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Interior Methods For a Class of Elliptic Variational Inequalities

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Large-Scale PDE-Constrained Optimization

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 30))

Abstract

We consider the applicat ion of primal-dual interior methods to the optimization of systems arising in the finite-element discretization of a class of elliptic variational inequalities. These problems lead to very large (possibly non-convex) optimization problems with upper and lower bound constraints.

When interior methods are applied to the discretized problem, the resulting linear systems have the same zero/nonzero structure as the finite-element equations solved for the unconstrained case. This crucial property allows the interior method to exploit existing efficient, robust and scalable multilevel algorithms for the solution of partial differential equat ions (PDEs).

We illustrate some of these ideas in th e context of the elliptic PDE package PLTMG.

Research supported by National Science Foundation grants DMS-9973276 and ACI-0082100.

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References

  1. R. E. Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, Users’ Guide 8.0, Software, Environments and Tools, Vol. 5, SIAM, Philadelphia, 1998.

    Google Scholar 

  2. R. E. Bank and K. R. Smith, The incomplete factorization multigraph algorithm, SIAM J. Sci. Comput., 20 (1999), pp. 1349–1364 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  3. D. P. Bertsekas, On the Goldstein-Levitin-Polyak gradient projection method, IEEE Trans. Automatic Control, AC-21 (1976), pp. 174–184.

    Google Scholar 

  4. -, Projected Newton methods for optimization problems with simple constraints, SIAM J. Control Optim., 20 (1982), pp. 221–246.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Bondarenko, D. Bortz, and J. J. Moré, COPS: Large-scale nonlin early constrained optimization problems, Technical Report ANL/MCS-TM-237, Mathematics and Computer Science division, Argonne National Laboratory, Argonne, IL, 1998. Revised October 1999.

    Google Scholar 

  6. D. Braess, Finite elements, Cambridge University Press, Cambridge, second ed., 2001. Theory, fast solvers, and applications in solid mechanics, Translated from the 1992 German edition by Larry L. Schumaker.

    Google Scholar 

  7. M. A. Branch, T. F. Coleman, and Y. Li, A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems, SIAM J. Sci. Comput., 21 (1999), pp. 1–23 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  8. T. D. Choi and C. T. Kelley, Estimates for the Nash-Sofer preconditioner for the reduced Hessian for some elliptic variational inequalities, SIAM J. Optim., 9 (1999), pp. 327–341.

    Article  MathSciNet  MATH  Google Scholar 

  9. T. F. Coleman and Y. Li, An interior trust region approach for nonlinear minimization subject to bounds, SIAM J. Optim., 6 (1996), pp. 418–445.

    Article  MathSciNet  MATH  Google Scholar 

  10. E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), pp. 201–213.

    Article  MathSciNet  MATH  Google Scholar 

  11. Z. Dostál, Box constrained quadratic programming with proportioning and projections, SIAM J. Optim., 7 (1997), pp. 871–887.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. V. Fiacco and G. P. Mccormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Classics in Applied Mathematics, SIAM, Philadelphia, 1990. ISBN 0-89871-254-8.

    Google Scholar 

  13. A. Forsgren and P. E. Gill, Primal-dual interior methods for nonconvex nonlinear programming, SIAM J. Optim., 8 (1998), pp. 1132–1152. (Electronic).

    Article  MathSciNet  MATH  Google Scholar 

  14. E. M. Gertz and P. E. Gill, A primal-dual trust region algorithm for nonlinear programming, Report to appear, Department of Mathematics, University of California, San Diego, 2002.

    Google Scholar 

  15. R. Glowinski, Numerical methods for nonlinear variational problems, Springer-Verlag, New York, 1984. ISBN 0-387-12434-9.

    MATH  Google Scholar 

  16. N. I. M. Gould, S. Lucidi, M. Roma, and P. L. Toint, Solving the trust-region subproblem using the Lanczos method, SIAM J. Optim., 9 (1999), pp. 504–525 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Heinkenschloss, M. Ulbrich, and S. Ulbrich, Superlinear and quadratic convergence of affine-scaling interior-point Newton methods for problems with simple bounds without strict complementarity assumption, Math. Program., 86 (1999), pp. 615–635.

    Article  MathSciNet  MATH  Google Scholar 

  18. R. H. W. Hoppe and R. Kornhuber, Adaptive multilevel methods for obstacle problems, SIAM J. Numer. Anal., 31 (1994), pp. 301–323.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980.

    MATH  Google Scholar 

  20. -, An introduction to variational inequalities and their applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. Reprint of the 1980 original.

    Book  MATH  Google Scholar 

  21. R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities. I, Numer. Math., 69 (1994), pp. 167–184.

    MathSciNet  MATH  Google Scholar 

  22. -, Monotone multigrid methods for elliptic variational inequalities. II, Numer. Math., 72 (1996), pp. 481–499.

    Article  MathSciNet  MATH  Google Scholar 

  23. C.-J. Lin and J. J. Moré, Newton’s method for large bound-constrained optimization problems, SIAM J. Optim., 9 (1999), pp. 1100–1127 (electronic). Dedicated to John E. Dennis, Jr., on his 60th birthday.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. J. Moré and G. Toraldo, On the solution of large quadratic programming problems with bound constraints, SIAM J. Optim., 1 (1991), pp. 93–113.

    Article  MathSciNet  MATH  Google Scholar 

  25. S. G. Nash, Newton-type minimization via the Lánczos method, SIAM J. Numer. Anal., 21 (1984), pp. 770–788.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Rauch, Partial differential equations, Springer-Verlag, New York, 1991.

    Book  MATH  Google Scholar 

  27. M. Ulbrich, S. Ulbrich, and M. Heinkenschloss, Global convergence of trust-region interior-point algorithms for infinite-dimensional nonconvex minimization subject to pointwise bounds, SIAM J. Control Optim., 37 (1999), pp. 731–764 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  28. S. J. Wright, Primal-dual interior-point methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. ISBN 0-89871-382-X.

    Google Scholar 

  29. -, On the convergence of the Newton/log-barrier method, Technical report MCS-P681-0897, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 1997, Revised April, 2000.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, La Jolla, CA

    Randolph E. Bank, Philip E. Gill & Roummel F. Marcia

Authors
  1. Randolph E. Bank
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  2. Philip E. Gill
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  3. Roummel F. Marcia
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Editor information

Editors and Affiliations

  1. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Lorenz T. Biegler

  2. Department of Computational and Applied Mathematics, Rice University, 6100 S. Main Street — MS134, Houston, TX, 77005, USA

    Matthias Heinkenschloss

  3. Department of Biomedical Engineering and Civil & Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Omar Ghattas

  4. Optimization & Uncertainty Estimation Dept., Sandia National Laboratories — MS 0847, P.O. Box 5800, Albuquerque, NM, 87185, USA

    Bart van Bloemen Waanders

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© 2003 Springer-Verlag Berlin Heidelberg

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Bank, R.E., Gill, P.E., Marcia, R.F. (2003). Interior Methods For a Class of Elliptic Variational Inequalities. In: Biegler, L.T., Heinkenschloss, M., Ghattas, O., van Bloemen Waanders, B. (eds) Large-Scale PDE-Constrained Optimization. Lecture Notes in Computational Science and Engineering, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55508-4_13

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  • DOI: https://doi.org/10.1007/978-3-642-55508-4_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-05045-2

  • Online ISBN: 978-3-642-55508-4

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