On the relation between bundle methods for maximal monotone inclusions and hybrid proximal point algorithms

Studies in Computational Mathematics

Volumn 8, Issue C, 2001, Pages 441-455

  Sagastizábal, Claudia A ^a   Solodov, Mikhail V ^a  

^a IMPA   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001492471     PISSN: 1570579X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S1570-579X(01)80026-4     Document Type: Chapter

References (41)

1. To be published in. Alber Ya.I. On average convergence of the iterative projection methods. Taiwanese Journal of Mathematics (1997)
2. Preprint of the Technion No. 1051. Haifa, Israel. Alber Ya.I. Proximal projection method for variational inequalities and Cesàro averaged approximations (1998)
3. Accepted for publication. Bacaud L., Lemaréchal C., Renaud A., and Sagastizábal C.A. Bundle methods in stochastic optimal power managment: a disaggregated approach using preconditioners. Computational Optimization and Applications (2001)
4. Bonnans J.F., Gilbert J.C., Lemaréchal C., and Sagastizábal C. A family of variable metric proximal point methods. Mathematical Programming 68 (1995) 15-47
5. Bonnans J.F., Gilbert J.Ch., Lemaréchal C., and Sagastizábal C. Optimisation Numérique: aspets théoriques et pratiques (1997), Springer-Verlag, Berlin 1997
6. Bruck R.E., and Reich S. Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston Journal of Mathematics 3 (1977) 459-470
7. Burachik R.S., Iusem A.N., and Svaiter B.F. Enlargement of monotone operators with applications to variational inequalities. Set-Valued Analysis 5 (1997) 159-180
8. Burachik R.S., Sagastizába; C.A., and Svaiter B.F. Bundle methods for maximal monotone operators. In: Tichatschke R., and Théra M. (Eds). Ill-posed variational problems and regularization techniques. Lecture Notes in Economics and Mathematical Systems No. 477 (1999), Springer-Verlag 49-64
9. Burachik R.S., Sagastizábal C.A., and Svaiter B.F. ε-enlargements of maximal monotone operators: Theory and applications. In: Fukushima M., and Qi L. (Eds). Reformulation-Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods (1999), Kluwer Academic Publishers 25-44
10. Burke J.V., and Qian M. A variable metric proximal point algorithm for monotone operators. SIAM Journal on Control and Optimization 37 (1998) 353-375
11. Cominetti R. Coupling the proximal point algorithms with approximation methods. Journal of Optimization Theory and Applications 95 (1997) 581-600
12. Cottle R.W., Pang J.-S., and Stone R.E. The Linear Complementarity Problem (1992), Academic Press, New York
13. Eckstein J. Approximate iterations in Bregman-function-based proximal algorithms. Mathematical Programming 83 (1998) 113-123
14. Harker P.T., and Pang J.-S. Finite-dimensional variational inequality problems: A survey of theory, algorithms and applications. Mathematical Programming 48 (1990) 161-220
15. Hiriart-Urruty J.-B., and Lemaréchal C. Convex Analysis and Minimization Algorithms (1993), Springer-Verlag, Berlin
16. to appear. Iusem A.N., and Lucambio Pérez L.R. An extragradient-type method for non-smooth variational inequalitie. Optimization (2000)
17. Kiwiel K.C. A method for solving certain quadratic programming problems arising in nonsmooth optimization. IMA Journal of Numerical Analysis 6 (1986) 137-152
18. Lemaire B. The proximal algorithm. In: Penot J.-P. (Ed). New Methods of Optimization and Their Industrial Use. International Seires of Numericla Mathematics 87 (1989), Birkhauser, Basel 73-87
19. Lemaréchal C., Pellegrino F., Renaud A., and Sagastizábal C. Bundle methods applied to the unit-commitment problem. In: Doležal J., and Fidler J. (Eds). System Modeling and Optimization (1995), Chapman and Hall 395-402
20. Lemaréchal C., and Sagstizábal C. An approach to variable metric bundle methods. In: Henry J., and Yvon J.P. (Eds). Lecture notes in Control and Information Sciences, No. 197, System Modelling and Optimization (1994), Springer-Verlag, Berlin 144-162
21. Lemaréchal C., and Sagastizábal C. Variable metric bundle methods: From conceptual to implementable forms. Mathematical Programming 76 (1997) 393-410
22. Lin Y.Y., and Pang J.-S. Iterative methods for large convex quadratic programs: A survey. SIAM Journal on Control and Optimization 25 (1987) 383-411
23. Luque F.J. Asymptotic convergence analysis of the proximal point algorithm. SIAM Journal on Control and Optimization 22 (1984) 277-293
24. Martinet B. Regularisation di'inequations variationelles par approximations successives. Revue Française d'Informatique et de Recherche Opérationelle 4 (1970) 154-159
25. Pennanen T. Dualization of generalized equations of maximal monotone type. SIAM Journal on Optimization 10 (2000) 809-835
26. Reich S. Weak convergence theorems for nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 67 (1979) 274-276
27. Revalski J.P., and Théra M. Variational and extended sums of monotone operators. In: Tichatschke R., and Théra M. (Eds). ILL-posed variational problems and regularization techniques. Lecture Notes in Economics and Mathematical Systems No. 477 (1999), Springer-Verlag 229-246
28. Rockafellar R.T. Local boundedness of nonlinear monotone operators. Michigan Mathematical Journal 16 (1969) 397-407
29. Rockafellar R.T. Monotone operators and the proximal point algorithm. SIAM Journal of Control and Optimization 14 (1976) 877-898
30. Shor N.Z. Minimization methods for nondifferentiable functions (1985), Springer-Verlag, Berlin 1985
31. Solodov M.V. A class of globally convergent algorithms for pseudomonotone variational inequalities. In: Ferris M.C., Mangasarian O.L., and Pang J.-S. (Eds). Complementarity: Applications, Algorithms and Extensions (2001), Kluwer Academic Publishers
32. Solodov M.V., and Svaiter B.F. A truly globally convergent Newton-type method for the monotone nonlinear complementarity problem. SIAM Journal on Optimization 10 (2000) 605-625
33. Solodov M.V., and Tseng P. Modified projection-type methods for monotone variational inequalities. SIAM Journal on Control and Optimization 34 (1996) 1814-1830
34. Solodov M.V., and Svaiter B.F. A globally convergent inexact Newton method for systems of monotone equations. In: Fukushima M., and Qi L. (Eds). Reformulation-Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods (1999), Kluwer Academic Publishers 355-369
35. Solodov M.V., and Svaiter B.F. A hybrid approximate extragradient-proximal point algorithm using the enlargemet of a maximal monotone operator. Set-Valued Analysis 7 (1999) 323-345
36. Solodov M.V., and Svaiter B.F. A hybrid projection-proxial point algorithm. Journal of Convex Analysis 6 (1999) 59-70
37. Solodov M.V., and Svaiter B.F. Error bounds for proximal point subproblems and associated inexact proximal point algorithms. Mathematical Programming 88 (2000) 371-389
38. Solodov M.V., and Svaiter B.F. An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions. Mathematics of Operations Research 25 (2000) 214-230
39. Tossing P. The perturbed proximal point algorithm and some of its applications. Applied Mathematics and Optimization 29 (1994) 125-159
40. Tseng P. A modified forward-backward splitting method for maximal monotone mappings. SIAM Journal on Control and Optimization 38 (2000) 431-446
41. Veselý L. Local uniform boundedness principle for families of ε-monotone operators. Nonlinear Analysis, Theory, Methods and Applications 24 (1995) 1299-1304