This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical
Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its
properties, eg, the Fenchel–Young inequality and the characterization of the convex
subdifferential using the analogue of the Fenchel–Moreau Theorem. These properties of the
Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm
and to prove its convergence for the case of Hadamard manifolds under appropriate …