Variational Inequalities and Normals to Convex Sets

Variational inequalities and even quasi-variational inequalities, as means of expressing constrained equilibrium, have utilized geometric properties of convex sets, but the theory of tangent cones and normal cones has yet to be fully exploited. Much progress has been made in that theory in recent years in understanding the variational geometry of nonconvex as well as convex sets and applying it to optimization problems. Parallel applications to equilibrium problems could be pursued now as well. This article explains how normal cone mappings and their calculus offer an attractive framework for many purposes, and how the variational geometry of the graphs of such mappings, as nonconvex sets of a special nature, furnishes powerful tools for use in ascertaining how an equilibrium is affected by perturbations. An application to aggregated equilibrium models, and in particular multi-commodity traffic equilibrium, is presented as an example.