Characterization of the subdifferentials of convex functions
R. TYRRELL ROCKAFELLAR · Pacific Journal of Mathematics · 1966
Each lower semi-continuous proper convex function / on a Banach space E defines a certain multivalued mapping df from E to E* called the subdifferential of /.It is shown here that the mappings arising this way are precisely the ones whose graphs are maximal "cyclically monotone" relations on E X E*, and that each of these is also a maximal monotone relation.Furthermore, it is proved that df determines / 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 / to the existence of approximate subgradients.Let E be a topological vector space over the real numbers R with dual E*.Let / be a proper convex function on E, i.e., an everywheredefined function with values in (-oo, +©o], not identically +°o, such