Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

TL;DR

This paper develops a theoretical framework for quasi-variational inequalities in Banach spaces, establishing solution existence, and introduces an augmented Lagrangian algorithm with convergence guarantees, supported by applications and numerical tests.

Contribution

It provides the first comprehensive analysis of QVIs in Banach spaces with pseudomonotonicity and Mosco-type continuity, along with a practical augmented Lagrangian solution method.

Findings

Established existence of solutions under new conditions.

Designed an augmented Lagrangian algorithm with proven convergence.

Demonstrated practical effectiveness through applications and numerical results.

Abstract

This paper deals with quasi-variational inequality problems (QVIs) in a generic Banach space setting. We provide a theoretical framework for the analysis of such problems which is based on two key properties: the pseudomonotonicity (in the sense of Brezis) of the variational operator and a Mosco-type continuity of the feasible set mapping. We show that these assumptions can be used to establish the existence of solutions and their computability via suitable approximation techniques. In addition, we provide a practical and easily verifiable sufficient condition for the Mosco-type continuity property in terms of suitable constraint qualifications. Based on the theoretical framework, we construct an algorithm of augmented Lagrangian type which reduces the QVI to a sequence of standard variational inequalities. A full convergence analysis is provided which includes the existence of solutions of the subproblems as well as the attainment of feasibility and optimality. Applications and numerical results are included to demonstrate the practical viability of the method.