Fast Interior Point Solution of Quadratic Programming Problems Arising from PDE-Constrained Optimization

Technical Report ERGO-2015-013

John W. Pearson and J. Gondzio

Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations.

Key words: Interior point methods, PDE-constrained optimization, Krylov subspace method, Preconditioning, Schur complement.

PDF ERGO-15-013.pdf.

Written: December 11, 2015, revised February 18, 2017.
Accepted for publication in Numerische Mathematik.