Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T22:17:53.304Z Has data issue: false hasContentIssue false
  • English
  • Français

Subgradient Criteria for Monotonicity, The Lipschitz Condition, and Convexity

Published online by Cambridge University Press:  20 November 2018

F. H. Clarke ,
R. J. Stern  and
P. R. Wolenski
F. H. Clarke
Affiliation:
Centre de recherches mathématiques Université de Montreal, Montreal, Québec H3C3J7
R. J. Stern
Affiliation:
Department of Mathematics and Statistics Concordia University Montreal, Quebec H4B 1R6
P. R. Wolenski
Affiliation:
Department of Mathematics Louisiana State University Baton Rouge, Louisiana, 70803 U.S.A.
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.

Let ƒ:H → (—∞,∞] be lower semicontinuous, where H is a real Hilbert space. An approach based upon nonsmooth analysis and optimization is used in order to characterize monotonicity of ƒ with respect to a cone, as well as Lipschitz behavior and constancy. The results, which involve hypotheses on the proximal subgradient ∂ πƒ, specialize on the real line to yield classical characterizations of these properties in terms of the Dini derivate. They also give new extensions of these results to the multidimensional case. A new proof of a known characterization of convexity in terms of proximal subgradient monotonicity is also given.

Keywords

26B05proximal subgradientlower Dini derivatecone monotonicityLipschitz behaviorconstancyconvexity
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 6 , 01 December 1993 , pp. 1167 - 1183
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Boas, R.P., A Primer of Real Functions, Cams Mathematical Monographs 13, Mathematical Association of America, Rahway, 1960.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. Bruckner, A.M., Differentiation of Real Functions, Springer-Verlag 659, Berlin, 1978.Google Scholar
4. Clarke, F.H., An indirect method in the calculus of variations, Trans. Amer. Math. Soc, in press.Google Scholar
5. Clarke, F.H., Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics 57, S.I.A.M., Philadelphia, 1989.Google Scholar
6. Clarke, F.H., Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983. Republished as Classics in Applied Mathematics 5, S.I.A.M., Philadelphia, 1990.Google Scholar
7. Clarke, F.H. and Redheffer, R.M., The proximal subgradient and constancy, Canad. Math. Bull. 36(1993. 30–32.Google Scholar
8. Hobson, E.W., Functions of a Real Variable and the Theory of Fourier's Series, Cambridge University Press, Cambridge, 1907.Google Scholar
9. Loewen, P., Optimal Control via Nonsmooth Analysis, CRM Lecture Notes Series, Amer. Math. Soc, Summer School on Control, CRM, Université de Montréal, (1992), Amer. Math. Soc, Providence, 1993.Google Scholar
10. McShane, E.J., Integration, Princeton University Press, Princeton, 1944.Google Scholar
11. Poliqu'm, R.A., Subgradient monotonicity and convex functions, Nonlinear Analysis 15(1990), 305–317.Google Scholar
12. Preiss, D., Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal 91 (1990), 312–345.Google Scholar
13. Riesz, F. and Nagy, B.Sz., Functional Analysis, Ungar, New York, 1955.Google Scholar
14. Rockafellar, R.T., Clarke's tangent cones and boundaries of closed sets in Rn, Nonlinear Analysis 3(1979), 145–154.Google Scholar
15. Saks, S., Theory of the Integral, Hafner, New York, 1937. L.C. Young, trans. Reprinted by Dover Press, 1964.Google Scholar
16. Treiman, J.S., Generalized gradients, Lipschitz behavior and directional derivatives, Can. J. Math. 37(1985), 1074.1084.Google Scholar
17. Zagrodny, D., Approximate mean value theorem for upper derivatives, Nonlinear Analysis 12(1988), 1413.1428.Google Scholar