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Two modified conjugate gradient methods for unconstrained optimization with applications in image restoration problems

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Abstract

The conjugate gradient methods (CGMs) are very effective iterative methods for solving unconstrained optimization problems. In this paper, the second inequality of the strong Wolfe line search is used to modify the conjugate parameters of the PRP and HS methods, and thereby two efficient conjugate parameters are presented. Under basic assumptions, we prove that the two modified CGMs satisfy sufficient descent condition and converge globally for unconstrained optimization problems. Finally, to verify the effectiveness of our presented methods, we perform medium-large-scale numerical experiments for the normal unconstrained optimization and image restoration problems. The numerical results show the encouraging efficiency and applicability of the proposed methods.

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References

  1. Hestenes, M.R., Stiefel, E.: Method of conjugate gradient for solving linear equations. J. Res. Natl. Bureau. Stand. 49, 409–436 (1952)

    Article  MATH  Google Scholar 

  2. Fletcher, R., Reeves, C.M.: Function minimization by conjugate gradients. J. Comput. 7, 149–154 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  3. Polak, E., Ribière, G.: Note surla convergence de directions conjugèes. Rev. Fr. Inf. Rech. Oper. 3(1), 35–43 (1969)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Dai, Y.H., Yuan, Y.X.: A nonlinear conjugate gradient with a strong global convergence property. SIAM J. Optim. 10(1), 177–182 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wei, Z.X., Yao, S.W., Liu, L.Y.: The convergence properties of some new conjugate gradient methods. Appl. Math. Comput. 183, 1341–1350 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Yao, S.W., Wei, Z.X., Huang, H.: A note about WYL’s conjugate gradient method and its application. Appl. Math. Comput. 191, 381–388 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Huang, H., Wei, Z.X., Yao, S.W.: The proof of the sufficient descent condition of the Wei-Yao-Liu conjugate gradient method under the strong Wolfe-Powell line search. Appl. Math. Comput. 189, 1241–1245 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Zhang, L.: An imporoved Wei–Yao–Liu nonlinear conjugate gradient method for optimization computation. Appl. Math. Comput. 215, 2269–2274 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Dai, Z.F., Wen, F.H.: Another improved Wei–Yao–Liu nonlinear conjugate gradient method with sufficient descent property. Appl. Math. Comput. 218, 7421–7430 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Yuan, G.L., Zhang, M.J.: A modified Hestenes–Stiefel conjugate gradient algorithm for large-scale optimization. Numer. Func. Anal. Opt. 34(8), 914–937 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, M., Liu, H.W., Liu, Z.X.: A new family of conjugate gradient methods for unconstrained optimization. J. Appl. Math. Comput. 58, 219–234 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jian, J.B., Chen, Q., Jiang, X.Z., Zeng, Y.F., Yin, J.H.: A new spectral conjugate gradient method for large-scale unconstrained optimization. Optim. Methods Softw. 32(3), 503–515 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiang, X.Z., Jian, J.B.: Improved Fletcher–Reeves and Dai–Yuan conjugate gradient methods with the strong Wolfe line search. J. Comput. Appl. Math. 328, 525–534 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lu, J.Y., Yuan, G.L., Wang, Z.: A modified Dai–Liao conjugate gradient method for solving unconstrained optimization and image restoration problems. J. Appl. Math. Comput. https://doi.org/10.1007/s12190-021-01548-3 (2021)

  17. Liu, Y.F., Zhu, Z.B., Zhang, B.X.: Two sufficient descent three-term conjugate gradient methods for unconstrained optimization problems with applications in compressive sensing. J. Appl. Math. Comput. https://doi.org/10.1007/s12190-021-01589-8 (2021)

  18. Andrei, N.: Hybrid conjugate gradient algorithm for unconstrained optimization. J. Optim. Theory Appl. 141, 249–264 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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 

  20. Zoutendijk, G.: Nonlinear programming, computational methods. In: Abadie, J. (Ed.) Integer and nonlinear programming. Amsterdam, North-holland, pp 37–86 (1970)

  21. Morè, J.J., Garbow, B.S.: Hillstrome KE testing unconstrained optimization software. ACM Trans. Math. Softw. 7, 17–41 (1981)

    Article  MATH  Google Scholar 

  22. Bongartz, I., Conn, A.R., Gould, N., Toint, P.L.: CUTE: constrained and unconstrained testing environment. ACM. Trans. Math. Softw. 21, 123–160 (1995)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Chan, R.H., Ho, C.W., Nikolova, M.: Salt-and-pepper noise removal by median-type noise detectors and detail preserving regularization. IEEE Trans. Image Process. 14(10), 1479–1485 (2005)

    Article  Google Scholar 

  25. Cai, J.F., Chan, R., Morini, B.: Minimization of an edge-preserving regularization functional by conjugate gradient type methods, image processing based on partial differential equations, pp. 109–122. Springer, Berlin, Heidelberg (2007)

    Google Scholar 

  26. Yu, G.H., Huang, J.H., Zhou, Y.: A descent spectral conjugate gradient method for impulse noise removal. Appl. Math. Lett. 23(5), 555–560 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Aminifard, Z., Babaie-Kafaki, S.: Dai-Liao extensions of a descent hybrid nonlinear conjugate gradient method with application in signal processing. Numer. Algorithms. https://doi.org/10.1007/s11075-021-01157-y (2021)

  28. Yuan, G.L., Wei, Z.X., Li, G.Y.: A modified Polak–Ribiére–Polyak conjugate gradient algorithm for nonsmooth convex programs. J. Comput. Appl. Math. 255, 86–96 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hwang, H., Haddad, R.A.: Adaptive median filters: new algorithms and results. IEEE Trans. Image Process. 4(4), 499–502 (1995)

    Article  Google Scholar 

  30. Bovik, A.: Handbook of image and video processing. Academic Press, San Diego (2000)

    MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Physics, Guangxi University for Nationalities, Nanning, Guangxi, 530006, China

    Guodong Ma, Hui Lin, Wenhui Jin & Daolan Han

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  1. Guodong Ma
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  2. Hui Lin
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  3. Wenhui Jin
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  4. Daolan Han
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Correspondence to Wenhui Jin.

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This work was supported by the Research Foundation of Guangxi University for Nationalities (2021KJQD04), NSFC (12171106), the Natural Science Foundation of Guangxi Province (2020GXNSFDA238017)

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Ma, G., Lin, H., Jin, W. et al. Two modified conjugate gradient methods for unconstrained optimization with applications in image restoration problems. J. Appl. Math. Comput. 68, 4733–4758 (2022). https://doi.org/10.1007/s12190-022-01725-y

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