Abstract
In this work we classify the at-point regularities of set-valued mappings into two categories and then we analyze their relationship through several implications and examples. After this theoretical tour, we use the subregularity properties to deduce implicit theorems for set-valued maps. Finally, we present some applications to the study of multicriteria optimization problems.
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Aragon Artacho, F.J., Mordukhovich, B.S.: Metric regularity and Lipscithian stability of parametric variational systems. Nonlinear Anal. 72, 1149–1170 (2010)
Aragon Artacho, F.J., Mordukhovich, B.S.: Enhanced metric regularity and Lipscithian stability of variational systems. J. Glob. Optim. 50, 145–167 (2010)
Arutyunov, A.V.: Covering mapping in metric spaces, and fixed points. Dokl. Math. 76, 665–668 (2007)
Arutyunov, A.V.: Stability of coincidence points and properties of covering mappings. Math. Notes 86, 153–158 (2009)
Arutyunov, A., Avakov, E., Gel’man, B., Dmitruk, A., Obukhovskii, V.: Locally covering maps in metric spaces and coincidence points. J. Fixed Point Theory Appl. 5, 105–127 (2009)
Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: Covering mappings and well-posedness of nonlinear Volterra equations. Nonlinear Anal. 75, 1026–1044 (2012)
Azé, D., Corvellec, J.-N.: Characterizations of error bounds for lower semicontinuous functions on metric spaces. ESAIM Control Optim. Calc. Var. 10, 409–425 (2004)
Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. Serie A, 122, 301–347 (2010)
Chuong, T.D., Kruger, A.Y., Yao, J.-C.: Calmness of efficient solution maps in parametric vector optimization. J. Glob. Optim. 51, 677–688 (2011)
De Giorgi, E., Marino, A., Tosques, M.: Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti. Accad. Naz. Lincei rend. Cl. Sci. Fis. Mat. Natur. 68, 180–187 (1980)
Dmitruk, A.V.: On a nonlocal metric regularity of nonlinear operators. Control Cybern. 34, 723–746 (2005)
Dmitruk, A.V., Milyutin, A.A., Osmolovskii, N.P.: Lyusternik’s theorem and the theory of extrema. Usp. Mat. Nauk 35, 11–46 (1980)
Dontchev, A.L., Frankowska, H.: Lyusternik–Graves theorem and fixed points. Proc. Am. Math. Soc. 139, 521–534 (2011)
Dontchev, A.L., Frankowska, H.: Lyusternik–Graves theorem and fixed points II. J. Convex Anal. (accepted)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009)
Durea, M., Nguyen, H.T., Strugariu, R.: Metric regularity of epigraphical multivalued mappings and applications to vector optimization. Math. Program. Serie B (accepted)
Durea, M., Strugariu, R.: On some Fermat rules for set-valued optimization problems. Optimization 60, 575–591 (2011)
Durea, M., Strugariu, R.: On parametric vector optimization via metric regularity of constraint systems. Math. Methods Oper. Res. 74, 409–425 (2011)
Durea, M., Strugariu, R.: Openness stability and implicit multifunction theorems: applications to variational systems. Nonlinear Anal. 75, 1246–1259 (2012)
Durea, M., Strugariu, R.: Chain rules for linear openness in general Banach spaces. SIAM J. Optim. (accepted)
Durea, M., Strugariu, R.: Chain rules for linear openness in metric spaces and applications. Applications to parametric variational systems (submitted)
Durea, M., Tammer, C.: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449–467 (2009)
Henrion, R., Outrata, J.V.: Calmness of constraint systems with applications. Math. Program. Serie B 104, 437–464 (2005)
Ioffe, A.D.: Towards variational analysis in metric spaces: metric regularity and fixed points. Math. Program. Serie B 123, 241–252 (2010)
Li, G., Mordukhovich, B.S.: Hölder metric subregularity with applications to proximal point method. (preprint, 2012)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. I: Basic Theory, vol. II: Applications, Springer, Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vols. 330 and 331. Berlin (2006)
Ngai, H.V., Nguyen, H.T., Théra, M.: Implicit multifunction theorems in complete metric spaces. Math. Program. Serie B (accepted)
Ngai, H.V., Nguyen, H.T., Théra, M.: Metric regularity of the sum of multifunctions and applications. Available at http://www.optimization-online.org/DB_HTML/2011/12/3291.html
Ngai, H.V., Théra, M.: Error bounds in metric spaces and application to the perturbation stability of metric regularity. SIAM J. Optim. 19, 1–20 (2008)
Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976)
Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)
Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)
Rockafellar, R.T., Wets, R.: Variational Analysis. Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vol. 317. Springer, Berlin (1998)
Ursescu, C.: Inherited openness. Rev. Roumaine Math. Pures Appl. 41, 5–6, 401–416 (1996)
Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)
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Apetrii, M., Durea, M. & Strugariu, R. On Subregularity Properties of Set-Valued Mappings. Set-Valued Var. Anal 21, 93–126 (2013). https://doi.org/10.1007/s11228-012-0213-4
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DOI: https://doi.org/10.1007/s11228-012-0213-4
Keywords
- Set-valued maps
- At-point regularity
- Around-point regularity
- Implicit multifunction theorems
- Solid vector optimization