We consider some metric regularity properties of order q for set-valued mappings and we establish several characterizations of these concepts in terms of Hölder-like properties of the inverses of the mappings considered. In addition, we show that even if these properties are weaker than the classical notions of regularity for set-valued maps, they allow us to solve variational inclusions under mild assumptions.
@article{bwmeta1.element.doi-10_2478_s11533-010-0087-3,
author = {Micha\"el Gaydu and Michel Geoffroy and C\'elia Jean-Alexis},
title = {Metric subregularity of order q and the solving of inclusions},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {147-161},
zbl = {1209.49053},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0087-3}
}
Michaël Gaydu; Michel Geoffroy; Célia Jean-Alexis. Metric subregularity of order q and the solving of inclusions. Open Mathematics, Tome 9 (2011) pp. 147-161. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0087-3/
[1] Alt W., Lipschitzian perturbations of infinite optimization problems, In: Mathematical Programming with Data Perturbations, II, Washington D.C., 1980, Lecture Notes in Pure and Appl. Math., 85, Dekker, New York, 1983, 7–21
[2] Aragón Artacho F.J., Dontchev A.L., Geoffroy M.H., Convergence of the proximal point method for metrically regular mappings, In: CSVAA 2004 - Control Set-Valued Analysis and Applications, ESAIM Proc., 17, EDP Sci., Les Ulis, 2007, 1–8 | Zbl 1235.90180
[3] Aragón Artacho F.J., Geoffroy M.H., Characterization of metric regularity of subdifferentials, J. Convex Anal., 2008, 15(2), 365–380 | Zbl 1146.49012
[4] Azé D., A unified theory for metric regularity of multifunctions, J. Convex Anal., 2006, 13(2), 225–252 | Zbl 1101.49013
[5] Banach S., Théorie des Opérations Linéaires, Monografje Matematyczne, Warsaw, 1932
[6] Borwein J.M., Zhuang D.M., Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps, J. Math. Anal. Appl., 1988, 134(2), 441–459 http://dx.doi.org/10.1016/0022-247X(88)90034-0 | Zbl 0654.49004
[7] Dontchev A.L., Local analysis of a Newton-type method based on partial linearization, In: The Mathematics of Numerical Analysis, Park City, 1995, Lectures in Appl. Math., 32, AMS, Providence, 1996, 295–306 | Zbl 0856.65064
[8] Dontchev A.L., Rockafellar R.T., Implicit Functions and Solution Mappings, Springer Monogr. Math., Springer, Dordrecht, 2009 | Zbl 1178.26001
[9] Ferris M.C., Pang J.S., Engineering and economic applications of complementarity problems, SIAM Rev., 1997, 39(4), 669–713 http://dx.doi.org/10.1137/S0036144595285963 | Zbl 0891.90158
[10] Fischer A., Local behavior of an iterative framework for generalized equations with nonisolated solutions, Math. Program., 2002, 94A(1), 91–124 http://dx.doi.org/10.1007/s10107-002-0364-4 | Zbl 1023.90067
[11] Frankowska H., An open mapping principle for set-valued maps, J. Math. Anal. Appl., 1987, 127(1), 172–180 http://dx.doi.org/10.1016/0022-247X(87)90149-1
[12] Frankowska H., Some inverse mapping theorems. Ann. Inst. H. Poincaré Anal. Non Linéaire, 1990, 7(3), 183–234 | Zbl 0727.26014
[13] Frankowska H., Quincampoix M., Hölder Metric Regularity of Set-Valued Maps, Math. Program. (in press), DOI: 10.1007/s10107-010-0401-7 | Zbl 1262.90173
[14] Geoffroy M.H., Jean-Alexis C., Piétrus A., A Hummel-Seebeck type method for variational inclusions, Optimization, 2009, 58(4), 389–399 http://dx.doi.org/10.1080/02331930701763223 | Zbl 1186.47053
[15] Geoffroy M.H., Pietrus A., A general iterative procedure for solving nonsmooth generalized equations, Comput. Optim. Appl., 2005, 31(1), 57–67 http://dx.doi.org/10.1007/s10589-005-1104-5 | Zbl 1114.90151
[16] Henrion R., Outrata J., Surowiec T., Strong stationary solutions to equilibrium problems with equilibrium constraints with applications to an electricity spot market model, preprint available at http://www.matheon.de/preprints/ 5511_sstat.pdf | Zbl 1281.90056
[17] Ioffe A.D., Metric regularity and subdifferential calculus, Russian Math. Surveys, 2000, 55(3), 501–558 http://dx.doi.org/10.1070/RM2000v055n03ABEH000292 | Zbl 0979.49017
[18] Klatte D., On quantitative stability for non-isolated minima, Control Cybernet., 1994, 23(1–2), 183–200 | Zbl 0808.90120
[19] Kummer B., Inclusions in general spaces: Hoelder stability, solution schemes and Ekeland's principle, J. Math. Anal. Appl., 2009, 358(2), 327–344 http://dx.doi.org/10.1016/j.jmaa.2009.04.060 | Zbl 1165.49017
[20] Leventhal D., Metric subregularity and the proximal point method, J. Math. Anal. Appl., 2009, 360(2), 681–688 http://dx.doi.org/10.1016/j.jmaa.2009.07.012 | Zbl 1175.49028
[21] Mordukhovich B.S., Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc., 1993, 340(1), 1–35 http://dx.doi.org/10.2307/2154544 | Zbl 0791.49018
[22] Mordukhovich B.S., Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren Math. Wiss., 330, Springer, Berlin, 2006
[23] Penot J.-P., Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear Anal., 1989, 13(6), 629–643 http://dx.doi.org/10.1016/0362-546X(89)90083-7
[24] Robinson S.M., Generalized equations, In: Mathematical Programming: the State of the Art, Bonn, 1982, Springer, Berlin, 1983, 346–367
[25] Robinson S.M., Newton's method for a class of nonsmooth functions, Set-Valued Anal., 1994, 2(1–2), 291–305 http://dx.doi.org/10.1007/BF01027107
[26] Walras L., Elements of Pure Economics, Alen and Unwin, London, 1954
[27] Wardrop J.G., Some theoritical aspects of road traffic research, In: Proceedings of the Institute of Civil Engineers, Part II, 1952, 325–378
[28] Yen N.D., Yao J.-C., Kien B.T., Covering properties at positive-order rates of multifunctions and some related topics, J. Math. Anal. Appl., 2008, 338(1), 467–478 http://dx.doi.org/10.1016/j.jmaa.2007.05.041 | Zbl 1137.47038