Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T05:59:10.268Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Construction of Hölder and Proximal Subderivatives

Published online by Cambridge University Press:  20 November 2018

J. M. Borwein ,
R. Girgensohn  and
Xianfu Wang
J. M. Borwein
Affiliation:
Department of Mathematics and Statistics Simon Fraser University Burnaby, BC V5A 1S6, email: [email protected]
R. Girgensohn
Affiliation:
GSF-Forschungszentrum Institut für Biomathematik und Biometrie Postfach 1129 85758 Neuherberg Germany, email: [email protected]
Xianfu Wang
Affiliation:
Department of Mathematics and Statistics Simon Fraser University Burnaby, BC V5A 1S6, email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct Lipschitz functions such that for all $s>0$ they are $s$-Hölder, and so proximally, subdifferentiable only on dyadic rationals and nowhere else. As applications we construct Lipschitz functions with prescribed Hölder and approximate subderivatives.

Keywords

49J5226A1626A24Lipschitz functionsHölder subdifferentialproximal subdifferentialapproximate subdifferentialsymmetric subdifferentialHölder smoothdyadic rationals
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 41 , Issue 4 , 01 December 1998 , pp. 497 - 507
Copyright
Copyright © Canadian Mathematical Society 1998

References

1. Benoist, J., Intégration du sous-différentiel proximal: un contre exemple. Canad. J. Math. (2) 50 (1998), 242–265.Google Scholar
2. Borwein, J. M. and Preiss, D., A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Amer.Math. Soc. 303 (1987), 517–527.Google Scholar
3. Borwein, J. M. and Fitzpatrick, S., Characterization of Clarke subgradients among one-dimensional multifunctions. In: Proc. of the Optimization Miniconference II (Eds. B. M. Glover and V. Jeyakumar), 1995, 61–73.Google Scholar
4. Borwein, J. M. and Wang, X., Lipschitz functions with prescribed Hölder subderivatives. In preparation.Google Scholar
5. Clarke, F. H., Ledyaev, Yu. S. and Wolenski, P. R., Proximal analysis and minimization principles. J. Math. Anal. Appl. 196 (1995), 722–735.Google Scholar
6. Clarke, F. H., Sternand, R. J., Wolenski, P. R., Subgradient criteria for monotonicity, the Lipschitz condition, and convexity. Canad. J. Math. (6) 45 (1993), 1167–1183.Google Scholar
7. Fleming, W., Functions of several variables. Springer-Verlag, 1977.Google Scholar
8. Ioffe, A. D., Approximate subdifferentials and applications I: The finite dimensional theory. Trans. Amer. Math. Soc. 281 (1984), 390–416.Google Scholar
9. Mordukhovich, B., Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40 (1976), 960–969.Google Scholar
10. Mordukhovich, B., Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems. Soviet Math. Dokl. 22 (1980), 526–530.Google Scholar
11. Rockafellar, R. T. and Wets, R. J-B., Variational analysis. Springer-Verlag, Berlin, 1998.Google Scholar