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A short note on the global convergence of the unmodified PRP method

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Abstract

To guarantee global convergence of the standard (unmodified) PRP nonlinear conjugate gradient method for unconstrained optimization, the exact line search or some Armijo type line searches which force the PRP method to generate descent directions have been adopted. In this short note, we propose a non-descent PRP method in another way. We prove that the unmodified PRP method converges globally even for nonconvex minimization by the use of an approximate descent inexact line search.

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References

  1. Dai, Y.: Nonlinear conjugate gradient methods. Wiley Encyclopedia of Operations Research and Management Science (2011)

  2. Dai Y.: Conjugate gradient methods with Armijo-type line searches. Acta Math. Appl. Sin. -Engl. Ser. 18, 123–130 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dai Y., Yuan Y.: Nonlinear conjugate gradient methods. Shanghai Science and Technology Publisher, Shanghai (2000)

    Google Scholar 

  4. Gilbert J.C., Nocedal J.: Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optim. 2, 21–42 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grippo L., Lucidi S.: A globally convergent version of the Polak-Ribière conjugate gradient method. Math. Program. 78, 375–391 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Hager W.W., Zhang H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2, 35–58 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Li D., Fukushima M.: A globally and superlinearly convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations. SIAM J. Numer. Anal. 37, 152–172 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Polak E., Ribière G.: Note sur la convergence de méthodes de directions conjuguées. Rev. Fr. Inform. Rech. Oper. 16, 35–43 (1969)

    Google Scholar 

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

    Article  Google Scholar 

  10. Zhang L., Zhou W., Li D.: A descent modified Polak–Ribière–Polyak conjugate gradient method and its global convergence. IMA J. Numer. Anal. 26, 629–640 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Changsha University of Science and Technology, Changsha, 410004, China

    Weijun Zhou

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  1. Weijun Zhou
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Correspondence to Weijun Zhou.

Additional information

This work was supported by the NSF foundation (10901026) of China, The Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities and the Key Project of the Scientific Research Fund of the Hunan Provincial Education Department.

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Zhou, W. A short note on the global convergence of the unmodified PRP method. Optim Lett 7, 1367–1372 (2013). https://doi.org/10.1007/s11590-012-0511-7

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  • DOI: https://doi.org/10.1007/s11590-012-0511-7

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