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Fixed point quasiconvex subgradient method

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  • Hishinuma, Kazuhiro
  • Iiduka, Hideaki

Abstract

Constrained quasiconvex optimization problems appear in many fields, such as economics, engineering, and management science. In particular, fractional programming, which models ratio indicators such as the profit/cost ratio as fractional objective functions, is an important instance. Subgradient methods and their variants are useful ways for solving these problems efficiently. Many complicated constraint sets onto which it is hard to compute the metric projections in a realistic amount of time appear in these applications. This implies that the existing methods cannot be applied to quasiconvex optimization over a complicated set. Meanwhile, thanks to fixed point theory, we can construct a computable nonexpansive mapping whose fixed point set coincides with a complicated constraint set. This paper proposes an algorithm that uses a computable nonexpansive mapping for solving a constrained quasiconvex optimization problem. We provide convergence analyses for constant diminishing step-size rules. Numerical comparisons between the proposed algorithm and an existing algorithm show that the proposed algorithm runs stably and quickly even when the running time of the existing algorithm exceeds the time limit.

Suggested Citation

  • Hishinuma, Kazuhiro & Iiduka, Hideaki, 2020. "Fixed point quasiconvex subgradient method," European Journal of Operational Research, Elsevier, vol. 282(2), pages 428-437.
    • Handle: RePEc:eee:ejores:v:282:y:2020:i:2:p:428-437
    DOI: 10.1016/j.ejor.2019.09.037
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    References listed on IDEAS

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    1. Stephen P. Bradley & Sherwood C. Frey, 1974. "Fractional Programming with Homogeneous Functions," Operations Research, INFORMS, vol. 22(2), pages 350-357, April.
    2. Iiduka, Hideaki, 2016. "Proximal point algorithms for nonsmooth convex optimization with fixed point constraints," European Journal of Operational Research, Elsevier, vol. 253(2), pages 503-513.
    3. Lee, Chia-Yen & Johnson, Andrew L. & Moreno-Centeno, Erick & Kuosmanen, Timo, 2013. "A more efficient algorithm for Convex Nonparametric Least Squares," European Journal of Operational Research, Elsevier, vol. 227(2), pages 391-400.
    4. Larsson, Torbjorn & Patriksson, Michael & Stromberg, Ann-Brith, 1996. "Conditional subgradient optimization -- Theory and applications," European Journal of Operational Research, Elsevier, vol. 88(2), pages 382-403, January.
    5. Hu, Yaohua & Yang, Xiaoqi & Sim, Chee-Khian, 2015. "Inexact subgradient methods for quasi-convex optimization problems," European Journal of Operational Research, Elsevier, vol. 240(2), pages 315-327.
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    Cited by:

    1. A. Kabgani & F. Lara, 2023. "Semistrictly and neatly quasiconvex programming using lower global subdifferentials," Journal of Global Optimization, Springer, vol. 86(4), pages 845-865, August.
    2. Alireza Kabgani, 2021. "Characterization of Nonsmooth Quasiconvex Functions and their Greenberg–Pierskalla’s Subdifferentials Using Semi-Quasidifferentiability notion," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 666-678, May.
    3. Hu, Yaohua & Li, Gongnong & Yu, Carisa Kwok Wai & Yip, Tsz Leung, 2022. "Quasi-convex feasibility problems: Subgradient methods and convergence rates," European Journal of Operational Research, Elsevier, vol. 298(1), pages 45-58.

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