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Subgradient representation of multifunctions

  • J. Borwein (a1), W. B. Moors (a2) and Y. Shao (a1)
Abstract

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.

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References
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[1]Apostol, T. M., Mathematical Analysis, first ed. (Addison-Wesley, Reading, MA, 1957).
[2]Apostol, T. M., Mathematical Analysis, second ed. (Addison-Wesley, Reading, MA, 1974).
[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), 6173.
[4]Borwein, J. M. and Moors, W. B., “Essentially smooth Lipschitz functions”, J. Functional Analysis 149 (1997) 305351.
[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), 512.
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[10]Janin, R., “Sur des multiapplications qui sont des gradients généralisés”, C. R. Acad. Sc. Paris 294 (1982) 115117.
[11]Minty, G., “Monotone (nonlinear) operators in Hilbert spaces”, Duke Math. J. 29 (1962) 341346.
[12]Phelps, R. R., Convex Functions, Monotone Operators and Differentiability (Springer-Verlag, New York, 1993).
[13]Poliquin, R. A., “A characterization of proximal subgradient set-valued mappings”, Can. Math. Bull. 36 (1993) 116122.
[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) 125144.
[15]Thibault, L., “On generalized differentials and subdifferentials of Lipschitz vector-valued functions”, Nonlinear Anal. TMA 6 (1982) 10371053.
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The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
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