Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds
Ronny Bergmann, Roland Herzog, Maurício Silva Louzeiro, Daniel Tenbrinck, José Vidal-Núñez
TL;DR
The paper develops a manifold-specific Fenchel duality framework and a pair of Riemannian primal–dual algorithms (exact and linearized) to solve non-smooth optimization problems on curved spaces. By defining an $m$-based Fenchel conjugate $F_m^*$ and establishing a manifold Fenchel–Moreau theory, it derives a saddle-point formulation and implements two Riemannian Chambolle–Pock variants, proving convergence for the linearized version on Hadamard manifolds. The methods are applied to ROF-type total variation models on manifolds, with extensive numerical experiments showing competitive performance against established manifold algorithms and even convergence on spaces with positive curvature. This work broadens non-smooth convex optimization techniques to manifold-valued data, enabling robust variational imaging and geometry-processing tasks in curved spaces.
Abstract
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 manifolds 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.