Abstract
We prove that the metric regularity of set-valued mappings is stable under some Wijsman-type perturbations. Then, we solve a variational inclusion viewed as a limit-problem using assumptions on a sequence of associated problems. Finally, we apply our results to classical methods for solving variational inclusions.
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References
Aubin J.-P. and Frankowska H. (1990). Set-Valued Analysis. Birkhäuser, Boston
Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers (1993)
Dontchev A.L. (1996). Local convergence of the Newton method for generalized equations. C. R. Acad. Sci., Paris, Ser. 1: 327–329
Dontchev A.L. and Hager W.W. (1994). An inverse mapping theorem for set-valued maps. Proc. Am. Math. Soc. 121: 481–489
Dontchev A.L., Lewis A.S. and Rockafellar R.T. (2002). The radius of metric regularity. Trans. AMS 355: 493–517
Dontchev A.L. and Rockafellar R.T. (2004). Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12: 79–109
Dontchev A.L., Quincampoix M. and Zlateva N. (2006). Aubin criterion for metric regularity. (English summary). J. Convex Anal. 13(2): 281–297
Geoffroy M.H., Hilout S. and Pietrus A. (2003). Acceleration of convergence in Dontchev’s iterative method for solving variationnal inclusions. Serdica. Math. J. 22: 385–398
Geoffroy M.H. and Pietrus A. (2005). A general iterative procedure for solving nonsmooth generalized equations. Comput. Optim. Appl. 31(1): 57–67
Geoffroy M.H., Hilout S. and Pietrus A. (2006). Stability of a cubically convergent method for generalized equations. Set-Valued Anal. 14(1): 41–54
Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Matematicheskikh Nauk. 55(3), 103–162 (2000); English translation in Russian Mathematical Surveys 55, 501–558 (2000)
Mordukhovich B.S. (1993). Complete characterization of openness, metric regularity and Lipschitzian properties of multifunctions. Trans. Am. Math. Soc. 340(1): 1–35
Pietrus A. (2000). Generalized equations under mild differentiability conditions. Rev. R. Acad. Cienc. Exact. Fis. Nat. 94(1): 15–18, Matematicas
Robinson S.M. (1976). Regularity and stability for convex multivalued functions. Math. Oper. Res. 1(2): 130–143
Robinson S.M. (1994). Newton’s method for a class of nonsmooth functions. Set-Valued Anal. 2: 291–305
Rockafellar, R.T., Wets, R.: Variational analysis. Ser. Comput. Stud. Math. 317, Springer (1998)
Sonntag Y. and Zalinescu C. (1993). Set convergences. An attempt of classification. Trans. Am. Math. Soc. 340(1): 199–226
Ursescu C. (1975). Multifunctions with convex closed graph. Czechoslovak Math. J. 25(100(3)): 438–441
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Geoffroy, M.H., Jean-Alexis, C. & Piétrus, A. Iterative solving of variational inclusions under Wijsman perturbations. J Glob Optim 42, 111–120 (2008). https://doi.org/10.1007/s10898-007-9253-y
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DOI: https://doi.org/10.1007/s10898-007-9253-y