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On the subdijferentiability of convex functions (1965)
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| Venue: | Bull. Amer. Math. Soc |
| Citations: | 84 - 2 self |
BibTeX
@ARTICLE{Rockafellar65onthe,
author = {R. T. Rockafellar},
title = {On the subdijferentiability of convex functions},
journal = {Bull. Amer. Math. Soc},
year = {1965},
pages = {605--611}
}
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Abstract
Each lower semi-continuous proper convex function f on a Banach space E defines a certain multivalued mapping of from E to E * called the subdifferential of f. It is shown here that the mappings arising this way are precisely the ones whose graphs are maximal "cyclically monotone " relations on Ex E*, and that each of these is also a maximal monotone relation. Furthermore, it is proved that of determines f uniquely up to an additive constant. These facts generally fail to hold when E is not a Banach space. The proofs depend on establishing a new result which relates the directional derivatives of f to the existence of approximate subgradients. Let E be a topological vector space over the real numbers R with dual E*. Let f be a proper convex function on E, i.e., an everywheredefined function with values in (- co, + co] ' not identically + co, such that (1.1) fC?,x + (1- tv)y) ~ tvf(x) + (1 + tv)f(y) for all x and y in E when 0 < tv < 1. A subgradient of f at x E E is an x * E E * such that f(y);;;;f(x) + <y- x, x*> for all y E E. (This says that f(x) is finite and that the graph of the affine function h(y) = f(x) + <y- x, x*> is a "nonvertical " supporting hyperplane at (x,f(x)) to the epigraph of J, which is the convex subset of EtfJR consisting of all the points lying above the graph of f.) For each x E E, we denote by ()f(x) the set of all subgradients of f at x, which is a weak " closed convex set in E*. If ()f(x)-=I=- 0, f is said to be s1tbdifferentiable at x. The subdifferential of f is the multivalued mapping (relation) ()f which assigns the set ()f(x) to each x. The notion of subdifferentiability has been developed recently in [3], [6/, [8], [11]. Much of the work has concerned the existence of subgradients. It is known, for example, that f is subdifferentiable wherever it is finite and continuous (see [6] or [8]). Results in [3] show among other things that, if E is a Banach space and f is lower semi-continuous (l.s.c.), then the set of points where f is subdiffer-
Keyphrases
banach space convex function proof depend directional derivative multivalued mapping maximal monotone relation certain multivalued mapping affine function convex set topological vector space semi-continuous proper convex function everywheredefined function approximate subgradients convex subset additive constant real number new result proper convex function
