Abstract

In this paper we compare two different approaches to analyse the second-order behaviour of a convex function. The first one is classical, we call it the horizontal approach; the second one is more recent, it is the vertical approach. We prove equivalences between horizontal and vertical growth conditions. Then we derive well-known directional results. Finally we show that the vertical approach is particularly interesting to get more than a first-order (and more than directional) analysis of the maximum eigenvalue function.

Author information

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

François Oustry

INRIA, Domaine de Voluceau-Rocquencourt, B.P. 105, 78153 Le Chesnay, France
and: ENSTA

Keywords

  • Convex analysis
  • second-order derivative
  • approximate subdifferential
  • semidefinite programming

References

  1. D. Azé: On the remainder of the first order development of convex functions. To appear.Google Scholar
  2. J.-B. Hiriart-Urruty: Limiting behaviour of the approximate first order and second order directional derivatives for a convex function, Nonlinear Analysis, Theory, Methods & Applications 6(12) (1982) 1309-1326.https://doi.org/10.1016/0362-546X(82)90106-7Google Scholar
  3. J.-B. Hiriart-Urruty, Y. S. Ledyaev: A note on the characterization of the global maxima of a (tangentially) convex function over a convex set, Journal of Convex Analysis 3(1) (1996) 55-61.https://doi.org/10.68381/jca03004Google Scholar
  4. J.-B. Hiriart-Urruty, A. Seeger: The second-order subdifferential and the Dupin indicatrices of a non-differentiable convex function, Proc. London. Math. Soc. 3(58) (1989) 351-365.https://doi.org/10.1112/plms/s3-58.2.351Google Scholar
  5. J.-B. Hiriart-Urruty, D. Ye: Sensivity analysis of all eigenvalues of a symmetric matrix, Numerische Mathematik 70 (1995) 45-72.https://doi.org/10.1007/s002110050109Google Scholar
  6. J.-B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimization Algorithms, Springer-Verlag, 1993. Two volumes.https://doi.org/10.1007/978-3-662-02796-7Google Scholar
  7. T. Kato: Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1980.Google Scholar
  8. C. Lemaréchal, J. Zowe: Some remarks on the construction of higher order algorithms in convex optimization, Applied Mathematics and Optimization 10 (1983) 51-68.https://doi.org/10.1007/BF01448379Google Scholar
  9. D. Noll: Second order epi derivatives and the Dupin indicatrix for nonsmooth functions, Nonlinear Analysis, Theory, Methods & Applications 24(4) (1995) 563-573.https://doi.org/10.1016/0362-546X(94)E0036-GGoogle Scholar
  10. F. Oustry: A global and superlinear algorithm to minimize the maximum eigenvalue function, 1997. In preparation.Google Scholar
  11. R. T Rockafellar: Convex Analysis, Princeton University Press, 1970.https://doi.org/10.1515/9781400873173Google Scholar
  12. R. T. Rockafellar: Generalized second derivatives of convex functions and saddle functions, Transactions of the American Mathematical Society 322(1) (1990) 51-78.https://doi.org/10.2307/2001522Google Scholar
  13. A. Seeger: Limiting behaviour of the approximate second-order subdifferential of a convex function, Journal of Optimization Theory and Applications 74(3) (1992) 527-544.https://doi.org/10.1007/BF00940325Google Scholar
  14. A. Shapiro: Perturbation theory of nonlinear programs when the set of optimal solutions is not a singleton, Applied Mathematics and Optimization 18 (1988) 215-229.https://doi.org/10.1007/BF01443623Google Scholar
  15. A. Shapiro: First and second order analysis of nonlinear semidefinite programs, Mathematical Programming (Series B), 1996. To appear.https://doi.org/10.1007/BF02614439Google Scholar
  16. M. Torki: Second order directional derivatives of all eigenvalues of a symmetric matrix. In preparation.Google Scholar
  17. M. Volle: Subdifferential of the upper envelope of convex functions, C. R. Acad. Sci. Paris 317 (1993) 845-849.Google Scholar

Suggested citation

F. Oustry. “Vertical Developments of a Convex Function.” Journal of Convex Analysis 5 (1998), No. 1, 153–170. https://doi.org/10.68381/jca05-12

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Published by Heldermann Verlag, 1998. Rights now held by Banach Press LLC.