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

TL;DR

This paper extends primal-dual extragradient methods to nonlinear nonsmooth PDE-constrained optimization, demonstrating convergence under certain conditions and applying the methods to inverse problems and control problems with numerical validation.

Contribution

It introduces a primal-dual extragradient algorithm for nonlinear nonsmooth PDE-constrained optimization and proves local convergence, including an accelerated variant with strong convexity.

Findings

Convergence is established under metric regularity conditions.

Accelerated algorithm converges with strong convexity assumptions.

Numerical examples validate the applicability to inverse and control problems.

Abstract

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 $L^1$- and $L^\infty$-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.