A Collection of Test Problems in PDEConstrained Optimization
Hans D. Mittelmann
Department of Math and Stats
Arizona State University
At the workshop
on
Optimization
in SimulationBased 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 PDEconstrained 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 [16] 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), 475491
PDF
[2] Bonettini, S., Galligani, E., and Ruggiero, V., An inexact Newton method
combined with Hestenes multipliers' scheme for the solution of indefinite
KarushKuhnTucker systems. Appl. Math. Comput. 168 (2005), 651676
PDF
[3] Schenk, O., Waechter, A., and Hagemann, M., Combinatorial Approaches
to the Solution of SaddlePoint Problems in LargeScale Parallel
InteriorPoint Methods. Comp. Opt. Applic. 36 (2007), 321341
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, 2955 (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, 175195 (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, 141160 (2001).
PS, PDF,
HTML
[4] H. D. Mittelmann, Verification of SecondOrder Sufficient
Optimality Conditions for Semilinear Elliptic and Parabolic Control Problems,
Comp. Optim. Applic. 20, 93110 (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
