Primal-dual symmetric intrinsic methods for finding antiderivatives of cyclically monotone operators

A fundamental result due to Rockafellar states that every cyclically monotone operator A admits an antiderivative f in the sense that the graph of A is contained in the graph of the subdifferential operator partial derivative f. Given a method m that assigns every finite cyclically monotone operator A some antiderivative m(A), 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.