On variational inequalities over polyhedral sets

The results on regularity behavior of solutions to variational inequalities over polyhedral sets proved in a series of papers by Robinson, Ralph and Dontchev-Rockafellar in the 90s has long become classics of variational analysis. But the available proofs are very complicated and practically do not use techniques of variational analysis. The only exception is the proof by Dontchev and Rockafellar of their “critical face” regularity criterion. In the paper we offer a different approach completely based on polyhedral geometry and a few basic principles of metric regularity theory. It leads to new proofs, that look simpler and shorter, and in addition gives some clarifying geometrical information.