# Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds

@article{Bergmann2021FenchelDT, title={Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds}, author={Ronny Bergmann and Roland Herzog and M. S. Louzeiro and Daniel Tenbrinck and Jos{\'e} Vidal-N{\'u}{\~n}ez}, journal={Found. Comput. Math.}, year={2021}, volume={21}, pages={1465-1504} }

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…

## 7 Citations

Fenchel Duality and a Separation Theorem on Hadamard Manifolds

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In this paper, we introduce a definition of Fenchel conjugate and Fenchel biconjugate on Hadamard manifolds based on the tangent bundle. Our definition overcomes the inconvenience that the conjugate…

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We propose a method for solving non-smooth optimization problems on manifolds. In order to obtain superlinear convergence, we apply a Riemannian Semi-smooth Newton method to a non-smooth non-linear…

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In this paper we propose a Riemannian version of the difference of convex algorithm (DCA) to solve a minimization problem involving the difference of convex (DC) function. Equivalence between the…

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An inexact version of the Riemannian Semismooth Newton method is proposed and conditions for local linear and superlinear convergence that hold independent of the sign of the curvature are proved.

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