Vol. 17, No. 3, 1966

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Characterization of the subdifferentials of convex functions

Ralph Tyrrell Rockafellar

Vol. 17 (1966), No. 3, 497–510

Each lower semi-continuous proper convex function f on a Banach space E defines a certain multivalued mapping ∂f 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 E × E, and that each of these is also a maximal monotone relation. Furthermore, it is proved that ∂f 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.

Mathematical Subject Classification
Primary: 46.45
Secondary: 47.80
Received: 7 January 1965
Published: 1 June 1966
Ralph Tyrrell Rockafellar