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An improved nonlinear conjugate gradient method with an optimal property

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Abstract

Conjugate gradient methods have played a special role in solving large scale nonlinear problems. Recently, the author and Dai proposed an efficient nonlinear conjugate gradient method called CGOPT, through seeking the conjugate gradient direction closest to the direction of the scaled memoryless BFGS method. In this paper, we make use of two types of modified secant equations to improve CGOPT method. Under some assumptions, the improved methods are showed to be globally convergent. Numerical results are also reported.

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References

  1. Dai Y H. Convergence properties of the BFGS algorithm. SIAM J Optim, 2002, 13: 693–701

    Article  MATH  MathSciNet  Google Scholar 

  2. Dai Y H, Kou C X. A nonlinear conjugate gradient algorithm with an optimal property and an improved wolfe line search. SIAM J Optim, 2013, 23: 296–320

    Article  MATH  MathSciNet  Google Scholar 

  3. Dai Y H, Liao L Z. New conjugate conditions and related nonlinear conjugate gradient methods. Appl Math Optim, 2001, 43: 87–101

    Article  MATH  MathSciNet  Google Scholar 

  4. Dai Y H, Yuan Y. A nonlinear conjugate gradient method with a strong global convergence property. SIAM J Optim, 1999, 10: 177–182

    Article  MATH  MathSciNet  Google Scholar 

  5. Dolan E D, Moré J J. Benchmarking optimization software with performance profiles. Math Program, 2002, 91: 201–213

    Article  MATH  MathSciNet  Google Scholar 

  6. Fletcher R, Reeves C. Function minimization by conjugate gradients. Comput J, 1964, 7: 149–154

    Article  MATH  MathSciNet  Google Scholar 

  7. Ford J A, Moghrabi I A. Alternative parameter choices for multi-step quasi-Newton methods. Optim Methods Softw, 1993, 2: 357–370

    Article  Google Scholar 

  8. Ford J A, Moghrabi I A. Multi-step quasi-Newton methods for optimization. J Comput Appl Math, 1994, 50: 305–323

    Article  MATH  MathSciNet  Google Scholar 

  9. Gilbert J G, Nocedal J. Global convergence properties of conjugate gradient methods for optimization. SIAM J Optim, 1992, 21: 21–42

    Article  MathSciNet  Google Scholar 

  10. Gould N I M, Orban D, Toint Ph L. CUTEr (and SifDec), a constrained and unconstrained testing environment, revisited. Technical Report TR/PA/01/04. CERFACS, Toulouse, France, 2001

    Google Scholar 

  11. Hager W W, Zhang H. A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J Optim, 2005, 16: 170–192

    Article  MATH  MathSciNet  Google Scholar 

  12. Hestenes M R, Stiefel E L. Methods of conjugate gradients for solving linear systems. J Research Nat Bur Standards, 1952, 49: 409–436

    Article  MATH  MathSciNet  Google Scholar 

  13. Kou C X. On improved Wolfe line search. Research Report. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 2011

    Google Scholar 

  14. Li D, Fukushima M. A modified BFGS method and its global convergence in nonconvex minimization. J Comput Appl Math, 2001, 129: 15–35

    Article  MATH  MathSciNet  Google Scholar 

  15. Li G Y, Tang C M, Wei Z X. New conjugacy condition and related new conjugate gradient methods for unconstrained optimization. J Comput Appl Math, 2007, 202: 523–539

    Article  MATH  MathSciNet  Google Scholar 

  16. Perry A. A modified conjugate gradient algorithm. Oper Res, 1978, 26: 1073–1078

    Article  MATH  MathSciNet  Google Scholar 

  17. Polak E, Ribière G. Note sur la convergence de méthodes de directions conjugées. Rev Francaise Informat Recherche Opértionelle, 1969, 3: 35–43

    MATH  Google Scholar 

  18. Polyak B T. The conjugate gradient method in extreme problems. USSR Comput Math and Math Phys, 1969, 9: 94–112

    Article  Google Scholar 

  19. Saman B K, Reza G, Nezam M A. Two new conjugate gradient methods based on modified secant equations. J Comput Appl Math, 2010, 234: 1374–1386

    Article  MATH  MathSciNet  Google Scholar 

  20. Wei Z, Li G, Qi L. New quasi-Newton methods for unconstrained optimization problems. Appl Math Comput, 2006, 175: 1156–1188

    Article  MATH  MathSciNet  Google Scholar 

  21. Yabe H, Takano M. Global convergence properties of nonlinear conjugate gradient methods with modified secant condition. Comput Optim Appl, 2004, 28: 203–225

    Article  MATH  MathSciNet  Google Scholar 

  22. Yuan Y X. A modified BFGS algorithm for unconstrained optimization. IMA J Numer Anal, 1991, 11: 325–332

    Article  MATH  MathSciNet  Google Scholar 

  23. Yuan Y X, Byrd R H. Non-quasi-Newton updates for unconstrained optimization. J Comput Math, 1995, 13: 95–107

    MATH  MathSciNet  Google Scholar 

  24. Zhang J Z, Deng N Y, Chen L H. New quasi-Newton equation and related methods for unconstrained optimization. J Optim Theory Appl, 1999, 102: 147–167

    Article  MATH  MathSciNet  Google Scholar 

  25. Zhang J Z, Xu C X. Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations. J Comput Appl Math, 2001, 137: 269–278

    Article  MATH  MathSciNet  Google Scholar 

  26. Zhou W, Zhang L. A nonlinear conjugate gradient method based on the MBFGS secant condition. Optim Methods Softw, 2006, 21: 707–714

    Article  MATH  MathSciNet  Google Scholar 

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  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    CaiXia Kou

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  1. CaiXia Kou
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Kou, C. An improved nonlinear conjugate gradient method with an optimal property. Sci. China Math. 57, 635–648 (2014). https://doi.org/10.1007/s11425-013-4682-1

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  • DOI: https://doi.org/10.1007/s11425-013-4682-1

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