Primal-Dual Symmetric Intrinsic Methods for Finding Antiderivatives of Cyclically Monotone Operators

A fundamental result due to Rockafellar states that every cyclically monotone operatorA admits an antiderivative f in the sense that the graph ofA is contained in the graph of the subdifferential operator ∂f . Given a method m that assigns every finite cyclically monotone operator A some antiderivative mA, we say that the method is primal-dual symmetric if m applied to the inverse of A produces the Fenchel conjugate of mA. Rockafellar’s antiderivatives do not possess this property. Utilizing Fitzpatrick functions and the proximal average, we present novel primal-dual symmetric intrinsic methods. The antiderivatives produced by these methods provide a solution to a problem posed by Rockafellar in 2005. The results leading to this solution are illustrated by various examples. 2000 Mathematics Subject Classification: Primary 47H05; Secondary 26B25, 52A41, 90C25.