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An Efficient Fifth-Order Method for Linear Optimization

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Abstract

In this paper, we present an efficient fifth-order method for solving the standard linear optimization problems. In order to do this, we show that solving this problem is equivalent to solving a system of nonlinear equations. Therefore, we build a sequence of functions giving an approximate solution of this system. To find this approximation, we give an algorithm, which is based on the idea of the Sharma’s method. Computational efficiency, in its general form, is discussed, and a comparison between the efficiency of proposed technique and existing one is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are verified in the examples.

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References

  1. El Ghami, M., Guennoun, Z.A., Boula, S., Steihaug, T.: Interior-point methods for linear optimization based on a kernel function with a trigonometric barrier term. J. Comput. Appl. Math. 236, 3613–3623 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Peyghamia, M.R., Hafshejania, S.F., Shirvanib, L.: Complexity of interior-point methods for linear optimization based on a new trigonometric kernel function. J. Comput. Appl. Math. 255, 74–85 (2014)

    Article  MathSciNet  Google Scholar 

  3. Bai, Y.Q., El Ghami, M., Roos, C.: A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization. SIAM J. Optim. 15, 101–128 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Peng, J., Roos, C., Terlaky, T. (eds.): Self-Regularity, A New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, Princeton (2002)

    MATH  Google Scholar 

  5. Roos, C., Terlaky, T., Vial, J.P. (eds.): Theory and Algorithms for Linear Optimization. An Interior-Point Approach. John Wiley and Sons, Chichester (1997)

    MATH  Google Scholar 

  6. Hertog, D.D. (eds.): Interior Point Approach to Linear, Quadratic and Convex Programming. Math. Appl., vol. 277, Kluwer Academic Publishers, Dordrecht (1994)

  7. Ye, Y. (ed.): Interior Point Algorithms. Theory and Analysis. John Wiley and Sons, Chichester (1997)

    MATH  Google Scholar 

  8. Monteiro, R.D.C., Adler, I.: Interior-path following primal-dual algorithms: part I: linear programming. Math. Program. 44, 27–41 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sharma, J.R., Gupta, P.: An efficient fifth order method for solving systems of nonlinear equations. Comput. Math. Appl. 67, 591–601 (2014)

    Article  MathSciNet  Google Scholar 

  10. Hime, A., Oliveira, J.E., Petraglia, A.: Solving nonlinear systems of functional equations with fuzzy adaptive simulated annealing. Appl. Soft Comput. 13, 4349–4357 (2013)

    Article  Google Scholar 

  11. Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, 2369–2374 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grau-Sánchez, M., Grau, À., Noguera, M.: Ostrowski type methods for solving systems of nonlinear equations. Appl. Math. Comput. 218, 2377–2385 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Noor, M.A., Waseem, M.: Some iterative methods for solving a system of nonlinear equations. Comput. Math. Appl. 57, 101–106 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Darvishi, M.T., Barati, A.: A fourth-order method from quadrature formulae to solve systems of nonlinear equations. Appl. Math. Comput. 188, 257–261 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Frontini, M., Sormani, E.: Third-order methods from quadrature formulae for solving systems of nonlinear equations. Appl. Math. Comput. 149, 771–782 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Homeier, H.H.H.: A modified Newton method with cubic convergence: the multivariable case. J. Comput. Appl. Math. 169, 161–169 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Frontini, M., Sormani, E.: Some variant of Newton’s method with third-order convergence. Appl. Math. Comput. 140, 419–426 (2003)

    MathSciNet  MATH  Google Scholar 

  19. Amat, S., Busquier, S., Gutiérrez, J.M.: Geometrical constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, L., Li, S.: A new kind of simple kennel function yielding good iteration bounds for primal-dual interior-point methods. J. Comput. Appl. Math. 235, 2944–2955 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are sincerely grateful to the anonymous referees and the Corresponding Editor for their valuable comments, which lead to a substantial improvement in the contents of this paper.

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Authors and Affiliations

  1. Analysis, Modeling and Simulation Laboratory, Hassan II University, Faculté des Sciences Ben M’Sik, Av Driss El Harti, B.P. 7955, Sidi Othmane, Casablanca, Morocco

    Youssef EL Foutayeni

  2. Unit for Mathematical and Computer Modeling of Complex Systems, IRD, Bondy, France

    Youssef EL Foutayeni

  3. Modeling Applied to Economics and Management Laboratory, Hassan II University, Casablanca, Morocco

    Hicham EL Bouanani

  4. Mathematical Populations Dynamics Laboratory, Cadi Ayyad University, Marrakech, Morocco

    Mohamed Khaladi

  5. UMMISCO UMI 209 IRD/UPMC, Bondy, France

    Mohamed Khaladi

Authors
  1. Youssef EL Foutayeni
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  2. Hicham EL Bouanani
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  3. Mohamed Khaladi
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Correspondence to Youssef EL Foutayeni.

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EL Foutayeni, Y., EL Bouanani, H. & Khaladi, M. An Efficient Fifth-Order Method for Linear Optimization. J Optim Theory Appl 170, 189–204 (2016). https://doi.org/10.1007/s10957-016-0896-z

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  • DOI: https://doi.org/10.1007/s10957-016-0896-z

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