Skip to main content

Advertisement

SpringerLink
Log in

An Optimal Algorithm for Bound and Equality Constrained Quadratic Programming Problems with Bounded Spectrum

  • Published:
Computing Aims and scope Submit manuscript

Abstract

An implementation of the recently proposed semi-monotonic augmented Lagrangian algorithm for solving the large convex bound and equality constrained quadratic programming problems is considered. It is proved that if the algorithm is applied to the class of problems with uniformly bounded spectrum of the Hessian matrix, then the algorithm finds an approximate solution at O(1) matrix-vector multiplications. The optimality results are presented that do not depend on conditioning of the matrix which defines the equality constraints. Theory covers also the problems with dependent constraints. Theoretical results are illustrated by numerical experiments.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • D. P. Bertsekas (1999) Nonlinear optimization Athena Scientific Belmont

    Google Scholar 

  • A. R. Conn N. I. M. Gould PL. Toint (1992) LANCELOT: A Fortran package for large scale nonlinear optimization Springer Berlin Occurrence Handle0761.90087

    MATH  Google Scholar 

  • Z. Dostál (1995) ArticleTitleDuality based domain decomposition with proportioning for the solution of free boundary problems J. Comput. Appl. Math. 63 203–208 Occurrence Handle10.1016/0377-0427(96)80912-5 Occurrence Handle1365560 Occurrence Handle0859.65056

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Dostál (1997) ArticleTitleBox constrained quadratic programming with proportioning and projections SIAM J. Optim. 7 871–887 Occurrence Handle10.1137/S1052623494266250 Occurrence Handle1462070 Occurrence Handle0912.65052

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Dostál (1999) ArticleTitleOn preconditioning and penalized matrices Numer. Linear Algebra Appl. 6 109–114 Occurrence Handle10.1002/(SICI)1099-1506(199903)6:2<109::AID-NLA150>3.0.CO;2-0 Occurrence Handle1695403 Occurrence Handle0982.65055

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Dostál (2003) ArticleTitleA proportioning based algorithm for bound constrained quadratic programming with the rate of convergence Numer. Algorith. 34 293–302 Occurrence Handle10.1023/B:NUMA.0000005347.98806.b2 Occurrence Handle1037.65061

    Article  MATH  Google Scholar 

  • Z. Dostál (2005) ArticleTitleSemi-monotonic inexact augmented Lagrangians for quadratic programming with equality constraints Optim. Meth. Softw. 20 IssueID6 715–727 Occurrence Handle10.1080/10556780410001733513 Occurrence Handle1127.90399

    Article  MATH  Google Scholar 

  • Z. Dostál (2005) ArticleTitleInexact semi-monotonic augmented Lagrangians with optimal feasibility convergence for quadratic programming with simple bounds and equality constraints SIAM J. Numer. Anal. 43 IssueID1 96–115 Occurrence Handle10.1137/S0036142903436393 Occurrence Handle2177136 Occurrence Handle1089.65047

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Dostál A. Friedlander S. A. Santos (1998) ArticleTitleSolution of coercive and semicoercive contact problems by FETI domain decomposition Contemp. Math. 218 83–93

    Google Scholar 

  • Dostál, Z., Friedlander, A., Santos S. A.: Adaptive precision control in quadratic programming with simple bounds and/or equalities. In: High performance software for nonlinear optimization (De Leone, R., Pardalos, P. M., Toraldo, G., eds.). Applied Optimization, vol. 24. Kluwer 1998, pp. 161–173.

  • Z. Dostál A. Friedlander S. A. Santos (1999) ArticleTitleAugmented Lagrangians with adaptive precision control for quadratic programming with equality constraints Comput. Optim. Appl. 14 37–53 Occurrence Handle10.1023/A:1008700911674 Occurrence Handle1704945 Occurrence Handle0958.90064

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Dostál A. Friedlander S. A. Santos K. F. Alesawi (2002) ArticleTitleAugmented Lagrangians with adaptive precision control for quadratic programming with equality constraints: corrigendum and addendum Comput. Optim. Appl. 23 127–133 Occurrence Handle10.1023/A:1019980809734 Occurrence Handle1931363 Occurrence Handle1046.90056

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Dostál A. Friedlander S. A. Santos (2003) ArticleTitleAugmented Lagrangians with adaptive precision control for quadratic programming with simple bounds and equality constraints SIAM J. Optim. 13 1120–1140 Occurrence Handle10.1137/S1052623499362573 Occurrence Handle2005920 Occurrence Handle1043.65076

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Dostál A. Friedlander S. A. Santos (2000) ArticleTitleSolution of contact problems by FETI domain decomposition with natural coarse space projection Comput. Meth. Appl. Mech. Engng. 190 1611–1627 Occurrence Handle10.1016/S0045-7825(00)00180-8 Occurrence Handle1005.74064

