Generalized Differentiation with Positively Homogeneous Maps: Applications in Set-Valued Analysis and Metric Regularity

Published Online:https://doi.org/10.1287/moor.1110.0497

References

  • Aubin J.-P. Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. (1984) 1:225–245LinkGoogle Scholar
  • Aubin J.-P.Viability Theory (1990) (Birkhäuser, Boston) Google Scholar
  • Aubin J.-P., Cellina A.Differential Inclusions: Set-Valued Maps and Viability Theory (1984) (Springer, New York) CrossrefGoogle Scholar
  • Aubin J.-P., Frankowska H.Set-Valued Analysis (1990) (Birkhäuser, Boston) Google Scholar
  • Beer G.Topologies on Closed and Convex Sets (1993) (Kluwer Academic Publishers, Dordrecht, The Netherlands) Google Scholar
  • 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) 134:441–459CrossrefGoogle Scholar
  • Burachik R. S., Iusem A. N.Set-Valued Mappings and Enlargements of Monotone Operators (2008) (Springer, New York) Google Scholar
  • Clarke F. H. Generalized gradients and applications. Trans. Amer. Math. Soc. (1975) 205:247–262CrossrefGoogle Scholar
  • Clarke F. H. On the inverse function theorem. Pacific J. Math. (1976) 64:97–102CrossrefGoogle Scholar
  • Clarke F. H.Optimization and Nonsmooth Analysis (1990) (SIAM, Philadelphia) CrossrefGoogle Scholar
  • Daniilidis A., Pang C. H. J. Continuity and differentiability of set-valued maps revisited in the light of tame geometry. J. London Math. Soc. (2011) . ForthcomingGoogle Scholar
  • Dmitruk A., Miliutin A. A., Osmolovskii N. Liusternik's theorem and the theory of extrema. Russian Math. Surveys (1981) 35:11–51CrossrefGoogle Scholar
  • Dontchev A. L., Rockafellar R. T. Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. (2004) 12:79–109CrossrefGoogle Scholar
  • Dontchev A. L., Rockafellar R. T.Implicit Functions and Solution Mappings: A View from Variational Analysis (2009) (Springer, New York) CrossrefGoogle Scholar
  • Frankowska H., Roxin E. Set-valued analysis and some control problems. Proc. Internat. Conf.: 30 Years of Modern Control Theory (1988) (Marcel Dekker, New York) 89–104 http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&S1=1013196Google Scholar
  • Henrion R. The approximate subdifferential and parametric optimization. (1997) . Habilitation Thesis, Humboldt University, BerlinGoogle Scholar
  • Ioffe A. D. Différentielles généralisées d'applications localement Lipschitziennes d'un espace de Banach dans un autre. C. R. Acad. Sci. Paris (1979) 289:637–640Google Scholar
  • Ioffe A. D. Nonsmooth analysis: Differential calculus of non-differentiable mappings. Trans. Amer. Math. Soc. (1981) 266:1–56CrossrefGoogle Scholar
  • Ioffe A. D. On the local surjection property. Nonlinear Anal. (1987) 11:565–592CrossrefGoogle Scholar
  • Ioffe A. D. Metric regularity and subdifferential calculus. Russian Math. Surveys (2000) 55(3):501–558CrossrefGoogle Scholar
  • Klatte D., Kummer B.Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications (2002) (Kluwer, Dordrecht, The Netherlands) Google Scholar
  • Klein E., Thompson A. C.Theory of Correspondences, Including Applications to Mathematical Economics (1984) (Wiley, New York) Google Scholar
  • Kruger A. Y. A covering theorem for set-valued mappings. Optimization (1988) 19:763–780CrossrefGoogle Scholar
  • Lebourg G. Valeur moyenne pour gradient généralisé. C. R. Acad. Sci. Paris (1975) 281:795–797Google Scholar
  • Li W. Sharp Lipschitz constants for basic optimal solutions and basic feasible solutions of linear programs. SIAM J. Control Optim. (1994) 32:140–153CrossrefGoogle Scholar
  • Mordukhovich B. S. Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mechanics (1976) 40:960–969 http://www.ams.org/mathscinet/search/journaldoc.html?j-JAPPM1CrossrefGoogle Scholar
  • Mordukhovich B. S. Metric approximations and necessary optimality conditions for general classes of extremal problems. Soviet Math. Doklady (1980) 22:526–530 http://www.ams.org/mathscinet/search/journaldoc.html?jc=SOVMDGoogle Scholar
  • Mordukhovich B. S.Approximation Methods in Problems of Optimization and Control (Russian) (1988) (Nauka, Moscow) Google Scholar
  • Mordukhovich B. S. Complete characterization of openness, metric regularity and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. (1993) 34:1–35CrossrefGoogle Scholar
  • Mordukhovich B. S. Generalized differential calculus for nonsmooth and set-valued maps. J. Math. Anal. Appl. (1994) 183:250–288CrossrefGoogle Scholar
  • Mordukhovich B. S.Variational Analysis and Generalized Differentiation I and II (2006) (Springer, Berlin) Google Scholar
  • Penot J.-P. Calcul sous-différentiel et optimization. J. Funct. Anal. (1978) 27:248–276CrossrefGoogle Scholar
  • Penot J.-P. Differentiability of relations and differential stability of perturbed optimization problems. SIAM J. Control Optim. (1984) 22:529–551CrossrefGoogle Scholar
  • Penot J.-P. Metric regularity, openness and Lipschitzean behavior of multifunctions. Nonlinear Anal. (1989) 13:629–643CrossrefGoogle Scholar
  • Robinson S. M. Some continuity properties of polyhedral multifunctions. Math. Programming Stud. (1981) 19:200–221CrossrefGoogle Scholar
  • Robinson S. M. Solution continuity in monotone affine variational inequalities. SIAM J. Optim. (2007) 18(3):1046–1060CrossrefGoogle Scholar
  • Rockafellar R. T. Lipschitzian properties of multifunctions. Nonlinear Anal. (1985) 9:867–885CrossrefGoogle Scholar
  • Rockafellar R. T., Attouch H., Aubin J.-P., Clarke F., Ekeland I. Proto-differentiability of set-valued mappings and its applications in optimization. Analyse Non Linéaire (1989) (Gauthier-Villars, Paris) 449–482Google Scholar
  • Rockafellar R. T., Wets R. J.-B.Variational Analysis (1998) (Springer, Berlin) CrossrefGoogle Scholar
  • Thibault L. On generalized differentials and subdifferentials of Lipschitz vector-valued functions. Nonlinear Anal. (1982) 6:1037–1053CrossrefGoogle Scholar
  • Warga J. Fat homeomorphisms and unbounded derivate containers. J. Math. Anal. Appl. (1981) 81:545–560CrossrefGoogle Scholar
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