| Details: |
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 un-regularized
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 assumptions. |