Abstract
The convergence of the proximal method of Martinet-Rockafellar, in exact or approximate form, is revisited in connexion to the asymptotical behaviour of the solutions to differential inclusions associated with maximal monotone operators. Actually, in the context of convex optimization the generated sequence is shown to be minimizing without any boundedness assumption. In the more general context of monotone inclusions, if the set of solutions has a non empty interior, then the generated sequence is strongly convergent.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alart P. & Lemaire B. Penalization in non-classical convex programming via variational convergence, Math. Progr. 51, 1991, 307–331.
Auslender A. Numerical methods for non differentiable convex optimization, in Math. Progr. Studies, n°30, Vial and al editors, 1985.
Auslender A. & Crouzeix J.P. Well behaved asymptotical convex functions, in Analyse non linéaire, Gauthier- Villars,1989, 101–122.
Baillon J.B. Quelques propriétés de convergence asymptotique pour les contrations impaires, C.R. Acad. Sci. Paris 283, 1976, 587–590.
Brézis H. Opérateurs maximaux monotones, Lecture Note n°5, North-Holland, 1973.
Brézis H. & Lions P.L. Produits infinis de résolvantes, Israël Journal of Mathematics, Vol 29, n°4, 1978.
Bruck Jr R.E. Asymptotic Convergence of Nonlinear Contra-tion Semigroups in Hubert Space, Journal of Functional Analysis 18, 1975, 15–26.
Güler O. On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control and Optimization, Vol 29, n°2, 1991, 403–419.
Lefebvre O. & Michelot C. About the finite convergence of the proximal point algorithm, in Trends in Mathematical Optimization, International Series of Numerical Mathematics, Vol 84(c), Birkhäuser Verlag, 1988, 153–161.
Lemaire B. Coupling optimization methods and variational covergence, in Trends in Mathematical Optimization, International Series of Numerical Mathematics, Vol 84(c), Birkhäuser Verlag, 1988, 163–179.
Lemaire B. The proximal algorithm, in New Methods in Optimization and their Industrial Uses, Proc. Symp. Pau and Paris 1987, 73–89.
Lemaire B. The proximal algorithm, in New Methods in Optimization and their Industrial Uses, Int. Ser. Numer. Math., Vol 87, Birkhäuser Verlag, 1989, 73–89.
Lemaire B. Quelques résultats récents sur l’algorithme proximal, Séminaire d’Analyse Numérique, Université de Toulouse, 1989.
Lemaire B. About the proximal algorithm, Premier Colloque Franco-Soviétique de Programmation Mathématique, Luminy, 1990.
Martinet B. Régularisation d’inéquations variationnelles par approximations successives, Rev. Française Inf. Rech. Oper., 1970, 154–159.
Martinet B. Algorithmes pour la résolution de problèmes d’optimisation et de minimax, Thèse d’Etat, Univ. Grenoble, 1972.
Moreau J.J. Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93, 1965, 273–299.
Moreau J.J. Un cas de convergence des itéréesce d’une contraction d’un espace hilbertien, C.R. Acad. Sci. Paris, 1978, 143–144.
Moudafi A. Thèse de Doctorat, Université des Sciences et des Techniques du Languedoc, 1991.
Polyak B.T. Sharp Minimum, International Workshop on Augmented Lagrangians, Vienna, 1979.
Reich S. On infinite products of resolvants, Lincei Rend. Sc. Fis. Mat. e Nat., Vol I.XIII, 1977.
Rockafellar R.T. Monotone Operators and the proximal point algorithm, SIAM J. Control and Optimisation, 1976, 877–898.
Rockafellar R.T. Augmented lagrangians and applications of the proximal point algorithm in convex programming, Math, of Operational Research, 1976, 97–116.
Spingarn J.E. Partial inverse of a monotone operator, Appl. Math. Opt. 10, 1983, 247–265.
Spingarn J.E. Applications of the method of partial inverses to convex programming:decomposition, Math. Prog. 32, 1985, 199–223.
Tikhonov A. & Arsenine V. Méthodes de résolution de problèmes mal posés, M.I.R, 1974.
Tossings P. Sur les zéros des opérateurs maximaux monotones et applications, Thèse de Doctorat, Université de Liège, 1990.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lemaire, B. (1992). About the Convergence of the Proximal Method. In: Oettli, W., Pallaschke, D. (eds) Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol 382. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51682-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-51682-5_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-55446-2
Online ISBN: 978-3-642-51682-5
eBook Packages: Springer Book Archive