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Proximal bundle methods based on approximate subgradients for solving Lagrangian duals of minimax fractional programs

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  • H. Boualam

    (Laboratoire MISI, Faculté des Sciences et Techniques)

  • A. Roubi

    (Laboratoire MISI, Faculté des Sciences et Techniques)

Abstract

We are interested in this work to solve the Lagrangian dual of a generalized fractional program (GFP), which gives the minimal value of the primal problem, thanks to some duality results. With the help of a general minimax equality assumption, we give duality results under minimal assumptions. Since the associated parametric programs of the dual of GFP are always concave, we use a general approximating proximal scheme to these subproblems and construct bundle methods by means of approximate values and approximate subgradients of the dual parametric function. As for all dual algorithms, the proposed methods generate a sequence of values that converges from below to the minimal value of the GFP and a sequence of approximate solutions that converges to a solution of the Lagrangian dual of the GFP. For certain classes of problems, the convergence is at least linear.

Suggested Citation

  • H. Boualam & A. Roubi, 2019. "Proximal bundle methods based on approximate subgradients for solving Lagrangian duals of minimax fractional programs," Journal of Global Optimization, Springer, vol. 74(2), pages 255-284, June.
    • Handle: RePEc:spr:jglopt:v:74:y:2019:i:2:d:10.1007_s10898-019-00757-2
    DOI: 10.1007/s10898-019-00757-2
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    References listed on IDEAS

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    1. Smail Addoune & Karima Boufi & Ahmed Roubi, 2018. "Proximal Bundle Algorithms for Nonlinearly Constrained Convex Minimax Fractional Programs," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 212-239, October.
    2. M. Gugat, 1998. "Prox-Regularization Methods for Generalized Fractional Programming," Journal of Optimization Theory and Applications, Springer, vol. 99(3), pages 691-722, December.
    3. 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.
    4. K. Boufi & A. Roubi, 2017. "Dual method of centers for solving generalized fractional programs," Journal of Global Optimization, Springer, vol. 69(2), pages 387-426, October.
    5. Robert Mifflin, 1977. "An Algorithm for Constrained Optimization with Semismooth Functions," Mathematics of Operations Research, INFORMS, vol. 2(2), pages 191-207, May.
    6. A. Roubi, 2000. "Method of Centers for Generalized Fractional Programming," Journal of Optimization Theory and Applications, Springer, vol. 107(1), pages 123-143, October.
    7. Frenk, J. B. G. & Kassay, G. & Kolumban, J., 2004. "On equivalent results in minimax theory," European Journal of Operational Research, Elsevier, vol. 157(1), pages 46-58, August.
    8. 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.
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    Cited by:

    1. Karima Boufi & Mostafa El Haffari & Ahmed Roubi, 2020. "Optimality Conditions and a Method of Centers for Minimax Fractional Programs with Difference of Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 105-132, October.

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