We consider the application of primal-dual interior methods to the optimization of systems arising in the finite-element discretization of a class of elliptic variational inequalities. These problems give rise to very large (possibly non-convex) optimization problems with upper and lower bound constraints.
When interior methods are applied to the discretized form of the problem, the zero/nonzero structure of the associated linear systems can be arranged to be identical to that of a discretization of the PDE constraints. This crucial property allows the interior method to exploit existing efficient, robust and scalable multilevel algorithms for the solution of PDEs.
We illustrate some of these ideas in the context of the elliptic PDE package PLTMG.