Abstract
In certain applications of PDE constrained optimization one would like to base an optimization method on an already existing contractive method (solver) for the forward problem. The forward problem consists of finding a feasible point with some parts of the variables (e.g., design variables) held fixed. This approach often leads to so-called simultaneous, all-at-once, or oneshot optimization methods. If only one iteration of the forward method per optimization iteration is necessary, a simultaneous method is called one-step. We present three illustrative linear examples in four dimensions with two constraints which highlight that in general there is only little connection between contraction of forward problem method and simultaneous one-step optimization method. We analyze the asymptotics of three prototypical regularization strategies to possibly recover convergence and compare them with Griewank’s One-Step One-Shot projected Hessian preconditioners. We present de facto loss of convergence for all of these methods, which leads to the conclusion that, at least for fast contracting forward methods, the forward problem solver must be used with adaptive accuracy controlled by the optimization method.
Keywords
Mathematics Subject Classification (2000). 65K05.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Battermann and M. Heinkenschloss. Preconditioners for Karush-Kuhn-Tucker matrices arising in the optimal control of distributed systems. In Control and estimation of distributed parameter systems (Vorau, 1996), volume 126 of Internat. Ser. Numer. Math., pages 15–32. Birkhäuser, Basel, 1998.
A. Battermann and E.W. Sachs. Block precondtioners for KKT sysems in PDE-governed optimal control problems. In Fast solution of discretized optimization problems (Berlin, 2000), volume 138 of Internat. Ser. Numer. Math., pages 1–18. Birkhäuser, Basel, 2001.
G. Biros and O. Ghattas. Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part I: The Krylov-Schur solver. SIAM Journal on Scientific Computing, 27(2):687–713, 2005.
G. Biros and O. Ghattas. Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part II: The Lagrange-Newton solver and its application to optimal control of steady viscous flows. SIAM Journal on Scientific Computing, 27(2):714–739, 2005.
H.G. Bock. Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen, volume 183 of Bonner Mathematische Schriften. Universität Bonn, Bonn, 1987.
H.G. Bock, W. Egartner, W. Kappis, and V. Schulz. Practical shape optimization for turbine and compressor blades by the use of PRSQP methods. Optimization and Engineering, 3(4):395–414, 2002.
R.H. Byrd, F.E. Curtis, and J. Nocedal. An inexact SQP method for equality constrained optimization. SIAM Journal on Optimization, 19(1):351–369, 2008.
N.I.M. Gould and Ph.L. Toint. Nonlinear programming without a penalty function or a filter. Mathematical Programming, Series A, 122:155–196, 2010.
A. Griewank. Projected Hessians for preconditioning in One-Step One-Shot design optimization. In Large-Scale Nonlinear Optimization, volume 83 of Nonconvex Optimization and Its Applications, pages 151–171. Springer US, 2006.
S.B. Hazra, V. Schulz, J. Brezillon, and N.R. Gauger. Aerodynamic shape optimization using simultaneous pseudo-timestepping. Journal of Computational Physics, 204(1):46–64, 2005.
M. Heinkenschloss and D. Ridzal. An Inexact Trust-Region SQP method with applications to PDE-constrained optimization. In K. Kunisch, G. Of, and O. Steinbach, editors, Proceedings of ENUMATH 2007, the 7th European Conference on Numerical Mathematics and Advanced Applications, Graz, Austria, September 2007. Springer Berlin Heidelberg, 2008.
M. Heinkenschloss and L.N. Vicente. Analysis of Inexact Trust-Region SQP algorithms. SIAM Journal on Optimization, 12(2):283–302, 2002.
A. Potschka, A. Küpper, J.P. Schlöder, H.G. Bock, and S. Engell. Optimal control of periodic adsorption processes: The Newton-Picard inexact SQP method. In Recent Advances in Optimization and its Applications in Engineering, pages 361–378. Springer Verlag Berlin Heidelberg, 2010.
A. Potschka, M.S. Mommer, J.P. Schlöder, and H.G. Bock. A Newton-Picard approach for efficient numerical solution of time-periodic parabolic PDE constrained optimization problems. Technical Report 2010-03-2570, Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, 2010. Preprint, http://www.optimization-online.org/DB_HTML/2010/03/2570.html.
A. Walther. A first-order convergence analysis of Trust-Region methods with inexact Jacobians. SIAM Journal on Optimization, 19(1):307–325, 2008.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel AG
About this chapter
Cite this chapter
Bock, H.G., Potschka, A., Sager, S., Schlöder, J.P. (2012). On the Connection Between Forward and Optimization Problem in One-shot One-step Methods. In: Leugering, G., et al. Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol 160. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0133-1_3
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0133-1_3
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0132-4
Online ISBN: 978-3-0348-0133-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)