# A Generalization of the Chambolle-Pock Algorithm to Banach Spaces with Applications to Inverse Problems

arXiv: https://arxiv.org/abs/1412.0126 · HTML: https://mlsanity.com/paper/1412.0126/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Nov 2014.

## Built on by (most cited first, 2 of 2)

- [Accelerated nonlinear primal-dual hybrid gradient methods with applications to supervised machine learning](https://mlsanity.com/paper/2109.12222/) (2021)
- [Efficient and robust high-dimensional sparse logistic regression via nonlinear primal-dual hybrid gradient algorithms](https://mlsanity.com/paper/2111.15426/) (2021)

## Abstract (arXiv)

For a Hilbert space setting Chambolle and Pock introduced an attractive first-order algorithm which solves a convex optimization problem and its Fenchel dual simultaneously. We present a generalization of this algorithm to Banach spaces. Moreover, under certain conditions we prove strong convergence as well as convergence rates. Due to the generalization the method becomes efficiently applicable for a wider class of problems. This fact makes it particularly interesting for solving ill-posed inverse problems on Banach spaces by Tikhonov regularization or the iteratively regularized Newton-type method, respectively.