Numerical Algorithms

, Volume 77, Issue 4, pp 1273–1282 | Cite as

A Dai-Liao conjugate gradient algorithm with clustering of eigenvalues

  • Neculai Andrei
Original Paper


A new value for the parameter in Dai and Liao conjugate gradient algorithm is presented. This is based on the clustering of eigenvalues of the matrix which determine the search direction of this algorithm. This value of the parameter lead us to a variant of the Dai and Liao algorithm which is more efficient and more robust than the variants of the same algorithm based on minimizing the condition number of the matrix associated to the search direction. Global convergence of this variant of the algorithm is briefly discussed.


Unconstrained optimization Conjugate gradient algorithms Eigenvalues clustering Condition number Wolfe conditions Convergence 


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  1. 1.
    Andrei, N.: Acceleration of conjugate gradient algorithms for unconstrained optimization. Appl. Math. Comput. 213, 361–369 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Andrei, N.: An unconstrained optimization test functions collection. Adv. Model. Optim. 10, 147–161 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Andrei, N.: Open problems in nonlinear conjugate gradient algorithms for unconstrained optimization. Bull. Malays. Math. Sci. Soc. 34, 319–330 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Axelsson, O.: A class of iterative methods for finite element equations. Comput. Methods Appl. Mech. Eng. 9, 123–137 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Axelsson, O., Lindskog, G.: On the rate of convergence of the preconditioned conjugate gradient methods. Numer. Math. 48, 499–523 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Babaie-Kafaki, S., Ghanbari, R.: The Dai-Liao nonlinear conjugate gradient method with optimal parameter choices. Eur. J. Oper. Res. 234, 625–630 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bongartz, I., Conn, A.R., Gould, N.I.M., Toint, Ph.L.: CUTEr: constrained and unconstrained testing environments. ACM Trans. Math. Softw. 21, 123–160 (1995)CrossRefzbMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dai, Y.H., Liao, L.Z.: New conjugate conditions and related nonlinear conjugate gradient methods. Appl. Math. Optim. 43, 87–101 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hager, W.W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16, 170–192 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hestenes, M.R., Steifel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. Sec. B 48, 409–436 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kaporin, I.E.: New convergence results and preconditioning strategies for the conjugate gradient methods. Numer. Linear Algebra Appl. 1, 179–210 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kratzer, D., Parter, S.V., Steuerwalt, M.: Block splittings for the conjugate gradient method. Comput. Fluids 11, 255–279 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Meurant, G.: Computer solution of large linear systems. Studies in Mathematics and its Applications, vol 28. North Holland, Elsevier, Amsterdam (1999)Google Scholar
  16. 16.
    Pestana, J., Wathen, A.J.: On the choice of preconditioner for minimum residual methods for non-Hermitian matrices. J. Comput. Appl. Math. 249, 57–68 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Reid, J.K.: On the method of conjugate gradients for solution of large sparse systems of linear equations. In: Reid, J.K. (ed.) Large Sparse Sets of Linear Equations, pp 231–254. Academic Press, London (1971)Google Scholar
  18. 18.
    Strakoš, Z.: On the real convergence rate of the conjugate gradient method. Linear Algebra Appl. 154–156, 535–549 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sun, W., Yuan, Y.X.: Optimization theory and methods. Nonlinear Programming. Springer Science + Business Media, New York (2006)zbMATHGoogle Scholar
  20. 20.
    Van der Sluis, A., Van der Vorst, H.A.: The rate of convergence of conjugate gradients. Numer. Math. 48, 543–560 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Winther, R.: Some superlinear convergence results for the conjugate gradient method. SIAM J. Numer. Anal. 17, 14–17 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Research Institute for InformaticsCenter for Advanced Modeling and OptimizationBucharest 1Romania
  2. 2.Academy of Romanian ScientistsSector 5. BucharestRomania

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