Nonlinear Fenchel Conjugates

Fenchel conjugates play an important role in nonsmooth optimization.
Especially in large scale, nonsmooth problems, like image processing, splitting methods
enable fast and efficient optimisation algorithms based on Fenchel duality.
Recently, a few different variants of Fenchel conjugates have been introduced on Manifolds.
The main challenge is, that the original Fenchel conjugate is defined as a supremum over
linear functions, which do not exist on manifolds.

In this talk we present a generalisation, the so-called nonlinear Fenchel conjugates.
We discuss, which properties can still be derived, for example how a biconjugate can be
introduced and how the relation between convolution and the Fenchel conjugate
are generalized to Lie groups.
As an application we present the Primal-Dual-Hybrid-Gradient or Chambolle-Pock algorithm
and illustrate its performance in an image denoising task for manifold-valued images.