Advertisement

Computational Optimization and Applications

, Volume 23, Issue 2, pp 143–169 | Cite as

Nonmonotone Globalization Techniques for the Barzilai-Borwein Gradient Method

  • L. Grippo
  • M. Sciandrone
Article

Abstract

In this paper we propose new globalization strategies for the Barzilai and Borwein gradient method, based on suitable relaxations of the monotonicity requirements. In particular, we define a class of algorithms that combine nonmonotone watchdog techniques with nonmonotone linesearch rules and we prove the global convergence of these schemes. Then we perform an extensive computational study, which shows the effectiveness of the proposed approach in the solution of large dimensional unconstrained optimization problems.

Barzilai-Borwein method gradient method steepest descent nonmonotone techniques unconstrained optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Barzilai and J.M. Borwein, “Two point step size gradient method,” IMA J. Numer. Anal., vol. 8, pp. 141–148, 1988.Google Scholar
  2. 2.
    D.P. Bertsekas, Nonlinear Programming, 2nd edn., Athena Scientific, 1999.Google Scholar
  3. 3.
    I. Bongartz, A. Conn, N. Gould, and P. Toint, “CUTE: constrained and unconstrained testing Environments,” ACM Transaction on Math. Software, vol. 21, pp. 123–160, 1995.Google Scholar
  4. 4.
    R.M. Chamberlain, M.J.D. Powell, C. Lemarechal, and H.C. Pedersen, “The watchdog technique for forcing convergence in algorithms for constrained optimization,” Math. Programming, vol. 16, pp. 1–17, 1982.Google Scholar
  5. 5.
    Y.H. Dai and L.Z. Liao, “R-Linear Convergence of the Barzilai and Borwein gradient method,” Research Report, 1999. Alsoin IMA J. Numer. Anal., vol. 22, pp. 1-10, 2002.Google Scholar
  6. 6.
    Y.H. Dai and H. Zhang, “An adaptive two-point stepsize gradient algorithm,” Research report, Chinese Academy of Sciences, 2000.Google Scholar
  7. 7.
    R. De Leone, M. Gaudioso, and L. Grippo, “Stopping criteria for linesearch methods without derivatives,” Math. Programming, vol. 30, pp. 285–300, 1984.Google Scholar
  8. 8.
    R. Fletcher, “Low storage methods for unconstrained optimization,” Lectures in Applied Mathematics (AMS), vol. 26, pp. 165–179, 1990.Google Scholar
  9. 9.
    R. Fletcher, “On the Barzilai-Borwein method,” Numerical Analysis Report NA/207, 2001.Google Scholar
  10. 10.
    A. Friedlander, J.M. Martinez, B. Molina, and M. Raydan, “Gradient method with retards and generalizations,” SIAM J. Numer. Anal., vol. 36, pp. 275–289, 1999.Google Scholar
  11. 11.
    A. Friedlander, J.M. Martinez, and M. Raydan, “A new method for large-scale box constrained convex quadratic minimization problems,” Optimization Methods and Software, vol. 5, pp. 55–74, 1995.Google Scholar
  12. 12.
    J. Gilbert and C. Lemaréchal, “Some numerical experiments with variable-storage quasi-Newton algorithms,” Math. Programming, Series B, vol. 45, pp. 407–435, 1989.Google Scholar
  13. 13.
    P. Gill, W. Murray, and M.H. Wright, Practical Optimization, Academic Press: San Diego, 1981.Google Scholar
  14. 14.
    W. Glunt, T.L. Hayden, and M. Raydan, “Molecular conformations from distances matrices,” J. Comput. Chem., vol. 14, pp. 114–120, 1993.Google Scholar
  15. 15.
    W. Glunt, T.L. Hayden, and M. Raydan, “Preconditioners for distance matrix algorithms,” J. Comput. Chem., vol. 15, pp. 227–232, 1994.Google Scholar
  16. 16.
    L. Grippo, F. Lampariello, and S. Lucidi, “Anonmonotone line search technique for Newton's method,” SIAM J. Numer. Anal., vol. 23, pp. 707–716, 1986.Google Scholar
  17. 17.
    L. Grippo, F. Lampariello, and S. Lucidi, “A class of nonmonotone stabilization methods in unconstrained optimization,” Numer. Math., vol, 59, pp. 779–805, 1991.Google Scholar
  18. 18.
    D.C. Liu and J. Nocedal, “On the limited-memory BFGS method for large scale optimization,” Math. Programming, vol. 45, pp. 503–528, 1989.Google Scholar
  19. 19.
    W. Liu and Y. H. Dai, “Minimization algorithms based on supervisor and searcher cooperation,” J. Optimization Theory and Applications, vol. 111, pp. 359–379, 1989.Google Scholar
  20. 20.
    B. Molina and M. Raydan, “Preconditioned Barzilai-Borwein method for the numerical solution of partial differential equations,” Numerical Algorithms, vol. 13, pp. 45–60, 1996.Google Scholar
  21. 21.
    J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press: San Diego, 1970.Google Scholar
  22. 22.
    M. Raydan, “On the Barzilai and Borwein choice of the steplength for the gradient method,” IMA J. Numer. Anal., vol. 13, pp. 618–622, 1993.Google Scholar
  23. 23.
    M. Raydan, “The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem,” SIAM J. Optim., vol. 7, pp. 26–33, 1997.Google Scholar
  24. 24.
    M. Raydan and B.F. Svaiter, “Relaxed steepest descent and Chauchy-Barzilai-Borwein method,” Computational Optimization and Applications, vol. 21, pp. 155–167, 2002.Google Scholar
  25. 25.
    D.F. Shanno and K.H. Phua, “Matrix conditioning and nonlinear optimization,” Math. Programming, vol. 14, pp. 149–160, 1978.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • L. Grippo
    • 1
  • M. Sciandrone
    • 2
  1. 1.Dipartimento di Informatica e SistemisticaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.Istituto di Analisi dei Sistemi ed Informatica del CNRRomaItaly

Personalised recommendations