    Article  MATH  Google Scholar 

  • Z. Dostál D. Horák (2003) ArticleTitleScalability and FETI based algorithm for large discretized variational inequalities Math. Comput. Simul. 61 347–357 Occurrence Handle10.1016/S0378-4754(02)00088-5 Occurrence Handle1015.65026

    Article  MATH  Google Scholar 

  • Z. Dostál D. Horák (2004) ArticleTitleScalable FETI with optimal dual penalty for a variational inequality Numer. Linear Algebra Appl. 11 455–472 Occurrence Handle10.1002/nla.355 Occurrence Handle2067815 Occurrence Handle1164.74526

    Article  MathSciNet  MATH  Google Scholar 

  • Dostál, Z., Horák, D.: On scalable algorithms for numerical solution of variational inequalities based on FETI and semi-monotonic augmented Lagrangians. In: Proc. Domain Decomposition Methods in Science and Engineering (Kornhuber R. et al., eds.). Lecture Notes in Computational Science and Engineering. Berlin: Springer 2004, pp. 487–494.

  • Z. Dostál D. Horák (2003) ArticleTitleScalable FETI with optimal dual penalty for semicoercive variational inequalities Contemp. Math. 329 293–302

    Google Scholar 

  • Z. Dostál D. Horák D. Stefanica (2005) ArticleTitleA Scalable FETI–DP algorithm for coercive variational inequalities IMACS J. Appl. Numer. Math. 54 IssueID3–4 378–390 Occurrence Handle10.1016/j.apnum.2004.09.009 Occurrence Handle1081.65064

    Article  MATH  Google Scholar 

  • Z. Dostál J. Schöberl (2005) ArticleTitleMinimizing quadratic functions over nonnegative cone with the rate of convergence and finite termination Comput. Optim. Appl. 30 IssueID1 23–43 Occurrence Handle10.1007/s10589-005-4557-7 Occurrence Handle2122181 Occurrence Handle1071.65085

    Article  MathSciNet  MATH  Google Scholar 

  • C. Farhat J. Mandel F.-X. Roux (1994) ArticleTitleOptimal convergence properties of the FETI domain decomposition method Comput. Meth. Appl. Mech. Engng. 115 365–385 Occurrence Handle1285024 Occurrence Handle10.1016/0045-7825(94)90068-X

    Article  MathSciNet  Google Scholar 

  • A. Friedlander J. M. Martínez M. Raydan (1995) ArticleTitleA new method for large scale box constrained quadratic minimization problems Optim. Meth. Softw. 5 57–74

    Google Scholar 

  • A. Friedlander J. M. Martínez S. A. Santos (1994) ArticleTitleA new trust region algorithm for bound constrained minimization Appl. Math. Optim. 30 235–266 Occurrence Handle10.1007/BF01183013 Occurrence Handle1288591 Occurrence Handle0821.90101

    Article  MathSciNet  MATH  Google Scholar 

  • J. Gondzio R. Sarkissian (2003) ArticleTitleParallel interior point solver for structured linear programs Math. Program. 96 561–584 Occurrence Handle10.1007/s10107-003-0379-5 Occurrence Handle1993463 Occurrence Handle1023.90039

    Article  MathSciNet  MATH  Google Scholar 

  • W. Hackbusch (1985) Multigrid methods and applications Springer Berlin

    Google Scholar 

  • W. W. Hager (1993) ArticleTitleAnalysis and implementation of a dual algorithm for constraint optimization JOTA 79 37–71 Occurrence Handle10.1007/BF00940552 Occurrence Handle1255280

    Article  MathSciNet  Google Scholar 

  • J. Nocedal S. J. Wright (2000) Numerical optimization Springer New York

    Google Scholar 

  • Y. Saad (2003) Iterative methods for sparse systems SIAM Philadelphia Occurrence Handle1031.65046

    MATH  Google Scholar 

  • J. Schöberl (1998) ArticleTitleSolving the Signorini problem on the basis of domain decomposition techniques Computing 60 323–344 Occurrence Handle1634831 Occurrence Handle0915.73077

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. VŠB Technical University of Ostrava, 17. Listopadu, 708 33, Ostrava, Czech Republic

    Z. Dostál

Authors
  1. Z. Dostál
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Z. Dostál.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dostál, Z. An Optimal Algorithm for Bound and Equality Constrained Quadratic Programming Problems with Bounded Spectrum. Computing 78, 311–328 (2006). https://doi.org/10.1007/s00607-006-0184-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-006-0184-0

AMS Subject Classifications

Keywords

Search

Navigation