Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds [article]

Ronny Bergmann and Roland Herzog and Maurício Silva Louzeiro and Daniel Tenbrinck and José Vidal-Núñez
2020 arXiv   pre-print
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, e.g.,~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
more » ... nifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas--Rachford algorithm on manifolds of nonpositive curvature. Furthermore, we show numerically that our novel algorithm even converges on manifolds of positive curvature.
arXiv:1908.02022v4 fatcat:hqf7c5dkzzea3mws4zyd6wiufq