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Fractional Programming. I, Duality

Author

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  • Siegfried Schaible

    (University of Cologne and Stanford University)

Abstract

This paper, which is presented in two parts, is a contribution to the theory of fractional programming, i.e., maximization of quotients subject to constraints. In Part I a duality theory for linear and concave-convex fractional programs is developed and related to recent results by Bector, Craven-Mond, Jagannathan, Sharma-Swarup, et al. Basic duality theorems of linear, quadratic and convex programming are extended. In Part II Dinkelbach's algorithm solving fractional programs is considered. The rate of convergence as well as a priori and a posteriori error estimates are determined. In view of these results the stopping rule of the algorithm is changed. Also the starting rule is modified using duality as introduced in Part I. Furthermore a second algorithm is proposed. In contrast to Dinkelbach's procedure the rate of convergence is still controllable. Error estimates are obtained too.

Suggested Citation

  • Siegfried Schaible, 1976. "Fractional Programming. I, Duality," Management Science, INFORMS, vol. 22(8), pages 858-867, April.
    • Handle: RePEc:inm:ormnsc:v:22:y:1976:i:8:p:858-867
    DOI: 10.1287/mnsc.22.8.858
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    Cited by:

    1. Wong, Man Hong, 2013. "Investment models based on clustered scenario trees," European Journal of Operational Research, Elsevier, vol. 227(2), pages 314-324.
    2. Yong Xia & Longfei Wang & Xiaohui Wang, 2020. "Globally minimizing the sum of a convex–concave fraction and a convex function based on wave-curve bounds," Journal of Global Optimization, Springer, vol. 77(2), pages 301-318, June.
    3. Denoyel, Victoire & Alfandari, Laurent & Thiele, Aurélie, 2017. "Optimizing healthcare network design under reference pricing and parameter uncertainty," European Journal of Operational Research, Elsevier, vol. 263(3), pages 996-1006.
    4. Oleksii Ursulenko & Sergiy Butenko & Oleg Prokopyev, 2013. "A global optimization algorithm for solving the minimum multiple ratio spanning tree problem," Journal of Global Optimization, Springer, vol. 56(3), pages 1029-1043, July.
    5. T Peña & P Lara & C Castrodeza, 2009. "Multiobjective stochastic programming for feed formulation," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 60(12), pages 1738-1748, December.
    6. Frenk, J.B.G. & Schaible, S., 2004. "Fractional Programming," Econometric Institute Research Papers ERS-2004-074-LIS, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    7. Xiaojun Lei & Zhian Liang, 2008. "Study on the Duality between MFP and ACP," Modern Applied Science, Canadian Center of Science and Education, vol. 2(6), pages 1-81, November.
    8. Ammar, E.E., 2007. "On optimality and duality theorems of nonlinear disjunctive fractional minmax programs," European Journal of Operational Research, Elsevier, vol. 180(3), pages 971-982, August.
    9. Frauke Liers & Lars Schewe & Johannes Thürauf, 2022. "Radius of Robust Feasibility for Mixed-Integer Problems," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 243-261, January.
    10. Castrodeza, Carmen & Lara, Pablo & Pena, Teresa, 2005. "Multicriteria fractional model for feed formulation: economic, nutritional and environmental criteria," Agricultural Systems, Elsevier, vol. 86(1), pages 76-96, October.
    11. X. L. Sun & H. Z. Luo & D. Li, 2007. "Convexification of Nonsmooth Monotone Functions1," Journal of Optimization Theory and Applications, Springer, vol. 132(2), pages 339-351, February.
    12. Paula Alexandra Amaral & Immanuel M. Bomze, 2019. "Nonconvex min–max fractional quadratic problems under quadratic constraints: copositive relaxations," Journal of Global Optimization, Springer, vol. 75(2), pages 227-245, October.
    13. C. Singh & M.A. Hanson, 1991. "Multiobjective fractional programming duality theory," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(6), pages 925-933, December.
    14. Lara, P. & Stancu-Minasian, I., 1999. "Fractional programming: a tool for the assessment of sustainability," Agricultural Systems, Elsevier, vol. 62(2), pages 131-141, November.
    15. Frenk, J.B.G. & Schaible, S., 2004. "Fractional Programming," ERIM Report Series Research in Management ERS-2004-074-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    16. Husain, I. & Hanson, Morgan A. & Jabeen, Z., 2005. "On nondifferentiable fractional minimax programming," European Journal of Operational Research, Elsevier, vol. 160(1), pages 202-217, January.

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