Skip to main content

Advertisement

SpringerLink
Log in

The inexact fixed matrix iteration for solving large linear inequalities in a least squares sense

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

A fixed matrix iteration algorithm was proposed by A. Dax (Numer. Algor. 50, 97–114 2009) for solving linear inequalities in a least squares sense. However, a great deal of computation for this algorithm is required, especially for large-scale problems, because a least squares subproblem should be solved accurately at each iteration. We present a modified method, the inexact fixed iteration method, which is a generalization of the fixed matrix iteration method. In this inexact iteration process, the classical LSQR method is implemented to determine an approximate solution of each least squares subproblem with less computational effort. The convergence of this algorithm is analyzed and several numerical examples are presented to illustrate the efficiency of the inexact fixed matrix iteration algorithm for solving large linear inequalities.

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. Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)

    Book  Google Scholar 

  2. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004)

  3. Bramley, R., Winnicka, B.: Solving linear inequalities in a least squares sense. SIAM J. Sci. Comput 17, 275–286 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Censor, Y., Altschuler, M.D., Powlis, W.D.: A computational solution of the inverse problem in radiation-therapy treatment planning. Appl. Math. Comput 25, 57–87 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  5. Censor, Y., Ben-Israel, A., Xiao, Y., Galvin, J.M.: On linear infeasibility arising in intensity modulated radiation therapy inverse planning. Linear Algebra Appl 428, 1406–1420 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chinneck, J.W.: Feasibility and infeasibility in optimization: algorithms and computational methods. In: International Series in Operations Research and Management Sciences, vol. 118. Springer-Verlag (2007)

  7. Davis, T.A.: Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization. ACM Trans. Math. Softw. 38, 1–22 (2011)

    Google Scholar 

  8. Dax, A.: The convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations. SIAM Rev. 32, 611–635 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dax, A.: The smallest correction of an inconsistent system of linear inequalities. Optimiz. Eng. 2, 349–359 (2001)

    Article  MATH  Google Scholar 

  10. Dax, A.: A hybrid algorithm for solving linear inequalities in a least squares sense. Numer. Algor. 50, 97–114 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gilbert, J.R., Moler, C., Schreiber, R.: Sparse matrices in MATLAB: design and implementation. SIAM J. Matrix Anal. Appl 13, 333–356 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Golub, G.H., Kahan, W.: Calculating the singular values and pseudoinverse of a matrix. SIAM J. Numer. Anal 2, 205–224 (1965)

    MATH  MathSciNet  Google Scholar 

  13. Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press (1983)

  14. Han, S.P.: Least squares solution of linear inequalities. Technical Report 2141, Math. Res. Center, University of Wisconsin-Madison (1980)

  15. He, B.S.: A projection and contraction method for a class of linear complementarity problem and its application in convex quadratic programming. Appl. Math. Optim. 25, 247–262 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. He, B.S.: Solving a class of linear projection equations. Numer. Math. 68, 71–80 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Herman, G.T.: A relaxation method for reconstructing object from noisy X-rays. Math. Progr. 8, 1–19 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  18. Herman, G.T.: Image Reconstruction from Projections: The Fundamentals of Computerized Tomography. Academic Press, New York (1982)

    Google Scholar 

  19. Moreau, J.J.: Decomposition orthogonale d’un espace Hilbertien selon deux cânes mutuellement polaires. C.R. Acad. Sci. Paris 225, 238–240 (1962)

    MathSciNet  Google Scholar 

  20. Morikuni, K., Hayami, K.: Inner-iteration Krylov subspace methods for least squares problems. SIAM J. Matrix Anal. Appl 34, 1–22 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Paige, C.C., Saunders, M.A.: LSQR, An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Software 8, 43–71 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  22. Pinar, M.C.: Newton’s method for linear inequality systems. Eur. J. Oper. Res 107, 710–719 (1998)

    Article  MATH  Google Scholar 

  23. Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  24. Stewart, G.W.: Matrix Algorithms, Volume 1: Basic Decompositions. SIAM, Philadelphia (1998)

    Book  Google Scholar 

  25. Stiefel, E.: Ausgleichung ohne Aufstellung der Gausschen Normalgleichungen. Wiss. Z. Tech. Hochsch. Dresden 2, 441–442 (1952–1953)

    MathSciNet  Google Scholar 

  26. Wang, R.S.: Functional Analysis and Optimization Theory. Beijing University of Aeronautics and Astronautics Press (2003). In Chinese

  27. Wierzbicki, A.P., Kurcyusz, S.: Projection on a cone, penalty functionals and duality theory for problems with inequality constraints in Hilbert space. SIAM J. Control Optim 15, 25–56 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  28. Xiu, N., Wang, C., Zhang, J.: Convergence properties of projection and contraction methods for variational inequality problems. Appl. Math. Optim 43, 147–168 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  29. Zarantonello, E.H.: Projections on Convex Sets in Hilbert Space and Spectral Theory. Academic Press (1971)

  30. National Institute of Standards and Technology, MatrixMarket, http://math.nist.gov/MatrixMarket (2002)

  31. Sparse QR factorization package, http://www.cise.ufl.edu/research/sparse/SPQR/ (2013)

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China

    Yuan Lei

Authors
  1. Yuan Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuan Lei.

Additional information

The work was supported by National Natural Science Foundations of China (No. 11201136) and Fundamental Research Funds for the Central Universities (No. 531107040014).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lei, Y. The inexact fixed matrix iteration for solving large linear inequalities in a least squares sense. Numer Algor 69, 227–251 (2015). https://doi.org/10.1007/s11075-014-9892-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-014-9892-2

Keywords

Search

Navigation