A Dantzig-Wolfe Decomposition Method for Quasi-Variational Inequalities

TL;DR

This paper introduces a Dantzig-Wolfe decomposition-based algorithm for solving large-scale quasi-variational inequalities, with proven convergence and applications to economic equilibrium problems, demonstrating effective performance in numerical tests.

Contribution

The paper develops a novel decomposition algorithm for quasi-variational inequalities, enabling efficient solutions for large-scale problems with convergence guarantees.

Findings

Algorithm demonstrates good performance on large-scale problems.

Proven global convergence under certain conditions.

Effective application to economic equilibrium modeling.

Abstract

We propose an algorithm to solve quasi-variational inequality problems, based on the Dantzig-Wolfe decomposition paradigm. Our approach solves in the subproblems variational inequalities, which is a simpler problem, while restricting quasi-variational inequalities in the master subproblems, making them generally (much) smaller in size when the original problem is large-scale. We prove global convergence of our algorithm, assuming that the mapping of the quasi-variational inequality is either single-valued and continuous or it is set-valued maximally monotone. Quasi-variational inequalities serve as a framework for several equilibrium problems, and we apply our algorithm to an important example in the field of economics, namely the Walrasian equilibrium problem formulated as a generalized Nash equilibrium problem. Our numerical assessment demonstrates good performance and usefullness of the approach for the large-scale cases.