Skip to main content
SpringerLink
Log in

Generalized hessian forC 1,1 functions in infinite dimensional normed spaces

  • Published:
Mathematical Programming Submit manuscript

Abstract

The subject of this paper is the systematic study of second order notions concerning differentiable functions with Lipschitz derivative. The results and notions are motivated by recent papers of Cominetti, Correa and Hiriart-Urruty. The first goal of this paper is the comparison of several known second order directional derivatives. The second goal is the introduction of a generalized Hessian which is a set of certain symmetric bilinear forms. The relation of this generalized Hessian to other existing second order derivatives is also described.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. Ben-Tal and J. Zowe, A unified theory of first and second order conditions for extremum problems in topological vector spaces,Mathematical Programming Study 19 (1982) 39–76.

    MathSciNet  Google Scholar 

  2. H. Buseman, Convex surfaces,Intersciences Tracts in Pure and Applied Mathematics 6 (Intersciences Publishers, New York, 1958).

    Google Scholar 

  3. W.L. Chan, L.R. Huang and K.F. Ng. On generalized second-order derivatives and Taylor expansions in nonsmooth optimization,SIAM Journal Control and Optimization 32 (1994) 591–611.

    Article  MATH  MathSciNet  Google Scholar 

  4. F. H. Clarke,Optimization and Nonsmooth Analysis (John Wiley and Sons, 1983).

  5. R. Cominetti and R. Correa, Sur une dérivée du second ordre en analyse non différentiable,Comptes Rendus de l'Académie des Sciences Série l 303 (1986) 761–764.

    MathSciNet  Google Scholar 

  6. R. Cominetti and R. Correa, A generalized second-order derivative in nonsmooth optimization.SIAM Journal Control and Optimization 28 (1990) 789–809.

    Article  MATH  MathSciNet  Google Scholar 

  7. J.-B. Hiriart-Urruty, Characterization of the plenary hull of the generalized Jacobian matrix,Mathematical Programming Study 17 (1982) 1–12.

    MATH  MathSciNet  Google Scholar 

  8. J.-B. Hiriart-Urruty, J-J. Strodiot and V. Hien Nguyen. Generalized Hessian Matrix and second order optimality conditions for problems withC 1,1 data,Applied Mathematics and Optimization 11 (1984) 43–56.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. D. Ioffe, Nonsmooth analysis: Differential calculus of nondifferentiable mappings,Transactions of the American Mathematical Society 266 (1981) 1–56.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. D. Ioffe, On some recent developments in the theory of second order optimality conditions, in: S. Dolecki ed.,Optimization, Proceedings Conf. Varetz, 1988. Lecture Notes in Mathematics 1405 (Springer-Verlag, Berlin, 1989) 55–68.

    Google Scholar 

  11. A. D. Ioffe, Variational analysis of a composite function: A formula for the lower second order epiderivative,Journal of Mathematical Analysis and its Applications 160 (1991) 379–405.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Lebourgh, Valeur moyenne pour gradient généralisé,Comptes Rendus de l'Académie des Sciences Série I 281 (1975) 795–797.

    Google Scholar 

  13. Zs. Páles, Separation with symmetric bilinear forms and symmetric selections of set-valued functions,Publicationes Mathematicae Debrecen 46 (1995) 321–331.

    MATH  MathSciNet  Google Scholar 

  14. Zs. Páles and V. Zeidan, Nonsmooth optimum problems with constraints,SIAM Journal Control and Optimization 32 (1994) 1476–1502.

    Article  MATH  Google Scholar 

  15. Zs. Páles and V. Zeidan, First and second order necessary conditions for control problems with constraints,Transactions of the American Mathematical Society 346 (1994) 421–453.

    Article  MATH  MathSciNet  Google Scholar 

  16. Zs. Páles and V. Zeidan, Separation via quadratic functions,Aequationes Mathematicae (to appear).

  17. Zs. Páles and V. Zeidan, Generalized Hessian forC 1,1 integral functionals (unpublished manuscript).

  18. R. T. Rockafellar, First and second order epi-differentiability in nonlinear programming,Transactions of the American Mathematical Society 307 (1988) 75–108

    Article  MATH  MathSciNet  Google Scholar 

  19. R. T. Rockafellar, Second order optimality conditions in nonlinear programming obtained by way of epi-derivatives,Mathematics of Operations Research 14 (1989) 462–484.

    Article  MATH  MathSciNet  Google Scholar 

  20. X. Q. Yang, Second order conditions inC 1,1 optimization with applications,Numerical Functional Analysis and Optimization 14 (1993) 621–632.

    MATH  MathSciNet  Google Scholar 

  21. X. Q. Yang and V. Jeyakumar, Generalized second order directional derivatives and optimization withC 1,1 functions,Optimization 26 (1992) 165–185.

    MATH  MathSciNet  Google Scholar 

  22. V. Zeidan, Sufficient conditions for optimal control and the generalized problem of Bolza, Thesis, University of British Columbia (1982).

  23. V. Zeidan, Sufficient conditions for the generalized problem of Bolza,Transactions of the American Mathematical Society 275 (1983) 561–583.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, L. Kossuth University, Pf. 12, H-4010, Debrecen, Hungary

    Zsolt Páles

  2. Department of Applied Mathematics, University of Waterloo, N2L 3GI, Waterloo, Ontario, Canada

    Vera Zeidan

Authors
  1. Zsolt Páles
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vera Zeidan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The research was supported by a grant from the National Science Foundation NSF-66-2270, which is gratefully acknowledged.

Research supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. T-016846 and by the Humboldt Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Páles, Z., Zeidan, V. Generalized hessian forC 1,1 functions in infinite dimensional normed spaces. Mathematical Programming 74, 59–78 (1996). https://doi.org/10.1007/BF02592147

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02592147

Keywords

Search

Navigation