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 may even converge on manifolds of positive curvature.
Highlights
Convex analysis plays an important role in optimization, and an elaborate theory on convex analysis and conjugate duality is available on locally convex vector spaces
We introduce a new notion of Fenchel duality for Riemannian manifolds, which allows us to derive a conjugate duality theory for convex optimization problems posed on such manifolds
After illustrating the correspondence to the Euclidean Chambolle–Pock algorithm, we compare the numerical performance of the Riemannian Chambolle–Pock algorithm (RCPA) to the cyclic proximal point algorithm (CPPA) and the parallel Douglas– Rachford algorithm (PDRA)
Summary
Convex analysis plays an important role in optimization, and an elaborate theory on convex analysis and conjugate duality is available on locally convex vector spaces. After illustrating the correspondence to the Euclidean (classical) Chambolle–Pock algorithm, we compare the numerical performance of the RCPA to the CPPA and the PDRA While the latter has only been shown to converge on Hadamard manifolds of constant curvature, it performs quite well on Hadamard manifolds in general. To this end, we extend some classical results from convex analysis and locally convex vector spaces to manifolds, like the Fenchel–Moreau Theorem ( known as the Biconjugation Theorem) and useful characterizations of the subdifferential in terms of the conjugate function. 4, we formulate the primal-dual hybrid gradient method ( referred to as the Riemannian Chambolle–Pock algorithm, RCPA) for general optimization problems on manifolds involving nonlinear operators.
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