Skip to main content
SpringerLink
Log in

Adaptive nonmonotone line search method for unconstrained optimization

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

In this paper, an adaptive nonmonotone line search method for unconstrained minimization problems is proposed. At every iteration, the new algorithm selects only one of the two directions: a Newton-type direction and a negative curvature direction, to perform the line search. The nonmonotone technique is included in the backtracking line search when the Newton-type direction is the search direction. Furthermore, if the negative curvature direction is the search direction, we increase the steplength under certain conditions. The global convergence to a stationary point with second-order optimality conditions is established. Some numerical results which show the efficiency of the new algorithm are reported.

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. Bunch J R, Parlett B N. Direct methods for solving symmetric indefinite systems of linear equations. SIAM J Nume Anal, 1971, 8: 639–655

    Article  MathSciNet  Google Scholar 

  2. Deng N Y, Xiao Y, Zhou F Y. A nonmonotonic trust region algorithm. Journal of Optimization Theory and Applications, 1993, 76: 259–285

    Article  MATH  MathSciNet  Google Scholar 

  3. Ferris M C, Lucidi S, Roma M. Nonmonotone curvilinear line search methods for unconstrained optimization. Computational Optimization and Applications, 1996, 6: 117–136

    MATH  MathSciNet  Google Scholar 

  4. Fu J, Sun W. Nonmonotone adaptive trust-region method for unconstrained optimization problems. Applied Mathematics and Computation, 2005, 163: 489–504

    Article  MATH  MathSciNet  Google Scholar 

  5. Grippo L, Lamparillo F, Lucidi S. A nonmonotone line search technique for Newton’s method. SIAM J Numer Anal, 1986, 23: 707–716

    Article  MATH  MathSciNet  Google Scholar 

  6. Gould N I M, Lucidi S, Roma M, et al. Exploiting negative curvature directions in linesearch methods for unconstrained optimization. Optimization Methods and Software, 2000, 14: 75–98

    Article  MATH  MathSciNet  Google Scholar 

  7. Lucidi S, Rochetich F, Roma M. Curvilinear stabilization techniques for truncate Newton methods in large scale unconstrained optimization. SIAM J Optimization, 1998, 3: 916–939

    Article  MathSciNet  Google Scholar 

  8. Lampariello F, Sciandrone M. Use of the minimum-norm search direction in a nonmonotone version of the Gauss-Newton method. Journal of Optimization Theory and Applications, 2003, 119: 65–82

    Article  MATH  MathSciNet  Google Scholar 

  9. McCormick G P. A modification of Armijo’s step-size rule for negative curvature. Mathematical Programming, 1977, 13: 111–115

    Article  MATH  MathSciNet  Google Scholar 

  10. Moré J J, Garbow B S, Hillstrom K E. Testing unconstrained optimization software. ACM Trans Math Software, 1981, 7: 17–41

    Article  MATH  MathSciNet  Google Scholar 

  11. Moré J J, Sorensen D C. On the use of directions of negative curvature in a modified Newton method. Mathematical Programming, 1979, 16: 1–20

    Article  MATH  MathSciNet  Google Scholar 

  12. Nocedal J, Wright S J. Numerical Optimization. New York: Springer, 1999

    MATH  Google Scholar 

  13. Sun W. Nonmonotone optimization methods: motivation and development. In: International Conference on Numerical Linear Algebra and Optimization, Guilin, China, October 7–10, 2003 (Invited talk)

  14. Sun W. Nonmonotone trust-region method for solving optimization problems. Applied Mathematics and Computation, 2004, 156: 159–174

    Article  MATH  MathSciNet  Google Scholar 

  15. Sun W, Han J, Sun J. On global convergence of nonmonotone decent method. J Comput Appl Math, 2002, 146: 89–98

    Article  MATH  MathSciNet  Google Scholar 

  16. Yuan Y, Sun W. Optimization Theory and Methods. Beijing: Science Press, 1997

    Google Scholar 

  17. Zhou H, Sun W. Nonmonotone descent algorithm for nonsmooth unconstrained optimization problems. International Journal of Pure and Applied Mathematics, 2003, 9: 153–163

    MATH  MathSciNet  Google Scholar 

  18. Zhou Q, Sun W. A nonmonotone second order steplength method for unconstrained optimization. Journal of Computational Mathematics, 2007, 23: 104–112

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Nanjing Normal University, Nanjing, 210097, China

    Qunyan Zhou & Wenyu Sun

  2. Department of Basic Science, Jiangsu Teacher University of Technology, Changzhou, 213001, China

    Qunyan Zhou

Authors
  1. Qunyan Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenyu Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenyu Sun.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, Q., Sun, W. Adaptive nonmonotone line search method for unconstrained optimization. Front. Math. China 3, 133–148 (2008). https://doi.org/10.1007/s11464-008-0001-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-008-0001-5

Keywords

MSC

Search

Navigation