Journal of Applied Mathematics and Computing

, Volume 23, Issue 1–2, pp 311–319 | Cite as

Global convergence properties of two modified BFGS-type methods

  • Qiang Guo
  • Jian-Guo Liu


This article studies a modified BFGS algorithm for solving smooth unconstrained strongly convex minimization problem. The modified BFGS method is based on the new quasi-Newton equation Bk+1sk=yk where yk*, =yk + Aksk andA k is a matrix. Wei, Li and Qi [WLQ] have proven that the average performance of two of those algorithms is better than that of the classical one. In this paper, we prove the global convergence of these algorithms associated to a general line search rule.

AMS Mathematics Subject Classification

65H10 65F10 

Key words and phrases

Unconstrained programme BFGS algorithm global convergence quasi-Newton method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. J. Dennis and J. J. Moré,Quasi-Newton methods, motivation and theory, SIAM REVIEW19 (1977) 46–89.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M. J. D. Powell.Some global convergence properties of a variable metric algorithm for minimization without exact line searches, InNonlinear programming (Proc. Sympos., New York, 1975), pages 53C72. SIAMCAMS Proc., Vol. IX. Amer. Math. Soc., Providence, R. I., 1976.Google Scholar
  3. 3.
    J. Werner,Uber die global konvergenze von variablemetric verfahren mit nichtexakter schrittweitenbestimmong, Numer. Math.31 (1978) 321–334.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. D. Pearson,Variable Metric Methods of Minimization, Computer J.12 (1969) 171–178.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J. Y. Han and G. H. Liu,General Form of Stepsize Selection Rules of Linesearch and Relevant Analysis of Global Convergence of BFGS Algorithm (Chinese), Acta Mathematicae Applicatae Sinica1 (1995) 112–122.MathSciNetGoogle Scholar
  6. 6.
    J. M. Ortega and W. C. Rheinboldt,Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970.MATHGoogle Scholar
  7. 7.
    R. H. Byrd and J. Nocedal,A Tool for the Analysis of Quasi-Newton Methods with Application to Unconstrained Minimization, SIAM J. Numer. Anal3 (1989) 727–739.CrossRefMathSciNetGoogle Scholar
  8. 8.
    L. Armijo,Minimization of functions having lipschitz-continuous first partial derivatives, Pacific J. of Mathematics16 1–3.Google Scholar
  9. 9.
    A. Goldstein and J. Price,An effective algorithm for minimization, Numerische Mathematik10 (1967) 184–189.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    P. Wolfe,Convergence conditions for ascent methods, SIAM Review11 (1969) 226–235.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    W. Warth and J. Werner,Effiziente schrittweiten-funktionen bei unrestringierten optimierungsaufgaben, Computing19 (1977) 59–72.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    R. De Leone, M. Gaudioso and L. Grippo,Stopping criteria for linesearch methods without derivative, Mathematical programming30 (1984) 285–300.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    N. I. Djiranovic-Milicic,On a modification of a stepsize algorithm, Eur. J. of Operational Research31 (1987) 66–70.CrossRefGoogle Scholar
  14. 14.
    J. M. Gorge, S. G. Burton and E. H. Kenneth,Testing unconstrained optimization software, ACM Trans. Math. Software7 (1981) 208–215.Google Scholar
  15. 15.
    J. -G. Liu and Q. Guo,Global Convergence Properties of the Modified BFGS Method, J. of Appl. Math. & Computing16 (2004) 195–205.MATHMathSciNetGoogle Scholar
  16. 16.
    J. -G. Liu, Z. -Q. Xia, R. -D. Ge and Q. Guo,An modified BFGS method for non-convex minimization problems (Chinese), Operation Research and Management13 (2004) 62–65.Google Scholar
  17. 17.
    Z. X. Wei, G. H. Yu, G. L. Yuan and Z. G. Lian,The superlinear convergence of a modifed BFGS-Type method for unconstrained optimization, Computational optimization and applications29 (2004) 315–332.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Z. Wei, G. Li and L. Qi,New quasi-Newton methods for unconstrained optimization problems. Math. Program. to appear.Google Scholar
  19. 19.
    Q. Guo and J. -G. Liu,Global convergence of a modified BFGS-type method for unconstrained non-convex minimization, J. of Appl. Math. & Computing21 (2006) 259–267.MATHMathSciNetGoogle Scholar
  20. 20.
    Z. Wei, L. Qi and X. Chen,An SQP-type method and its application in stochastic programming, J. Optim. Theory Appl.116 (2003) 205–228.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    R. -D. Ge and Z. Q. Xia,An ABS Algorithm for Solving Singular Nonlinear System with Rank One Defect, J. Appl. Math.and Computing(old:KJCAM)9 (2002) 167–184.MATHMathSciNetGoogle Scholar
  22. 22.
    R. -D. Ge and Z. -Q. Xia,An ABS Algorithm for Solving Singular Nonlinear System with Rank Defects, J. of Appl. Math. & Computing5 (2003) 1–20.MathSciNetCrossRefGoogle Scholar

Copyright information

© Korean Society for Computational & Applied Mathematics and Korean SIGCAM 2007

Authors and Affiliations

  1. 1.Science College of Dalian Nationalities UniversityDalianP.R. China
  2. 2.Institute of System EngineeringDalian University of TechnologyDalianP. R. China

Personalised recommendations