Approximation and exact penalization in simple bilevel variational problems

TL;DR

This paper investigates a bilevel variational problem with an inexact lower level, employing penalization and cutting plane techniques to develop algorithms and analyze their convergence.

Contribution

It introduces an inexact approach to bilevel variational problems, combining penalization and cutting plane methods, with detailed convergence analysis.

Findings

Algorithms effectively solve inexact bilevel problems.

Penalization techniques ensure regularity for solution methods.

Convergence analysis accounts for inexactness effects.

Abstract

A simple bilevel variational problem where the lower level is a variational inequality while the upper level is an optimization problem is studied. We consider an inexact version of the lower problem, which guarantees enough regularity to allow the exploitation of techniques of exact penalization. Moreover, cutting planes are used to approximate the Minty gap function of the lower level. Algorithms to solve the resulting inexact bilevel problem are devised relying on these techniques and approximations. Finally, their convergence is studied in details by analysing also the effect of the given inexactness.