Advertisement

Afrika Matematika

, Volume 30, Issue 3–4, pp 597–609 | Cite as

An efficient conjugate gradient trust-region approach for systems of nonlinear equation

  • Farzad RahpeymaiiEmail author
Article
  • 38 Downloads

Abstract

In this paper, we introduce a combination of family of some conjugate gradient methods (CG) with the trust-region method. Whenever the trust-region algorithm is unsuccessful, a family of CG methods is used to prevent resolving the trust-region subproblem. The computational cost for such a family is trivial. The global theory of the new approach is proved and numerical experiments are reported.

Keywords

Nonlinear equations Trust-region framework Conjugate gradient Global theory Derivative-free optimization 

Mathematics Subject Classification

90C30 93E24 34A34 

Notes

References

  1. 1.
    Ahookhosh, M., Amini, K.: A nonmonotone trust-region method with adaptive radius for unconstrained optimization. Comput. Math. Appl. 60, 411–422 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ahookhosh, M., Amini, K., Peyghami, M.R.: A nonmonotone trust-region line search method for large-scale unconstrained optimization. Appl. Math. Modell. 36, 478–487 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Amini, K., Shiker Mushtak, A.K., Kimiaei, M.: A line search trust-region algorithm with nonmonotone adaptive radius for a system of nonlinear equations. 4OR-Q. J. Oper. Res. 4(2), 132–152 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bellavia, S., Macconi, M., Morini, B.: STRSCNE: a scaled trust-region solver for constrained nonlinear equations. Comput. Optim. Appl. 28, 31–50 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bouaricha, A., Schnabel, R.B.: Tensor methods for large sparse systems of nonlinear equations. Math. Program. 82, 377–400 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Broyden, C.G.: The convergence of an algorithm for solving sparse nonlinear systems. Math. Comput. 25(114), 285–294 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Conn, A.R., Gould, N.I.M., Toint, PhL: Trust-Region Methods. Society for Industrial and Applied Mathematics SIAM, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  8. 8.
    Dai, Y.H., Yuan, Y.: A nonlinear conjugate gradient method with a strong global convergence property. SIAM J. Optim. 10, 177–182 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Esmaeili, H., Kimiaei, M.: A new adaptive trust-region method for system of nonlinear equations. Appl. Math. Model. 38(11–12), 3003–3015 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fan, J.Y.: Convergence rate of the trust region method for nonlinear equations under local error bound condition. Comput. Optim. Appl. 34, 215–227 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fan, J.Y.: An improved trust region algorithm for nonlinear equations. Comput. Optim. Appl. 48(1), 59–70 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fan, J.Y., Pan, J.Y.: A modified trust region algorithm for nonlinear equations with new updating rule of trust region radius. Int. J. Comput. Math. 87(14), 3186–3195 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fasano, G., Lampariello, F., Sciandrone, M.: A truncated nonmonotone Gauss-Newton method for large-scale nonlinear least-squares problems. Comput. Optim. Appl. 34(3), 343–358 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fletcher, R., Reeves, C.: Function minimization by conjugate gradients. Comput. J. 7, 149–154 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grippo, L., Sciandrone, M.: Nonmonotone derivative-free methods for nonlinear equations. Comput. Optim. Appl. 37, 297–328 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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
  18. 18.
    Hestenes, M.R., Stiefel, E.L.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49, 409–436 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kimiaei, M., Rahpeymaii, F.: A new nonmonotone line-search trust-region approach for nonlinear systems. TOP (2019).  https://doi.org/10.1007/s11750-019-00497-2 Google Scholar
  20. 20.
    La Cruz, W., Venezuela, C., Martínez, J.M., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments, Technical Report RT–04–08, July (2004)Google Scholar
  21. 21.
    Li, D.H., Fukushima, M.: A derivative-free line search and global convergence of Broyden-like method for nonlinear equations. Optim. Methods Softw. 13, 181–201 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lukšan, L., Vlček, J.: Sparse and partially separable test problems for unconstrained and equality constrained optimization, Technical Report. No. 767, (1999)Google Scholar
  23. 23.
    Moré, J.J., Garbow, B.S., Hillström, K.E.: Testing unconstrained optimization software. ACM Trans. Math. Softw. 7, 17–41 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (2006)zbMATHGoogle Scholar
  25. 25.
    Toint, PhL: Numerical solution of large sets of algebraic nonlinear equations. Math. Comput. 46(173), 175–189 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yuan, G., Lu, S., Wei, Z.: A new trust-region method with line search for solving symmetric nonlinear equations. Int. J. Comput. Math. 88(10), 2109–2123 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yuan, G.L., Wei, Z.X., Lu, X.W.: A BFGS trust-region method for nonlinear equations. Computing. 92(4), 317–333 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Yuan, Y.: Trust region algorithm for nonlinear equations. Information. 1, 7–21 (1998)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Zhang, J., Wang, Y.: A new trust region method for nonlinear equations. Math. Methods Oper. Res. 58, 283–298 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsPayame Noor UniversityTehranIran

Personalised recommendations