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On Modified DFP Update for Unconstrained Optimization

Received: 25 December 2016     Accepted: 9 January 2017     Published: 6 February 2017
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Abstract

In this paper, we propose a new modify of DFP update with a new extended quasi-Newton condition for unconstrained optimization problem so called update. This update is based on a new Zhang Xu condition we show that update preserves the value of determinant of the next Hessian matrix equal to the value of determinant of current Hessian matrix theoretically and practically. Global convergence of the modify is established. Local and super linearly convergence are obtained for the proposed method. Numerical results are given to compare a performance of the modify method with the standard DFP method on same function is selected.

Published in American Journal of Applied Mathematics (Volume 5, Issue 1)
DOI 10.11648/j.ajam.20170501.13
Page(s) 19-30
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Quasi-Newton Equation, the DFP Updating Formula, Global Convergence and Super Linearly Convergence

References
[1] Al-Bayati, A., (1991), A new family of self-scaling variable metric Algorithms for unconstraint optimization, Journal of Educ. and Sci., Iraq, Vol. 12, pp. 25-54.
[2] Dennis J. E., More J., (1974), "A characterization of super linear Convergence and its application to quasi-Newton methods Math and Computation 28 (6) 549-60.
[3] F. Freudenstein and B. Roth,(1962), Numerical solution of system of nonlinear equations, Journal of ACM, Vol. 10, No. 4, pp. 550-556.
[4] H. H. Rosen brock, (1960), An automatic method for finding the greatest least value of a function, Computer Journal, Vol. 3, No. 3, pp. 175-184.
[5] J. E. Dennis, Jr. and Robert B. Schnabel, (1996), Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Classics in Applied Mathematics.
[6] Oren, S. S (1973), Self-scaling Variable Metric Algorithms without line search for Unconstrained minimization. Mathematics of computation, 27: 873-885.
[7] Saad S. Mahmood, (2011), α-BFGS update for Unconstrained Opt-imization, Journal of college of Education, No. 1, Almustansiriya University.
[8] Todd M. J., (1984), Quasi-Newton updates in abstract spaces, SIAM Review, 26: 367-377.
[9] W. Sun and Y. Yuan, (2006), "Optimization Theory and Method: Nonlinear Programing", Vol. 1 of Springer optimization and its Applications, Springer, New York, NY, USA.
[10] Yuan, Y., (1990), On a Modified Algorithm for Unconstrained Optimization, Computing Center, Academia Sinica, Beijing, China.
Cite This Article
  • APA Style

    Saad Shakir Mahmood, Samira Hassan Shnywer. (2017). On Modified DFP Update for Unconstrained Optimization. American Journal of Applied Mathematics, 5(1), 19-30. https://doi.org/10.11648/j.ajam.20170501.13

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    ACS Style

    Saad Shakir Mahmood; Samira Hassan Shnywer. On Modified DFP Update for Unconstrained Optimization. Am. J. Appl. Math. 2017, 5(1), 19-30. doi: 10.11648/j.ajam.20170501.13

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    AMA Style

    Saad Shakir Mahmood, Samira Hassan Shnywer. On Modified DFP Update for Unconstrained Optimization. Am J Appl Math. 2017;5(1):19-30. doi: 10.11648/j.ajam.20170501.13

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  • @article{10.11648/j.ajam.20170501.13,
      author = {Saad Shakir Mahmood and Samira Hassan Shnywer},
      title = {On Modified DFP Update for Unconstrained Optimization},
      journal = {American Journal of Applied Mathematics},
      volume = {5},
      number = {1},
      pages = {19-30},
      doi = {10.11648/j.ajam.20170501.13},
      url = {https://doi.org/10.11648/j.ajam.20170501.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20170501.13},
      abstract = {In this paper, we propose a new modify of DFP update with a new extended quasi-Newton condition for unconstrained optimization problem so called  update. This update is based on a new Zhang Xu condition we show that  update preserves the value of determinant of the next Hessian matrix equal to the value of determinant of current Hessian matrix theoretically and practically. Global convergence of the modify is established. Local and super linearly convergence are obtained for the proposed method. Numerical results are given to compare a performance of the modify  method with the standard DFP method on same function is selected.},
     year = {2017}
    }
    

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  • TY  - JOUR
    T1  - On Modified DFP Update for Unconstrained Optimization
    AU  - Saad Shakir Mahmood
    AU  - Samira Hassan Shnywer
    Y1  - 2017/02/06
    PY  - 2017
    N1  - https://doi.org/10.11648/j.ajam.20170501.13
    DO  - 10.11648/j.ajam.20170501.13
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 19
    EP  - 30
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20170501.13
    AB  - In this paper, we propose a new modify of DFP update with a new extended quasi-Newton condition for unconstrained optimization problem so called  update. This update is based on a new Zhang Xu condition we show that  update preserves the value of determinant of the next Hessian matrix equal to the value of determinant of current Hessian matrix theoretically and practically. Global convergence of the modify is established. Local and super linearly convergence are obtained for the proposed method. Numerical results are given to compare a performance of the modify  method with the standard DFP method on same function is selected.
    VL  - 5
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics, College of Education, Almustansiryah University, Baghdad, Iraq

  • Department of Mathematics, College of Education, Almustansiryah University, Baghdad, Iraq

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