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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

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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

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