  We construct an algebraic multigrid (AMG) based preconditioner for the
reduced Hessian of a linear-quadratic optimization problem constrained by an
elliptic partial differential equation. While the preconditioner generalizes a
geometric multigrid preconditioner introduced in earlier works, its
construction relies entirely on a standard AMG infrastructure built for solving
the forward elliptic equation, thus allowing for it to be implemented using a
variety of AMG methods and standard packages. Our analysis establishes a clear
connection between the quality of the preconditioner and the AMG method used.
The proposed strategy has a broad and robust applicability to problems with
unstructured grids, complex geometry, and varying coefficients. The method is
implemented using the Hypre package and several numerical examples are
presented.
