The size of the Dini subdifferential
HTML articles powered by AMS MathViewer
- by Joël Benoist PDF
- Proc. Amer. Math. Soc. 129 (2001), 525-530 Request permission
Abstract:
Given a lower semicontinuous function $f:\mathbb {R}^h \rightarrow \mathbb {R} \cup \{+\infty \}$, we prove that the points of $\mathbb {R}^h$, where the lower Dini subdifferential contains more than one element, lie in a countable union of sets which are isomorphic to graphs of some Lipschitzian functions defined on $\mathbb {R}^{h-1}$. Consequently, the set of all these points has a null Lebesgue measure.References
- Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
- J. Benoist, Intégration du sous-différentiel proximal: un contre exemple, Can. J. Math., 50 (2), 1988, p. 242-265.
- Bernard Cornet (ed.), General equilibrium theory and increasing returns, Elsevier Science B.V., Amsterdam, 1988. J. Math. Econom. 17 (1988), no. 2-3. MR 1001684
- F. H. Clarke and R. M. Redheffer, The proximal subgradient and constancy, Canad. Math. Bull. 36 (1993), no. 1, 30–32. MR 1205891, DOI 10.4153/CMB-1993-005-9
- F. H. Clarke, R. J. Stern, and P. R. Wolenski, Subgradient criteria for monotonicity, the Lipschitz condition, and convexity, Canad. J. Math. 45 (1993), no. 6, 1167–1183. MR 1247540, DOI 10.4153/CJM-1993-065-x
- R. T. Rockafellar, Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981), no. 3, 424–436. MR 629642, DOI 10.1287/moor.6.3.424
Additional Information
- Joël Benoist
- Affiliation: Maître de Conférences, LACO, CNRS-ESA 6090, Université de Limoges, 87 060 Limoges, France
- Email: benoist@unilim.fr
- Received by editor(s): October 7, 1998
- Received by editor(s) in revised form: May 3, 1999
- Published electronically: September 18, 2000
- Communicated by: David R. Larson
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 525-530
- MSC (2000): Primary 26A16, 26A24
- DOI: https://doi.org/10.1090/S0002-9939-00-05549-0
- MathSciNet review: 1707505