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An efficient adaptive scaling parameter for the spectral conjugate gradient method

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Abstract

This paper deals with the choice of the scaling parameter in the spectral conjugate gradient (SCG) method proposed by Birgin and Martínez (in Appl Math Optim 43:117–128, 2001). Theoretical analyses show that the scaling parameter selection not only influences the numerical stability, but also plays an important role in ensuring the descent property of the SCG method. Based on these analyses, an adaptive scaling parameter is proposed to overcome the drawback of the original choice. Global convergence of the SCG method with our new parameter is established for both convex and nonconvex objective functions. Numerical results on CUTEr problems indicate that the proposed scaling parameter is very efficient and promising.

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References

  1. Andrei, N.: Scaled conjugate gradient algorithms for unconstrained optimization. Comput. Optim. Appl. 38, 401–416 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Babaie-Kafaki, S., Reza, G.: A descent family of Dai-Liao conjugate gradient methods. Optim. Methods Softw. 29, 583–591 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barzilai, J., Borwein, J.M.: Two point step size gradient method. IMA J. Numer. Anal. 8, 141–148 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Birgin, E.G., Martínez, J.M.: A spectral conjugate gradient method for unconstrained optimization. Appl. Math. Optim. 43, 117–128 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Progr. 91(2), 201–213 (2002)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Hager, W.W., Zhang, H.: Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent. ACM Trans. Math. Softw. 32, 113–137 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Powell, M.J.D.: Restart procedures of the conjugate gradient method. Math. Program. 12, 241–254 (1977)

    Article  MATH  Google Scholar 

  12. Raydan, M.: The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7, 26–33 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Shanno, D.F., Phua, K.H.: Minimization of unconstrained multivariate functions. ACM Trans. Math. Softw. 2, 87–94 (1976)

    Article  MATH  Google Scholar 

  14. Yu, G.H., Guan, L.T., Chen, W.F.: Spectral conjugate gradient methods with sufficient descent property for large-scale unconstrained optimization. Optim. Methods Softw. 23, 275–293 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

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Acknowledgments

This research was supported by the Natural Science Foundation of China (Grant No. 71272086). The authors are grateful to Prof. E.G. Birgin for the SCG code and Profs. W.W. Hager and H. Zhang for the CG_DESCENT code. We also thank the two anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. School of Economics and Business Administration, Chongqing University, Chongqing, 400044, China

    Yang Zhang & Bin Dan

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  1. Yang Zhang
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  2. Bin Dan
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Correspondence to Bin Dan.

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Zhang, Y., Dan, B. An efficient adaptive scaling parameter for the spectral conjugate gradient method. Optim Lett 10, 119–136 (2016). https://doi.org/10.1007/s11590-015-0865-8

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  • DOI: https://doi.org/10.1007/s11590-015-0865-8

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