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Ukrainian Mathematical Journal

, Volume 69, Issue 11, pp 1689–1709 | Cite as

Multiobjective Nonlinear Sum of Fractional Optimization Problems with Nonconvex Constraints with the Use of the Duality-Based Branch and Bound Algorithm

  • D. Bhati
  • P. Singh
Article
  • 14 Downloads

We study the solution of a multiobjective nonlinear sum of fractional optimization problems. A dualitybased branch and bound cut method is developed for the efficient solution of this problems. The proposed methodology is validated by proving the required theoretical assertions for the solution. The present method is an extension of the work P. P. Shen, Y. P. Duan, and Y. G. Pei [J. Comput. Appl. Math., 223, 145–158 (2009)] developed for a single-objective sum of ratios of nonlinear optimization problems. The proposed method is realized in MatLab (version 2014b). Two numerical problems are considered and solved by using the proposed method and the global optimal solution is obtained.

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References

  1. 1.
    F. H. Mathis and L. J. Mathis, “A nonlinear programming algorithm for hospital management,” SIAM Rev., 37, 230–234 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    S. Schaible, “A note on the sum of a linear and linear fractional functions,” Naval Res. Logist., 24, 61–963 (1977).CrossRefGoogle Scholar
  3. 3.
    H. P. Benson, “Global optimization algorithm for the nonlinear sum of ratios problem,” J. Math. Anal. Appl., 263, 301–315 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    H. P. Benson, “Global optimization algorithm for the nonlinear sum of ratios problem,” J. Optim. Theory Appl., 112(1), 1–129 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    H. P. Benson, “Generating sum of ratios test problems in global optimization,” J. Optim. Theory Appl., 119(3), 615–621 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    H. P. Benson, “On the global optimization of sums of linear fractional functions over a convex set,” J. Optim. Theory Appl., 121(1), 19–39 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H. P. Benson, “A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratio problem,” Eur. J. Oper. Res., 182, 597–611 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    H. P. Benson, “Solving sum of ratios fractional programs via concave minimization,” J. Optim. Theory Appl., 135, 1–17 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    H. P. Benson, “Branch and bound outer approximation algorithms for sum-of-ratios fractional programs,” J. Optim. Theory Appl., 146, 1–18 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    C. H. Scott and T. R. Jefferson, “Duality of nonconvex sum of ratios,” J. Optim. Theory Appl., 98(1), 151–159 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    P. Shen and L. Jin, “Using canonical partition to globally maximizing the nonlinear sum of ratios,” Appl. Math. Model., 34, 2396–2413 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Y. J. Wang and K. C. Zhang, “Global optimization of nonlinear sum of ratios problem,” Appl. Math. Comput., 158, 319–330 (2004).MathSciNetzbMATHGoogle Scholar
  13. 13.
    P. P. Shen and C. F.Wang, “Global optimization for sum of ratios problem with with coefficient,” Appl. Math. Comput., 176, 219–229 (2006).MathSciNetzbMATHGoogle Scholar
  14. 14.
    H. Jiao and P. Shen, “A note on the paper global optimization of nonlinear sum of ratios,” Appl. Math. Comput., 188, 1812–1815 (2007).MathSciNetzbMATHGoogle Scholar
  15. 15.
    S. J. Qu, K. C. Zhang, and J. K. Zhao, “An efficient algorithm for globally minimizing sum of quadratics ratios problem with nonconvex quadratics constraints,” Appl. Math. Comput., 189, 1624–1636 (2007).MathSciNetzbMATHGoogle Scholar
  16. 16.
    P. Shen, Y. Chen, and M. Yuan, “Solving sum of quadratic ratios fractional programs via monotonic function,” Appl. Math. Comput., 212, 234–244 (2009).MathSciNetzbMATHGoogle Scholar
  17. 17.
    P. Shen, W. Li, and X. Bai, “Maximizing for the sum of ratios of two convex functions over a convex set,” Comput. Oper. Res., 40, 2301–2307 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    P. Shen and C. F. Wang, “Global optimization for sum of generalization fractional functions,” J. Comput. Appl. Math., 214, 1–12 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    P. P. Shen, Y. P. Duan, and Y. G. Pei, “A simplicial branch and duality bound algorithm for the sum of convex-convex ratios problem,” J. Comput. Appl. Math., 223, 145–158 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    M. Jaberipour and E. Khorram, “Solving the sum-of-ratios problem by a harmony search algorithm,” J. Comput. Appl. Math., 234, 733–742 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    L. Jin and X. P. Hou, “Global optimization for a class nonlinear sum of ratios problems,” Math. Probl. Eng., Article ID: 103569 (2014).Google Scholar
  22. 22.
    Y. Gao and S. Jin, “A global optimization algorithm for sum of linear ratios problem,” J. Appl. Math., Article ID: 276245 (2013).Google Scholar
  23. 23.
    R. W. Freund and F. Jarre, “Solving the sum-of-ratios problem by an interior-point method,” J. Global Optim., 19, 83–102 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    S. Schaible and J. Shi, “Fractional programming: the sum-of-ratio case,” Optim. Methods Softw., 18(2), 219–229 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    J. G. Carlsson and J. Shi, “A linear relaxation algorithm for solving the sum of linear ratios problem with lower dimension,” Oper. Res. Lett., 41, 381–389 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    A. M. Ashtiani and A. V. Ferreira, “A branch and cut algorithm for a class of sum of ratios problems,” Appl. Math. Comput., 268, 596–608 (2015).MathSciNetGoogle Scholar
  27. 27.
    R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer, Berlin (1996).CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • D. Bhati
    • 1
  • P. Singh
    • 1
  1. 1.Motilal Nehru National Institute of TechnologyAllahabadIndia

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