Abstract

Locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon-Nikodym property, a generalized core-Jacobian, Δ f(p) is introduced, and its fundamental properties are established. Primarily, it is shown that the β-closure of its convex hull is exactly the generalized Jacobian. Furthermore, the nonemptiness, the β-compactness, the β-upper semicontinuity, and even another representation are obtained. Connections with known notions are derived and chain rules are proved using key results developed. Therefore, the generalized core-Jacobian introduced in this paper is proved to enjoy all the properties that allow this set to be the nucleus of the generalized Jacobian.

Author information

Contact details are reproduced from the original publication and may be historical.

Zsolt Páles

Institute of Mathematics, University of Debrecen, 4010 Debrecen, Pf. 12, Hungary

pales@math.klte.hu

Vera Zeidan

Dept. of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A.

zeidan@math.msu.edu

Keywords

  • Generalized Jacobian

Mathematics Subject Classification

49J52, 49A52, 58C20

References

  1. Y. Benyamini, J. Lindenstrauss: Geometric Nonlinear Functional Analysis. Vol. 1, Colloquium Publications 48, American Mathematical Society, Providence (2000).https://doi.org/10.1090/coll/048Google Scholar
  2. F. H. Clarke: On the inverse function theorem, Pacific J. Math. 64(1) (1976) 97–102.https://doi.org/10.2140/pjm.1976.64.97Google Scholar
  3. F. H. Clarke: Optimization and Nonsmooth Analysis, John Wiley & Sons, New York (1983).Google Scholar
  4. M. Fabián, D. Preiss: On the Clarke’s generalized Jacobian, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14 (1987) 305–307.Google Scholar
  5. R. B. Holmes: Geometric Functional Analysis and its Applications, Graduate Texts in Mathematics 24, Springer, Berlin (1975).https://doi.org/10.1007/978-1-4684-9369-6Google Scholar
  6. A. D. Ioffe: Nonsmooth analysis: differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc. 266(1) (1981) 1–56.https://doi.org/10.1090/S0002-9947-1981-0613784-7Google Scholar
  7. A. D. Ioffe: Directional compactness, scalarization and nonsmooth semi-Fredholm mappings, Nonlinear Anal., Theory Methods Appl. 29(2) (1997) 201–219.https://doi.org/10.1016/S0362-546X(96)00046-6Google Scholar
  8. H. Th. Jongen, D. Pallaschke: On linearization and continuous selections of functions, Optimization 19(3) (1988) 343–353.https://doi.org/10.1080/02331938808843350Google Scholar
  9. D. Klatte, B. Kummer: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications, Nonconvex Optimization and its Applications 60, Kluwer, Dordrecht (2002).Google Scholar
  10. L. Kuntz, S. Scholtes: Structural analysis of nonsmooth mappings, inverse functions, and metric projections, J. Math. Anal. Appl. 188(2) (1994) 346–386.https://doi.org/10.1006/jmaa.1994.1431Google Scholar
  11. L. Kuntz, S. Scholtes: Qualitative aspects of the local approximation of a piecewise differentiable function, Nonlinear Anal., Theory Methods Appl. 25(2) (1995) 197–215.https://doi.org/10.1016/0362-546X(94)00202-SGoogle Scholar
  12. B. S. Mordukhovich: Metric approximations and necessary conditions for optimality for general classes of nonsmooth extremal problems, Dokl. Akad. Nauk SSSR 254(5) (1980) 1072–1076.Google Scholar
  13. B. S. Mordukhovich: Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl. 183(1) (1994) 250–288.https://doi.org/10.1006/jmaa.1994.1144Google Scholar
  14. B. S. Mordukhovich: Coderivatives of set-valued mappings: calculus and applications, Nonlinear Anal., Theory Methods Appl. 30(5) (1997) 3059–3070.https://doi.org/10.1016/S0362-546X(96)00377-XGoogle Scholar
  15. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. I: Basic Theory, Grundlehren der Mathematischen Wissenschaften 330, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31247-1Google Scholar
  16. Zs. Páles, V. Zeidan: Infinite dimensional generalized Jacobian: properties and calculus rules, J. Math. Anal. Appl. 344(1) (2008) 55–75.https://doi.org/10.1016/j.jmaa.2008.02.044Google Scholar
  17. Zs. Páles, V. Zeidan: Generalized Jacobian for functions with infinite dimensional range and domain, Set-Valued Anal. 15(4) (2007) 331–375.https://doi.org/10.1007/s11228-007-0043-yGoogle Scholar
  18. Zs. Páles, V. Zeidan: Infinite dimensional Clarke generalized Jacobian, J. Convex Analysis 14(2) (2007) 433–454.https://doi.org/10.68381/jca14028Google Scholar
  19. J.-S. Pang, D. Ralph: Piecewise smoothness, local invertibility, and parametric analysis of normal maps, Math. Oper. Res. 21(2) (1996) 401–426.https://doi.org/10.1287/moor.21.2.401Google Scholar
  20. B. H. Pourciau: Analysis and optimization of Lipschitz continuous mappings, J. Optimization Theory Appl. 22(3) (1977) 311–351.https://doi.org/10.1007/BF00932859Google Scholar
  21. D. Ralph, S. Scholtes: Sensitivity analysis of composite piecewise smooth equations, Math. Program., Ser. B 76(3) (1997) 593–612.https://doi.org/10.1007/BF02614400Google Scholar
  22. D. Ralph, H. Xu: Implicit smoothing and its application to optimization with piecewise smooth equality constraints, J. Optim. Theory Appl. 124(3) (2005) 673–699.https://doi.org/10.1007/s10957-004-1180-1Google Scholar
  23. R. T. Rockafellar: Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, Princeton (1970).Google Scholar
  24. R. T. Rockafellar: A property of piecewise smooth functions, Comput. Optim. Appl. 25(1-3) (2003) 247–250.https://doi.org/10.1023/A:1022921624832Google Scholar
  25. S. Scholtes: Introduction to Piecewise Differentiable Equations, Habilitation Thesis, University of Karlsruhe, Karlsruhe (1994).Google Scholar
  26. L. Thibault: Subdifferentials of compactly Lipschitzian vector-valued functions, Ann. Mat. Pura Appl., IV. Ser. 125 (1980) 157–192.https://doi.org/10.1007/BF01789411Google Scholar
  27. L. Thibault: On generalized differentials and subdifferentials of Lipschitz vector-valued functions, Nonlinear Anal., Theory Methods Appl. 6(10) (1982) 1037–1053.https://doi.org/10.1016/0362-546X(82)90074-8Google Scholar
  28. J. Warga: Fat homeomorphisms and unbounded derivate containers, J. Math. Anal. Appl. 81(2) (1981) 545–560; Errata: ibid. 82(2) (1982) 582–583.https://doi.org/10.1016/0022-247X(81)90081-0https://doi.org/10.1097/00000446-198282040-00025Google Scholar

Suggested citation

Z. Páles, V. Zeidan. “The Core of the Infinite Dimensional Generalized Jacobian.” Journal of Convex Analysis 16 (2009), No. 2, 321–349. https://doi.org/10.68381/jca16017

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press.