Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials

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 regular 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.

Primal-Dual Extragradient Methods for Nonlinear Nonsmooth PDE-Constrained Optimization

“…Hence, for the sake of consistency with [9], we denote the infimum over valid constants by R −1 ( w| q), or R −1 for short when there is no ambiguity about the point ( w, q). Metric regularity is then equivalent to R −1 ( w| q) > 0.…”

“…Metric regularity-whose verification is the main difficulty in function spaces and will be investigated based on the results of [9] at the end of this section-allows one to remove the squares from ( D 2 -loc-γ-F * ) and bridge from the perturbed local solutions q i to local solutions q i . This is done through a sequence of technical lemmas in [20, Theorem 2.6.…”

“…We wish to apply the results from [9]. Toward this end, we consider the MoreauYosida regularization (1.4) of F for some parameter γ > 0 and assume (using (1.5)) that the convexified graphical derivative of the regularized subdifferential satisfies at least at nondegenerate points for some cone V ∂F * (v|η) and a pointwise-defined selfadjoint positive semidefinite linear superposition operator…”

“…This verification is significantly more involved in infinite dimensions. For problems of the form (1.1) where F and G are given by integral functionals for regular integrands, we can apply the theory from [9] to obtain an explicit, verifiable condition for metric regularity to hold. While our analysis will show that for problems such as (1.2) this condition does in fact not hold in general unless a Moreau-Yosida regularization is introduced-or the data y δ and the fitting term are finite-dimensional-we do obtain convergence for an arbitrarily small regularization c 2017 SIAM.…”

“…where the indicator function is to be understood pointwise almost everywhere), which was not considered in [9]. The analysis there could be extended to cover this case; specifically, all nondegenerate cases would be covered by improving [9,Lem.…”