# Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization

arXiv: https://arxiv.org/abs/1606.06219 · HTML: https://mlsanity.com/paper/1606.06219/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2016.

## Built on by (most cited first, 1 of 1)

- [Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems](https://mlsanity.com/paper/2307.00296/) (2023)

## Abstract (arXiv)

We study the extension of the Chambolle--Pock primal-dual algorithm to nonsmooth optimization problems involving nonlinear operators between function spaces. Local convergence is shown under technical conditions including metric regularity of the corresponding primal-dual optimality conditions. We also show convergence for a Nesterov-type accelerated variant provided one part of the functional is strongly convex. We show the applicability of the accelerated algorithm to examples of inverse problems with L1- and L^∞-fitting terms as well as of state-constrained optimal control problems, where convergence can be guaranteed after introducing an (arbitrary small, still nonsmooth) Moreau--Yosida regularization. This is verified in numerical examples.