Geodesic
Subgradient Criteria for Monotonicity, The Lipschitz Condition, and Convexity
Canadian journal of mathematics, Tome 45 (1993) no. 6, pp. 1167-1183

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-1993-065-x
Mots-clés : 26B05, proximal subgradient, lower Dini derivate, cone monotonicity, Lipschitz behavior, constancy, convexity 
Clarke, F. H.; Stern, R. J.; Wolenski, P. R. Subgradient Criteria for Monotonicity, The Lipschitz Condition, and Convexity. Canadian journal of mathematics, Tome 45 (1993) no. 6, pp. 1167-1183. doi: 10.4153/CJM-1993-065-x
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