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Globally minimizing the sum of a convex–concave fraction and a convex function based on wave-curve bounds

Author

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  • Yong Xia

    (Beihang University)

  • Longfei Wang

    (Henan University)

  • Xiaohui Wang

    (Beihang University)

Abstract

We consider the problem of minimizing the sum of a convex–concave function and a convex function over a convex set (SFC). It can be reformulated as a univariate minimization problem, where the objective function is evaluated by solving convex optimization. The optimal Lagrangian multipliers of the convex subproblems are used to construct sawtooth curve lower bounds, which play a key role in developing the branch-and-bound algorithm for globally solving (SFC). In this paper, we improve the existing sawtooth-curve bounds to new wave-curve bounds, which are used to develop a more efficient branch-and-bound algorithm. Moreover, we can show that the new algorithm finds an $$\epsilon $$ϵ-approximate optimal solution in at most $$O\left( \frac{1}{\epsilon }\right) $$O1ϵ iterations. Numerical results demonstrate the efficiency of our algorithm.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:77:y:2020:i:2:d:10.1007_s10898-019-00870-2
    DOI: 10.1007/s10898-019-00870-2
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    References listed on IDEAS

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    1. Siegfried Schaible, 1976. "Duality in Fractional Programming: A Unified Approach," Operations Research, INFORMS, vol. 24(3), pages 452-461, June.
    2. 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.
    3. Siegfried Schaible, 1976. "Fractional Programming. I, Duality," Management Science, INFORMS, vol. 22(8), pages 858-867, April.
    4. Lei-Hong Zhang, 2013. "On optimizing the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphere," Computational Optimization and Applications, Springer, vol. 54(1), pages 111-139, January.
    5. 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.
    6. Werner Dinkelbach, 1967. "On Nonlinear Fractional Programming," Management Science, INFORMS, vol. 13(7), pages 492-498, March.
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