An Optimization Approach to Parameter Identification in Variational Inequalities of Second Kind -- II

TL;DR

This paper advances the inverse problem of parameter identification in second-kind variational inequalities by introducing regularization techniques for nonsmooth parts, enabling smoother solution maps and providing convergence and optimality analysis.

Contribution

It develops a regularization-based optimization framework for parameter identification in nonsmooth variational inequalities, extending previous work to include nonlinear non-smooth functions.

Findings

Regularized variational inequality leads to a smooth parameter-to-solution map.

Convergence analysis of the regularized approach is established.

Optimality conditions for the regularized problem are derived.

Abstract

This paper continues earlier work and is concerned with the inverse problem of parameter identification in variational inequalities of the second kind that does not only treat the parameter linked to a bilinear form, but importantly also the parameter linked to a nonlinear non-smooth function. The optimization approach in the earlier work on the inverse problem using the output-least squares formulation involves the variational inequality of the second kind as constraint. Here we use regularization technics of nondifferentiable optimization, regularize the nonsmooth part in the variational inequality and arrive at an optimization problem for which the constraint variational inequality is replaced by the regularized variational equation. For this case, the smoothness of the parameter-to-solution map is studied and convergence analysis and optimality conditions are given.