Skip to main content
SpringerLink
Log in

Solving the minimal least squares problem subject to bounds on the variables

  • Part II Numerical Mathematics
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

A computational procedure is developed for determining the solution of minimal length to a linear least squares problem subject to bounds on the variables. In the first stage, a solution to the least squares problem is computed and then in the second stage, the solution of minimal length is determined. The objective function in each step is minimized by an active set method adapted to the special structure of the problem.

The systems of linear equations satisfied by the descent direction and the Lagrange multipliers in the minimization algorithm are solved by direct methods based on QR decompositions or iterative preconditioned conjugate gradient methods. The direct and the iterative methods are compared in numerical experiments, where the solutions are sought to a sequence of related, minimal least squares problems subject to bounds on the variables. The application of the iterative methods to large, sparse problems is discussed briefly.

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

  1. Å. Björck and T. Elfving,Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations. BIT 19 (1979), 145–163.

    Google Scholar 

  2. P. Businger and G. H. Golub,Linear least squares solutions by Householder transformations. Num. Math. 7 (1965), 269–276.

    Google Scholar 

  3. A. K. Cline, C. B. Moler, G. W. Stewart and J. H. Wilkinson,An estimate for the condition number of a matrix. SIAM J. Numer. Anal. 16 (1979), 368–375.

    Google Scholar 

  4. C. Cryer,The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control 9 (1971), 385–392.

    Google Scholar 

  5. U. Eckhardt,Quadratic programming by successive overrelaxation. Report Jül-1064-MA, Kernforschungsanlage, Jülich (1974).

  6. L. Eldén,Numerical analysis of regularization and constrained least squares methods. Ph.D. thesis, Dept. of Mathematics, Linköping University, Linköping (1977).

    Google Scholar 

  7. R. Fletcher,Practical Methods of Optimization, vol. 2, Constrained Optimization. Wiley, Chichester-New York (1981).

    Google Scholar 

  8. R. Fletcher and M. P. Jackson,Minimization of a quadratic function of many variables subject only to lower and upper bounds. J. Inst. Maths Applics 14 (1974), 159–174.

    Google Scholar 

  9. P. E. Gill, G. H. Golub, W. Murray and M. A. Saunders,Methods for modifying matrix factorizations. Math. Comp. 28 (1974), 505–535.

    Google Scholar 

  10. P. E. Gill and W. Murray,Minimization subject to bounds on the variables. NPL report NAC 72, National Physical Laboratory, Teddington (1976).

    Google Scholar 

  11. P. E. Gill and W. Murray,Numerically stable methods for quadratic programming. Math. Prog. 14 (1978), 349–372.

    Google Scholar 

  12. P. E. Gill, W. Murray and M. H. Wright,Practical Optimization. Academic Press, London-New York (1981).

    Google Scholar 

  13. K. H. Haskell, and R. J. Hanson,An algorithm for linear least squares problems with equality and nonnegativity constraints. Math. Prog. 21 (1981), 98–118.

    Google Scholar 

  14. I. Karasalo,A criterion for truncation of the QR-decomposition algorithm for the singular linear least squares problem. BIT 14 (1974), 156–166.

    Google Scholar 

  15. C. L. Lawson and R. J. Hanson,Solving Least Squares Problems. Prentice-Hall, Englewood Cliffs, NJ (1974).

    Google Scholar 

  16. P. Lötstedt,Numerical simulation of time-dependent contact and friction problems in rigid body mechanics. Report TRITA-NA-8214, Dept. of Num. Anal. and Comp. Science, Royal Institute of Technology, Stockholm (1982) (to appear in SIAM J. Sci. Stat. Comput 5 (1984)).

    Google Scholar 

  17. R. Mifflin,A stable method for solving certain constrained least squares problems. Math. Prog. 16 (1979), 141–158.

    Google Scholar 

  18. D. P. O'Leary,A generalized conjugate gradient algorithm for solving a class of quadratic programming problems. Lin. Alg. Appl. 34 (1980), 371–399.

    Google Scholar 

  19. K. Schittkowski and J. Stoer.A factorization method for the solution of constrained linear least squares problems allowing subsequent data changes. Num. Math. 31 (1979), 431–463.

    Google Scholar 

  20. J. Stoer,On the numerical solution of constrained least-squares problems. SIAM J. Num. Anal. 8 (1971), 382–411.

    Google Scholar 

Download references

Author information

Author notes

  1. Per Lötstedt

    Present address: TKLUB2, Aerospace Division, SAAB-Scania, S-581 88, Linköping, Sweden

Authors and Affiliations

  1. Department of Numerical Analysis and Computing Science, The Royal Institute of Technology, S-100 44, Stockholm, Sweden

    Per Lötstedt

Authors
  1. Per Lötstedt
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by The National Swedish Board for Technical Development under contract dnr 80-3341.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lötstedt, P. Solving the minimal least squares problem subject to bounds on the variables. BIT 24, 205–224 (1984). https://doi.org/10.1007/BF01937487

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01937487

Keywords

Search

Navigation