Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T18:22:05.178Z Has data issue: false hasContentIssue false
  • English
  • Français

A Characterization of Proximal Subgradient Set-Valued Mappings

Published online by Cambridge University Press:  20 November 2018

R. A. Poliquin
R. A. Poliquin*
Affiliation:
Department of Mathematics University of Alberta Edmonton, Alberta T6G 2G1
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.

In this paper we tackle the problem of identifying set-valued mappings that are subgradient set-valued mappings. We show that a set-valued mapping is the proximal subgradient mapping of a lower semicontinuous function bounded below by a quadratic if and only if it satisfies a monotone selection property.

Keywords

49A5258C0658C2065K10proximal subgradientsnonsmooth analysisquadratic conjugateset-valued mappingsmonotone mappinggeneralized subgradientsstrictly submonotonecyclically submonotone
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 1 , 01 March 1993 , pp. 116 - 122
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Clarke, F. H., Generalized gradients and applications, Trans. Amer. Math. Soc. 205(1975), 247–262.Google Scholar
2. Clarke, F. H., Nonsmooth Analysis and Optimization, John Wiley, New York, 1983.Google Scholar
3. Ioffe, A. D., Approximate subdifferentials and applications I: the finite dimensional case, Trans. Amer. Math. Soc. 281(1984), 389–416.Google Scholar
4. Ioffe, A. D., Calculus of Dini sub differentials of functions and contingent coderivatives of set-valued maps, Nonlinear Analysis 8( 1984), 517–539.Google Scholar
5. Janin, R., Sur des multiapplications qui sont des gradients généralisés, C.R. Acad. Sc. Paris 294(1982), 115–117.Google Scholar
6. Loewen, P. D., The proximal normal formula in Hilbert space, Nonlinear Analysis 11(1987), 979–995.Google Scholar
7. Poliquin, R. A., Integration of subdifferential of nonconvexfunctions, Nonlinear Analysis, 17(1991), 385– 398.Google Scholar
8. Poliquin, R. A., Subgradient monotonicity and convex functions, Nonlinear Analysis 14(1990), 305–317.Google Scholar
9. Rockafellar, R. T., Convex Analysis, Princeton University Press, 1970 .Google Scholar
10. Rockafellar, R. T., Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math, of Op. Res, 6( 1981 ), 427–437.Google Scholar
11. Rockafellar, R. T., Favorable classes of Lipschitz continuous functions in sub gradient optimization. In: Progress in Nondifferentiable Optimization, Institute of Applied Systems Analysis, Laxenburg, Austria, (1982), 125— 144.Google Scholar
12. Rockafellar, R. T., Extensions of sub gradient calculus with applications to optimization, Nonlinear Analysis 9( 1985), 665–698.Google Scholar
13. Rockafellar, R. T. and R. J.-Wets, B., Variational Analysis, to appear.Google Scholar
14. Spingarn, J. E., Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc. 264(1981), 77–89.Google Scholar