The least slope of a convex function and the maximal monotonicity of its subdifferential |
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Authors: | S. Simons |
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Affiliation: | (1) Department of Mathematics, University of California, 93106 Santa Barbara, California |
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Abstract: | In this paper, we give a direct proof of Rockafellar's result that the subdifferential of a proper convex lower-semicontinuous function on a Banach space is maximal monotone. Our proof is simpler than those that have appeared to date. In fact, we show that Rockafellar's result can be embedded in a more general situation in which we can quantify the degree of failure of monotonicity in terms of a quotient like the one that appears in the definition of Fréchet differentiability. Our analysis depends on the concepts of the least slope of a convex function, which is related to the steepest descent of optimization theory.The author would like to express his thanks to R. R. Phelps for reading a preliminary version of this paper and making some very valuable suggestions. |
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Keywords: | Subdifferential of a convex function maximal monotone operator separation theorem sandwich theorem |
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