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arXiv · 1908.02022

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

Ronny Bergmann·Roland Herzog·Maurício Silva Louzeiro·Daniel Tenbrinck·José Vidal-Núñez
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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.

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Ronny Bergmann, Roland Herzog, Maurício Silva Louzeiro, Daniel Tenbrinck, José Vidal-Núñez. 2020-10-09. Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds. https://doi.org/10.1007/s10208-020-09486-5

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