Skip to main content
SpringerLink
Log in
Book cover

Large-Scale Nonlinear Optimization pp 61–82Cite as

On implicit-factorization constraint preconditioners

  • Chapter

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 83))

Summary

Recently Dollar and Wathen [14] proposed a class of incomplete factorizations for saddle-point problems, based upon earlier work by Schilders [40]. In this paper, we generalize this class of preconditioners, and examine the spectral implications of our approach. Numerical tests indicate the efficacy of our preconditioners.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Avriel. Nonlinear Programming: Analysis and Methods. Prentice-Hall, Englewood Cliffs, New Jersey, 1976.

    Google Scholar 

  2. R. H. Bartels and G. H. Golub. The simplex method of linear programming using lu decompositions. Communications of the ACM, 12:266–268, 1969.

    Article  Google Scholar 

  3. L. Bergamaschi, J. Gondzio, and G. Zilli. Preconditioning indefinite systems in interior point methods for optimization. Computational Optimization and Applications, 28:149–171, 2004.

    Article  MathSciNet  Google Scholar 

  4. G. Biros and O. Ghattas. A Lagrange-Newton-Krylov-Schur method for pde-constrained optimization. SIAG/Optimiztion Views-and-News, 11(2): 12–18, 2000.

    Google Scholar 

  5. J. H. Bramble and J. E. Pasciak. A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Mathematics of Computation, 50:1–17, 1988.

    Article  MathSciNet  Google Scholar 

  6. R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9(4):877–900, 2000.

    Article  MathSciNet  Google Scholar 

  7. Y. Chabrillac and J.-P. Crouzeix. Definiteness and semidefiniteness of quadratic forms revisited. Linear Algebra and its Applications, 63:283–292, 1984.

    Article  MathSciNet  Google Scholar 

  8. T. F. Coleman and A. Pothen. The null space problem I: complexity. SIAM Journal on Algebraic and Discrete Methods, 7(4):527–537, 1986.

    MathSciNet  Google Scholar 

  9. T. F. Coleman and A. Pothen. The null space problem II: algorithms. SIAM Journal on Algebraic and Discrete Methods, 8(4):544–563, 1987.

    MathSciNet  Google Scholar 

  10. A. R. Conn, N. I. M. Gould, D. Orban, and Ph. L. Toint. A primal-dual trust-region algorithm for non-convex nonlinear programming. Mathematical Programming, 87(2):215–249, 2000.

    Article  MathSciNet  Google Scholar 

  11. R. W. Cottle. Manifestations of the Schur complement. Linear Algebra and its Applications, 8:189–211, 1974.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Debreu. Definite and semidefinite quadratic forms. Econometrica, 20(2):295–300, 1952.

    MATH  MathSciNet  Google Scholar 

  13. H. S. Dollar, N. I. M. Gould, W. H. A. Schilders, and A. J. Wathen. On iterative methods and implicit-factorization preconditioners for regularized saddle-point systems. Technical Report RAL-TR-2005-011, Rutherford Appleton Laboratory, Chilton, Oxfordshire, England, 2005.

    Google Scholar 

  14. H. S. Dollar and A. J. Wathen. Incomplete factorization constraint preconditioners for saddle point problems. Technical Report 04/01, Oxford University Computing Laboratory, Oxford, England, 2004.

    Google Scholar 

  15. I. S. Duff. MA57-a code for the solution of sparse symmetric definite and indefinite systems. ACM Transactions on Mathematical Software, 30(2):118–144, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  16. I. S. Duff and J. K. Reid. The multifrontal solution of indefinite sparse symmetric linear equations. ACM Transactions on Mathematical Software, 9(3):302–325, 1983.

    Article  MathSciNet  Google Scholar 

  17. I. S. Duff and J. K. Reid. The design of MA48: a code for the direct solution of sparse unsymmetric linear systems of equations. ACM Transactions on Mathematical Software, 22(2):187–226, 1996.

    Article  MathSciNet  Google Scholar 

  18. C. Durazzi and V. Ruggiero. Indefinitely constrained conjugate gradient method for large sparse equality and inequality constrained quadratic problems. Numerical Linear Algebra with Applications, 10(8):673–688, 2002.

    Article  MathSciNet  Google Scholar 

  19. H. C. Elman, D. J. Silvester, and A. J. Wathen. Finite-Elements and Fast Iterative Solvers: with applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford, 2005, to appear.

    Google Scholar 

  20. J. J. H. Forrest and J. A. Tomlin. Updating triangular factors of the basis to maintain sparsity in the product form simplex method. Mathematical Programming, 2(3):263–278, 1972.

    Article  MathSciNet  Google Scholar 

  21. D. M. Gay. Electronic mail distribution of linear programming test problems. Mathematical Programming Society COAL Newsletter, December 1985. See http://www.netlib.org/lp/data/.

    Google Scholar 

  22. P. E. Gill, W. Murray, D. B. Ponceleón, and M. A. Saunders. Preconditioners for indefinite systems arising in optimization. SIAM Journal on Matrix Analysis and Applications, 13(1):292–311, 1992.

    Article  MathSciNet  Google Scholar 

  23. P. E. Gill, W. Murray, and M. A. Saunders. SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM Journal on Optimization, 12(4):979–1006, 2002.

