Abstract
In this paper preconditioners for linear systems arising in interior-point methods for the solution of distributed control problems are derived and analyzed. The matrices K in these systems have a block structure with blocks obtained from the discretization of the objective function and the governing differential equation. The preconditioners have a block structure with blocks being composed of preconditioners for the subblocks of the system matrix K. The effectiveness of the preconditioners is analyzed and numerical examples for an elliptic model problem are shown.
This author was supported by the NSF DMS-9403699, AFOSR F49620-93-1-0280, and in part by the DoE DE FG03-95ER25257.
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References
H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Systems & Control: Foundations & Applications, Birkhôuser-Verlag, Boston, Basel, Berlin, 1989.
R. Barrett, M. Berry, T. F. Chan, J. D. J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, AND H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1993.
A. Battermann,Preconditioners for Karush-Kuhn-Tucker systems arising in optimal control, Master’s thesis, Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1996.
J. Bonnans, C. Pola, and R. RebaI, Perturbed path following interior point algorithm, Tech. Rep. No. 2745, INRIA, Domaine de Voluceau, 78153 Rocquencourt, France, 1995.
D. Braess and P. Peisker, On the numerical solution of the biharmonic equation and the role of squaring matrices, Ima J. Numer. Anal, 6 (1986), pp. 393–404.
L. Collatz AND W. Wetterling, Optimization Problems, Springer-Verlag, Berlin, Heidelberg, New York, 1975.
J. E. Dennis, M. Heinkenschloss, and L. N. Vicente, Trust-region interior-point algorithms for a class of nonlinear programming problems, Tech. Rep. TR94-45, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892, 1994. Available electronically at http://www.caam.rice.edu/~trice/trice_soft.html./~trice/trice_soft.html.
A. S. El-Bakry, R. A. Tapia, T. Tsuchiya, and Y. Zhang, On the formulation and theory of the primal-dual Newton interior-point method for nonlinear programming, Journal of Optimization Theory and Applications, 89 (1996), pp. 507–541.
R. W. Freund and F. Jarre, A qmr-based interior-point method for solving linear programs, Mathematical Programming, Series B, (to appear).
P. E. Gill, W. Murray, D. B. Ponceleön, and M. A. Saunders, Preconditioned for indefinite systems arising in optimization, Siam J. Matrix Anal. Appl., 13 (1992), pp. 292–311.
G. Golub AND C. F. van Loan, Matrix Computations, John Hopkins University Press, Baltimore, London, 1989.
S. Ito, C. T. Kelley, and E. W. Sachs, Inexact primal-dual interior-point iteration for linear programs in function spaces, Computational Optimization and Applications, 4 (1995), pp. 189–201.
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1971.
C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, Siam J. Numer. Anal, 12 (1975), pp. 617–629.
T. Rusten And R. Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl, 13 (1992), pp. 887–904.
J.Stoer,Solution of large linear systems of equations by conjugate gradient type methods, in Mathematical Programming, The State of The Art, A.Bachem, M. Grötschel, and B.Korte, eds. Springer-Verlag, Berlin, Heidelberg, New-York, 1983, pp. 540–565.
D. Sylvester and A. Wathen, Fast iterative solution of stabilized Stokes systems part II: using general block preconditioners, SIAM J. Numer. Anal, 31 (1994), pp. 1352–1367.
L. N. Vicente, On interior point Newton algorithms for discretized optimal control problems with state constraints, tech. rep, Departamento de Matematica, Universidade de Coimbra, 3000 Coimbra, Portugal, 1996.
M. H. Wright, Interior point methods for constrained optimization, in Acta Numerica 1992, A. Iserles, ed, Cambridge University Press, Cambridge, London, New York, 1992, pp. 341–407.
S. J. Wright, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, PA, 1996.
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Battermann, A., Heinkenschloss, M. (1998). Preconditioners for Karush-Kuhn-Tucker Matrices Arising in the Optimal Control of Distributed Systems. In: Desch, W., Kappel, F., Kunisch, K. (eds) Control and Estimation of Distributed Parameter Systems. International Series of Numerical Mathematics, vol 126. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8849-3_2
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