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 $\partial f$. Given a method $\mathfrak{m}$ that assigns every finite cyclically monotone operator $A$ some antiderivative $\mathfrak{m}_A$, we say that the method is primal-dual symmetric if $\mathfrak{m}$ applied to the inverse of $A$ produces the Fenchel conjugate of $\mathfrak{m}_A$. 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.