Stability of saddle points via explicit coderivatives of pointwise subdifferentials

TL;DR

This paper develops stability criteria for saddle points in nonsmooth PDE-constrained optimization problems using explicit coderivative characterizations, enhancing understanding of solution stability under perturbations.

Contribution

It introduces an explicit pointwise coderivative characterization of subdifferentials for convex integral functionals in infinite-dimensional spaces.

Findings

Derived stability criteria for saddle points in nonsmooth PDE problems

Applied criteria to parameter identification in elliptic PDEs with non-differentiable data fitting

Provided explicit coderivative formulas for convex integral functionals

Abstract

We derive stability criteria for saddle points of a class of nonsmooth optimization problems in Hilbert spaces arising in PDE-constrained optimization, using metric regularity of infinite-dimensional set-valued mappings. A main ingredient is an explicit pointwise characterization of the Fr\'echet coderivative of the subdifferential of convex integral functionals. This is applied to several stability properties for parameter identification problems for an elliptic partial differential equation with non-differentiable data fitting terms.