Download PDF - Properties of projection and penalty methods for discretized elliptic control problemsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Properties of projection and penalty methods for discretized elliptic control problems

Authors 1, 2

Affiliations

  1. Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland
  2. Institute of Numerical Mathematics, Dresden University of Technology, D-01062 Dresden, Germany

Abstract

In this paper, properties of projection and penalty methods are studied in connection with control problems and their discretizations. In particular, the convergence of an interior-exterior penalty method applied to simple state constraints as well as the contraction behavior of projection mappings are analyzed. In this study, the focus is on the application of these methods to discretized control problem.

Keywords

ENG
convex programming, control of PDE, projection methods, penalty methods

Bibliography

  1. N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl. 23 (2002), 201-229.
  2. E. Casas and F. Tröltzsch, Error estimates for linear-quadratic elliptic control problems, in: Barbu, V. (ed.) et al., Analysis and optimization of differential systems. IFIP TC7/WG 7.2 International Working Conference. Kluwer, Boston (2003), 89-100.
  3. A. Cegielski, A method of projection onto an acute cone with level control in convex minimization, Math. Programming 85 (1999), 469-490.
  4. K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state constrained elliptic control problem, Preprint MATH-NM-01-2006, TU Dresden 2006.
  5. K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambrigde Univ. Press, Cambridge 1990.
  6. Ch. Grossmann and H.-G. Roos, Numerische Behandlung partieller Differentialgleichungen (3-rd edition), B.G. Teubner, Stuttgart 2005.
  7. M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl. 30 (2005), 45-61.
  8. A.A. Kaplan, Convex programming algorithms using the smoothing of exact penalty functions, (Russian), Sib. Mat. Zh. 23 (1982), 53-64.
  9. S. Kim, H. Ahn and S.-C. Cho, Variable target value subgradient method, Math. Programming 49 (1991), 359-369.
  10. K.C. Kiwiel, The efficiency of subgradient projection methods for convex optimization, part I: General level methods, SIAM J. Control Optim. 34 (1996), 660-676.
  11. A. Rösch, Error estimates for linear-quadratic control problems with control constraints, Optim. Methods Softw. 21 (2006), 121-134.
  12. F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen. Theorie, Verfahren und Anwendungen, Vieweg, Wiesbaden 2005.
  13. M. Weiser, T. Gänzler and A. Schiela, A control reduced primal interior point method for PDE constrained optimization, ZIB Report 04-38, Zuse-Zentrum Berlin 2004.
  14. E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A: Linear Monotone Operators, Springer, New York 1990.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
2007/27/1
Export to PBN, Opens in new tab
Pages:
23-41
Main language of publication
English
Received
2006-07-24
Published
2007

DOI: 10.7151/dmdico1074

Exact and natural sciences