Lagrange Multipliers, (Exact) Regularization and Error Bounds for Monotone Variational Inequalities

TL;DR

This paper investigates regularization methods for monotone variational inequalities, establishing the role of Lagrange multipliers and error bounds, and proposing iterative schemes with convergence criteria based on solution set properties.

Contribution

It introduces the concept of exact regularization linked to Lagrange multipliers and provides error bounds and stopping criteria for iterative schemes solving variational inequalities.

Findings

Exact regularization is characterized by the existence of Lagrange multipliers.

Error bounds and stopping criteria are derived for regularized schemes.

The analysis applies to solution sets with weak sharpness of order greater than one.

Abstract

We examine two central regularization strategies for monotone variational inequalities, the first a direct regularization of the operative monotone mapping, and the second via regularization of the associated dual gap function. A key link in the relationship between the solution sets to these various regularized problems is the idea of exact regularization, which, in turn, is fundamentally associated with the existence of Lagrange multipliers for the regularized variational inequality. A regularization is said to be exact if a solution to the regularized problem is a solution to the unregularized problem for all parameters beyond a certain value. The Lagrange multipliers corresponding to a particular regularization of a variational inequality, on the other hand, are defined via the dual gap function. Our analysis suggests various conceptual, iteratively regularized numerical schemes, for which we provide error bounds, and hence stopping criteria, under the additional assumption that the solution set to the unregularized problem is what we call weakly sharp of order greater than one.