Abstract
In this paper, we consider the extension of the notion of well-posedness by perturbations, introduced by Zolezzi for optimization problems, to other related variational problems like inclusion problems and fixed-point problems. Then, we study the conditions under which there is equivalence of the well-posedness in the above sense between different problems. Relations with the so-called diagonal well-posedness are also given. Finally, an application to staircase iteration methods is presented.
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TYKHONOV, A. N., On the Stability of the Functional Optimization Problem, USSR Journal of Computational Mathematics and Mathematical Physics, Vol. 6, pp. 631–634, 1966.
AUSLENDER, A., CROUZEIX, J.P., and COMINETTI, R., Convex Functions with Unbounded Level Sets and Applications to Duality Theory, SIAM Journal on Optimization, Vol. 3, pp. 669–687, 1993.
BEDNARCZUK, E., and PENOT, J. P., Metrically Well-Set Minimization Problems, Applied Mathematics and Optimization, Vol. 26, pp. 273–285, 1992.
BEER, G., and LUCCHETTI, R., Well-Posed Optimization Problems and a New Topology for the Closed Subsets of a Metric Space, Rocky Mountain Journal of Mathematics, Vol. 23, pp. 1197–1220, 1993.
DONTCHEV, A. L., and ZOLEZZI, T., Well-Posed Optimization Problems, Lectures Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 1543, 1993.
FURI, M., and VIGNOLI, A., About Well-Posed Optimization Problems for Functionals in Metric Spaces, Journal of Optimization Theory and Applications, Vol. 5, pp. 225–229, 1970.
LEMAIRE, B., Bonne Position, Conditionnement, et Bon Comportement Asymptotique, Report 5, Séinaire d'Analyse Convexe de Montpellier, 1992.
LEMAIRE, B., Well-Posedness, Conditioning, and Regularization of Minimization, Inclusion, and Fixed-Point Problems, Pliska Studia Mathematica Bulgarica, Vol. 12, pp. 71–84, 1998.
OULD AHMED SALEM, C., Approximation de Points Fixes d'une Contraction, Thèse de Doctorat, Université Montpellier II, 1998.
REVALSKI, J. P., Generic Well-Posedness in Some Classes of Optimization Problems, Acta Universitatis Carolinae-Mathematica et Physica, Vol. 28, pp. 117–125, 1987.
REVALSKI, J. P., and Zhivkov, N. V., Well-Posed Constrained Optimization Problems in Metric Spaces, Journal of Optimization Theory and Applications, Vol. 76, pp. 143–163, 1993.
ZOLEZZI, T., On Equiwellset Minimum Problems, Applied Mathematics and Optimization, Vol. 4, pp. 209–223, 1978.
BAHRAOUI, M.A., and LEMAIRE, B., Convergence of Diagonally Stationary Sequences in Convex Optimization, Set-Valued Analysis, Vol. 2, pp. 1–13, 1994.
BOYER, R., Quelques Algorithmes Diagonaux en Optimisation Convexe, Thèse de Doctorat, Université de Provence, 1974.
LEMAIRE, B., Bounded Diagonally Stationary Sequences in Convex Optimization, Journal of Convex Analysis (to appear).
LORIDAN, P., Sur un Procédé d'Optimisation Utilisant Simultanément les Méthodes de Pénalisation et des Variations Locales, SIAM Journal on Control, Vol. 11, pp. 159–172, 1973 and Vol. 11, pp. 173–184, 1973.
ZOLEZZI, T., Well-posed Criteria in Optimization with Application to the Calculus of Variations, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 25, pp. 437–453, 1995.
ZOLEZZI, T., Extended Well-Posedness of Optimization Problems, Journal of Optimization Theory and Applications, Vol. 91, pp. 257–268, 1996.
LUCCHETTI, R., and ZOLEZZI, T., On Well-Posedness and Stability Analysis in Optimization, Mathematical Programming with Data Perturbations, Edited by A. V. Fiacco, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, Vol. 195, pp. 223–251, 1998.
BRØNDSTED, A., and ROCKAFELLAR, R. T., On the Subdifferentiability of Convex Functions, Proceedings of the American Mathematical Society, Vol. 16, pp. 605–611, 1965.
DIESTEL, J., Geometry of Banach Spaces: Selected Topics, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, 1975.
BROWDER, F. E., Nonlinear Maximal Monotone Operators in Banach Space, Mathematische Annalen, Vol. 175, pp. 89–113, 1968.
ROCKAFELLAR, R. T., On the Maximality of Sums of Nonlinear Monotone Operators, Transactions of the American Mathematical Society, Vol. 149, pp. 75–88, 1970.
ROCKAFELLAR, R. T., Monotone Operators and the Proximal Point Algorithm, SIAM Journal on Control and Optimization, Vol. 14, pp. 877–898, 1976.
KAPLAN, A., and TICHATSCHKE, R., Stable Methods for Ill-Posed Problems, Akademie Verlag, Berlin, Germany, 1994.
LEMAIRE, B., Regularization of Fixed-Point Problems and Staircase Iteration, Ill-Posed Variational Problems and Regularization Techniques, Edited by M. Théra and R. Tichatschke, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 477, pp. 151–166, 1999.
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Lemaire, B., Ould Ahmed Salem, C. & Revalski, J.P. Well-Posedness by Perturbations of Variational Problems. Journal of Optimization Theory and Applications 115, 345–368 (2002). https://doi.org/10.1023/A:1020840322436
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DOI: https://doi.org/10.1023/A:1020840322436