    Article  MathSciNet  Google Scholar 

  24. P. E. Gill and M. A. Saunders. private communication, 1999.

    Google Scholar 

  25. G. H. Golub and C. Greif. On solving block-structured indefinite linear systems. SIAM Journal on Scientific Computing, 24(6):2076–2092, 2003.

    Article  MathSciNet  Google Scholar 

  26. N. I. M. Gould. On practical conditions for the existence and uniqueness of solutions to the general equality quadratic-programming problem. Mathematical Programming, 32(1):90–99, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  27. N. I. M. Gould, M. E. Hribar, and J. Nocedal. On the solution of equality constrained quadratic problems arising in optimization. SIAM Journal on Scientific Computing, 23(4):1375–1394, 2001.

    Article  MathSciNet  Google Scholar 

  28. N. I. M. Gould, S. Lucidi, M. Roma, and Ph. L. Toint. Solving the trust-region subproblem using the Lanczos method. SIAM Journal on Optimization, 9(2):504–525, 1999.

    Article  MathSciNet  Google Scholar 

  29. N. I. M. Gould, D. Orban, and Ph. L. Toint. CUTEr (and SifDec), a Constrained and Unconstrained Testing Environment, revisited. ACM Transactions on Mathematical Software, 29(4):373–394, 2003.

    Article  MathSciNet  Google Scholar 

  30. N. I. M. Gould, D. Orban, and Ph. L. Toint. GALAHAD—a library of thread-safe fortran 90 packages for large-scale nonlinear optimization. ACM Transactions on Mathematical Software, 29(4):353–372, 2003.

    Article  MathSciNet  Google Scholar 

  31. C. Greif, G. H. Golub, and J. M. Varah. Augmented Lagrangian techniques for solving saddle point linear systems. Technical report, Computer Science Department, University of British Columbia, Vancouver, Canada, 2004.

    Google Scholar 

  32. HSL. A collection of Fortran codes for large-scale scientific computation, 2004. See http://hsl.rl.ac.uk/.

    Google Scholar 

  33. C. Keller, N. I. M. Gould, and A. J. Wathen. Constraint preconditioning for indefinite linear systems. SIAM Journal on Matrix Analysis and Applications, 21(4):1300–1317, 2000.

    Article  MathSciNet  Google Scholar 

  34. L. Lukšan and J. Vlček. Indefinitely preconditioned inexact Newton method for large sparse equality constrained nonlinear programming problems. Numerical Linear Algebra with Applications, 5(3):219–247, 1998.

    Article  MathSciNet  Google Scholar 

  35. B. A. Murtagh and M. A. Saunders. Large-scale linearly constrained optimization. Mathematical Programming, 14(1):41–72, 1978.

    Article  MathSciNet  Google Scholar 

  36. B. A. Murtagh and M. A. Saunders. A projected Lagrangian algorithm and its implementation for sparse non-linear constraints. Mathematical Programming Studies, 16:84–117, 1982.

    MathSciNet  Google Scholar 

  37. J. Nocedal and S. J. Wright. Large sparse numerical optimization. Series in Operations Research. Springer Verlag, Heidelberg, Berlin, New York, 1999.

    Google Scholar 

  38. L. A. Pavarino. Preconditioned mixed spectral finite-element methods for elasticity and Stokes problems. SIAM Journal on Scientific Computing, 19(6):1941–1957, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  39. I. Perugia and V. Simoncini. Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations. Numerical Linear Algebra with Applications, 7(7–8):585–616, 2000.

    Article  MathSciNet  Google Scholar 

  40. W. Schilders. A preconditioning technique for indefinite systems arising in electronic circuit simulation. Talk at the one-day meeting on preconditioning methods for indefinite linear systems, TU Eindhoven, December 9, 2002.

    Google Scholar 

  41. R. B. Schnabel and E. Eskow. A revised modified Cholesky factorization algorithm. SIAM Journal on Optimization, 9(4):1135–1148, 1999.

    Article  MathSciNet  Google Scholar 

  42. R. J. Vanderbei and D. F. Shanno. An interior point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231–252, 1999.

    Article  MathSciNet  Google Scholar 

  43. R. C. Whaley, A. Petitet, and J. Dongarra. Automated empirical optimization of software and the ATLAS project. Parallel Computing, 27(1–2):3–35, 2001.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Numerical Analysis Group, Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD, England

    H. Sue Dollar & Andrew J. Wathen

  2. Rutherford Appleton Laboratory, Computational Science and Engineering Department, Chilton, Oxfordshire, OX11 0QX, England

    Nicholas I. M. Gould

Authors
  1. H. Sue Dollar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicholas I. M. Gould
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrew J. Wathen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Rome “La Sapienza”, Italy

    G. Di Pillo  & M. Roma  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer Science+Business Media, Inc.

About this chapter

Cite this chapter

Dollar, H.S., Gould, N.I.M., Wathen, A.J. (2006). On implicit-factorization constraint preconditioners. In: Di Pillo, G., Roma, M. (eds) Large-Scale Nonlinear Optimization. Nonconvex Optimization and Its Applications, vol 83. Springer, Boston, MA. https://doi.org/10.1007/0-387-30065-1_5

Download citation

Publish with us

Policies and ethics

Search

Navigation