Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-11T00:21:49.128Z Has data issue: false hasContentIssue false
  • English
  • Français

Subgradient representation of multifunctions

Published online by Cambridge University Press:  17 February 2009

J. Borwein ,
W. B. Moors  and
Y. Shao
J. Borwein
Affiliation:
CECM, Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
W. B. Moors
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
Y. Shao
Affiliation:
CECM, Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
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 provide necessary and sufficient conditions for a minimal upper semicontinuous multifunction defined on a separable Banach space to be the subdifferential mapping of a Lipschitz function.

Type
Research Article
Information
The ANZIAM Journal , Volume 40 , Issue 3 , January 1999 , pp. 301 - 313
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Apostol, T. M., Mathematical Analysis, first ed. (Addison-Wesley, Reading, MA, 1957).Google Scholar
[2]Apostol, T. M., Mathematical Analysis, second ed. (Addison-Wesley, Reading, MA, 1974).Google Scholar
[3]Borwein, J. M. and Fitzpatrick, S., “Characterization of Clarke subgradients among one-dimensional multifunctions”, in Proceedings of the Optimization Miniconference II, (Univ. Ballarat Press, 1995), 61–73.Google Scholar
[4]Borwein, J. M. and Moors, W. B., “Essentially smooth Lipschitz functions”, J. Functional Analysis 149 (1997) 305–351.CrossRefGoogle Scholar
[5]Borwein, J. M. and Moors, W. B., “Lipschitz functions with minimal Clarke subdifferential mappings”, in Proceedings of the Optimization Miniconference III, (Univ. Ballarat Press, 1997), 5–12.Google Scholar
[6]Borwein, J. M. and Moors, W. B., “Null sets and essentially smooth Lipschitz functions”, SIAM J. Optimization 8 (1998), 309–323.CrossRefGoogle Scholar
[7]Christensen, J. P. R., Topological and Borel Structure (American Elsevier, New York, 1974).Google Scholar
[8]Clarke, F. H., Optimization and Nonsmooth Analysis (Wiley Inter-science, New York, 1983).Google Scholar
[9]Correa, R., Jofré, A. and Thibault, L., “Subdifferential monotonicity as characterization of convex functions”, Numer. Funct. Anal. Optimiz. 15 (1994) 531–535.CrossRefGoogle Scholar
[10]Janin, R., “Sur des multiapplications qui sont des gradients généralisés”, C. R. Acad. Sc. Paris 294 (1982) 115–117.Google Scholar
[11]Minty, G., “Monotone (nonlinear) operators in Hilbert spaces”, Duke Math. J. 29 (1962) 341–346.CrossRefGoogle Scholar
[12]Phelps, R. R., Convex Functions, Monotone Operators and Differentiability (Springer-Verlag, New York, 1993).Google Scholar
[13]Poliquin, R. A., “A characterization of proximal subgradient set-valued mappings”, Can. Math. Bull. 36 (1993) 116–122.CrossRefGoogle Scholar
[14]Rockafellar, R. T., “Favorable classes of Lipschitz continuous functions in subgradient optimization”, in Progress in Nondifferentiable Optimization, (Institute of Applied Systems Analysis, Laxenburg, Austria, 1982) 125–144.Google Scholar
[15]Thibault, L., “On generalized differentials and subdifferentials of Lipschitz vector-valued functions”, Nonlinear Anal. TMA 6 (1982) 1037–1053.CrossRefGoogle Scholar