. Burachik R.S., Iusem A.N. Set-Valued Mappings and Enlargements of Monotone Operators 2008, Springer-Verlag.
18. Carlier G. Remarks on Toland's duality, convexity constraint and optimal transport. Pacific Journal of Optimization 2008, 4:423-432.
19. Censor Y., Iusem A.N., Zenios S.A. An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Mathematical Programming (Series A) 1998, 81:373-400.
20. Combettes P.L. Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 2004, 53:475-504.
21. Combettes P.L., Pesquet J.-C. Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued and Variational Analysis 2011, published online, Optimization, 53 (2004) 475-504. 10.1007/s11228-011-0191-y.
22. J. Eckstein, Splitting Methods for Monotone Operators with Applications to Parallel Optimization, Ph.D. Thesis, MIT, 1989.
23. Eckstein J., Bertsekas D.P. On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming (Series A) 1992, 55:293-318.
24. Eckstein J., Ferris M.C. Smooth methods of multipliers for complementarity problems. Mathematical Programming (Series A) 1999, 86:65-90.
25. Eckstein J., Svaiter B.F. A family of projective splitting methods for the sum of two maximal monotone operators. Mathematical Programming (Series B) 2008, 111:173-199.
26. Eckstein J., Svaiter B.F. General projective splitting methods for sums of maximal monotone operators. SIAM Journal on Control and Optimization 2009, 48:787-811.
27. Hadjisavvas N., Schaible S. On a generalization of paramonotone maps and its application to solving the Stampacchia variational inequality. Optimization 2006, 55:593-604.
28. Halpern B. Fixed points of nonexpanding maps. Bulletin of the AMS 1967, 73:957-961.
29. Y. Haugazeau, Sur les Inéquations Variationnelles et la Minimisation de Fonctionnelles Convexes, thèse de doctorat, Université de Paris, 1968.
30. Iusem A.N. On some properties of paramonotone operators. Journal of Convex Analysis 1998, 5:269-278.
31. Lawrence J., Spingarn J.E. On fixed points of non-expansive piecewise isometric mappings. Proceedings of the London Mathematical Society (Series 3) 1987, 55:605-624.
32. J. Luque, Convolutions of maximal monotone mappings, technical report LIDS-P-1597, MIT Libraries, Cambridge, MA, 1986. http://mit.dspace.org/handle/1721.1/2953.
33. Martínez-Legaz J.-E. Some generalizations of Rockafellar's surjectivity theorem. Pacific Journal of Optimization 2008, 4:527-535.
34. B. Mercier, Inéquations Variationnelles de la Mécanique (Publications Mathématiques d'Orsay, no. 80.01), Orsay, France: Université de Paris-XI, 1980. http://portail.mathdoc.fr/PMO/PDF/M_MERCIER-87.pdf.
35. Minty G.J. Monotone (nonlinear) operators in Hilbert spaces. Duke Mathematical Journal 1962, 29:341-346.
36. Mosco U. Dual variational inequalities. Journal of Mathematical Analysis and Applications 1972, 40:202-206.
37. Moudafi A. On the stability of the parallel sum of maximal monotone operators. Journal of Mathematical Analysis and Applications 1996, 199:478-488.
38. Ogura N., Yamada I. Nonstrictly convex minimization over the bounded fixed point set of a nonexpansive mapping. Numerical Functional Analysis and Optimization 2003, 24:129-135.
39. Passty G.B. The parallel sum of nonlinear monotone operators. Nonlinear Analysis 1986, 10:215-227.
40. Pennanen T. Dualization of generalized equations of maximal monotone type. SIAM Journal on Optimization 2000, 10:809-835.
41. Robinson S.M. Composition duality and maximal monotonicity. Mathematical Programming (Series A) 1999, 85:1-13.
42. Rockafellar R.T. Convex Analysis 1970, Princeton University Press, Princeton.
43. Rockafellar R.T., J-B Wets R. Variational Analysis 2009, Springer-Verlag, corrected 3rd printing.
44. Simons S. Minimax and Monotonicity 1998, Springer-Verlag.
45. Simons S. From Hahn-Banach to Monotonicity 2008, Springer-Verlag.
46. Svaiter B.F. On weak convergence of the Douglas-Rachford method. SIAM Journal on Control and Optimization 2011, 49:280-287.
47. Wittmann R. Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 1992, 58:486-491.
48. Zǎlinescu C. Convex Analysis in General Vector Spaces 2002, World Scientific Publishing.
49. Zǎlinescu C. A new convexity property for monotone operators. Journal of Convex Analysis 2006, 13:883-887.
50. Zeidler E. Nonlinear Functional Analysis and Its Applications II/A: Linear Monotone Operators 1990, Springer-Verlag.
51. Zeidler E. Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators 1990, Springer-Verlag.
52. Zeidler E. Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems 1993, Springer-Verlag.