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Canadian Journal of Mathematics Canadian Journal of Mathematics

Article contents

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.
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Abstract

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 ∂ π ƒ, 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

26B05 proximal subgradient lower Dini derivate cone monotonicity Lipschitz behavior constancy convexity
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 6 , 01 December 1993 , pp. 1167 - 1183
Copyright
Copyright © Canadian Mathematical Society 1993

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