Interior Methods For a Class of Elliptic Variational Inequalities

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 lead to very large (possibly non-convex) optimization problems with upper and lower bound constraints. When interior methods are applied to the discretized problem, the resulting linear systems have the same zero/nonzero structure as the finite-element equations solved for the unconstrained case. This crucial property allows the interior method to exploit existing efficient, robust and scalable multilevel algorithms for the solution of partial differential equations (PDEs). We illustrate some of these ideas in the context of the elliptic PDE package PLTMG.