A Collection of Test Problems in
PDE-Constrained Optimization

Hans D. Mittelmann
Department of Math and Stats
Arizona State University

At the workshop on

Optimization in Simulation-Based Models

at the

Institute for Mathematics and its Applications

it was suggested to provide a collection of such models in a form that researchers can not only learn about such problems but even test some of them. This website addresses this request. More information may be added later.

While PDE-constrained optimization problems arise in various contexts, for example, in parameter identification and shape optimization, an important class is that of control problems.

In our papers [1-6] we had considered such models.

All the problems were fully discretized and then coded in the modeling language AMPL.

This way, the approximations can be fed to all NLP solvers that have an AMPL interface. Several such solvers are implemented at the NEOS Server.

Our selection of AMPL models includes elliptic control problems and parabolic problems. See the README in each directory for the characteristics and special features of each test problem. The mathematical formulations of all problems are given in a PDF file.

An AMPL model can be retrieved and be submitted as is to the NEOS server, for example, through web submission to solvers such as KNITRO or LOQO. Various parameters (discretization, model) can be changed and the behavior of both the solution and the algorithm can be explored.

We report the results of these and other NLP solvers as part of our benchmark effort.
See sources 3 (elliptic problems) and 4 (parabolic problems) here.

Also, see these papers in which interior point algorithms are applied to these problems

[1] Bonettini, Silvia, A nonmonotone inexact Newton method. Optim. Methods Softw. 20 (2005), 475--491 PDF

[2] Bonettini, S., Galligani, E., and Ruggiero, V., An inexact Newton method combined with Hestenes multipliers' scheme for the solution of indefinite Karush-Kuhn-Tucker systems. Appl. Math. Comput. 168 (2005), 651--676 PDF

[3] Schenk, O., Waechter, A., and Hagemann, M., Combinatorial Approaches to the Solution of Saddle-Point Problems in Large-Scale Parallel Interior-Point Methods. Comp. Opt. Applic. 36 (2007), 321--341

And see this webpage with additional information:
http://dm.unife.it/~bonettini/ip_pcg/controllo.htm

References

[1] H. Maurer and H. D. Mittelmann, Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints. Part 1: Boundary Control, Comp. Optim. Applic. 16, 29-55 (2000) PS, PDF, HTML

[2] H. D. Mittelmann and H. Maurer, Solving Elliptic Control Problems with Interior Point and SQP Methods: Control and State Constraints, J. Comp. Appl. Math. 120, 175-195 (2000). PS, PDF, HTML

[3] H. Maurer and H. D. Mittelmann, Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints. Part 2: Distributed Control, Comp. Optim. Applic. 18, 141-160 (2001). PS, PDF, HTML

[4] H. D. Mittelmann, Verification of Second-Order Sufficient Optimality Conditions for Semilinear Elliptic and Parabolic Control Problems, Comp. Optim. Applic. 20, 93-110 (2001). PS, PDF, HTML

[5] H. D. Mittelmann, Sufficient Optimality for Discretized Parabolic and Elliptic Control Problems, in Fast solution of discretized optimization problems, K.-H. Hoffmann, R.H.W. Hoppe, and V. Schulz (eds.), ISNM 138, Birkhäuser, Basel, 2001 PS, PDF, HTML

[6] H. D. Mittelmann and F. Tröltzsch, Sufficient Optimality in a Parabolic Control Problem, in: Trends in Industrial Mathematics, Applied Optimization, vol. 72, A.H. Siddiqi and M. Kocvara (eds), Kluwer, Dordrecht, The Netherlands, 2002 PS, PDF, HTML