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A modified descent Polak–Ribiére–Polyak conjugate gradient method with global convergence property for nonconvex functions

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Abstract

Following the modification scheme of Dong et al. made on the Hestenes–Stiefel method, we suggest a modified Polak–Ribiére–Polyak technique which satisfies the sufficient descent condition. We show that the method is globally convergent with the Wolfe line search conditions as well as the backtracking Armijo-type line search strategy proposed by Grippo and Lucidi, without convexity assumption on the objective function. Numerical experiments on some test functions of the CUTEr collection show that the method performs promisingly.

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References

  1. Andrei, N.: Numerical comparison of conjugate gradient algorithms for unconstrained optimization. Stud. Inform. Control 16(4), 333–352 (2007)

    Google Scholar 

  2. Andrei, N.: A modified Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization. Optimization 60(12), 1457–1471 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babaie-Kafaki, S., Ghanbari, R.: A descent extension of the Polak–Ribière–Polyak conjugate gradient method. Comput. Math. Appl. 68(12), 2005–2011 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babaie-Kafaki, S., Ghanbari, R.: An optimal extension of the Polak–Ribière–Polyak conjugate gradient method. Numer. Funct. Anal. Optim. 38(9), 1115–1124 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dai, Y.H.: Analyses of conjugate gradient methods. In: Ph.D. Thesis. Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences (1997)

  6. Dai, Y.H., Han, J.Y., Liu, G.H., Sun, D.F., Yin, H.X., Yuan, Y.X.: Convergence properties of nonlinear conjugate gradient methods. SIAM J. Optim. 10(2), 348–358 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Dai, Y.H., Liao, L.Z.: New conjugacy conditions and related nonlinear conjugate gradient methods. Appl. Math. Optim. 43(1), 87–101 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dai, Z.: Two modified Polak–Ribière–Polyak-type nonlinear conjugate methods with sufficient descent property. Numer. Funct. Anal. Optim. 31(8), 892–906 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dai, Z.F., Wen, F.: Some improved sparse and stable portfolio optimization problems. Financ. Res. Lett. 27, 46–52 (2018)

    Article  Google Scholar 

  10. Dai, Z.F., Wen, F.: Two nonparametric approaches to mean absolute deviation portfolio selection model. J. Ind. Manag. Optim. (2019), (Accepted)

  11. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2, Ser. A), 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dong, X.L., Liu, H.W., He, Y.B., Babaie-Kafaki, S., Ghanbari, R.: A new three-term conjugate gradient method with descent direction for unconstrained optimization. Math. Model. Anal. 21(3), 399–411 (2016)

    Article  MathSciNet  Google Scholar 

  13. Dong, X.L., Liu, H.W., He, Y.B., Yang, X.M.: A modified Hestenes–Stiefel conjugate gradient method with sufficient descent condition and conjugacy condition. J. Comput. Appl. Math. 281(1), 239–249 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gilbert, J.C., Nocedal, J.: Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optim. 2(1), 21–42 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gould, N.I.M., Orban, D., Toint, PhL: CUTEr: a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Software 29(4), 373–394 (2003)

    Article  MATH  Google Scholar 

  16. Grippo, L., Lucidi, S.: A globally convergent version of the Polak–Ribière gradient method. Math. Program. 78(3), 375–391 (1997)

    Article  MATH  Google Scholar 

  17. Hager, W.W., Zhang, H.: Algorithm 851: \(\text{ CG }_{-}\)Descent, a conjugate gradient method with guaranteed descent. ACM Trans. Math. Software 32(1), 113–137 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Hager, W.W., Zhang, H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2(1), 35–58 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49(6), 409–436 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (2006)

    MATH  Google Scholar 

  21. Polak, E., Ribière, G.: Note sur la convergence de méthodes de directions conjuguées. Rev. Française Informat. Recherche Opérationnelle 3(16), 35–43 (1969)

    MathSciNet  MATH  Google Scholar 

  22. Polyak, B.T.: The conjugate gradient method in extreme problems. USSR Comput. Math. Math. Phys. 9(4), 94–112 (1969)

    Article  MATH  Google Scholar 

  23. Sun, W., Yuan, Y.X.: Optimization Theory and Methods: Nonlinear Programming. Springer, New York (2006)

    MATH  Google Scholar 

  24. Yu, G., Guan, L., Li, G.: Global convergence of modified Polak–Ribière–Polyak conjugate gradient methods with sufficient descent property. J. Ind. Manag. Optim. 4(3), 565–579 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yu, G.H.: Nonlinear Self-Scaling Conjugate Gradient Methods for Large-Scale Optimization Problems, Ph.D. Thesis. Sun Yat–Sen University (2007)

  26. Yuan, G.L.: Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems. Optim. Lett. 3(1), 11–21 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, L., Zhou, W., Li, D.H.: A descent modified Polak–Ribière–Polyak conjugate gradient method and its global convergence. IMA J. Numer. Anal. 26(4), 629–640 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zoutendijk, G.: Nonlinear programming, computational methods. In: Abadie, J. (ed.) Integer and Nonlinear Programming, pp. 37–86. North-Holland, Amsterdam (1970)

    MATH  Google Scholar 

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Acknowledgements

This research was in part supported by the grant 97022259 from Iran National Science Foundation (INSF), and in part by the Research Council of Semnan University. The authors thank the anonymous reviewers for their valuable comments and suggestions helped to improve the quality of this work. They are also grateful to Professor Michael Navon for providing the line search code.

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  1. Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, P.O. Box: 35195–363, Semnan, Iran

    Zohre Aminifard & Saman Babaie-Kafaki

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  1. Zohre Aminifard
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  2. Saman Babaie-Kafaki
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Correspondence to Saman Babaie-Kafaki.

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Aminifard, Z., Babaie-Kafaki, S. A modified descent Polak–Ribiére–Polyak conjugate gradient method with global convergence property for nonconvex functions. Calcolo 56, 16 (2019). https://doi.org/10.1007/s10092-019-0312-9

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