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Transformation of Quasiconvex Functions to Eliminate Local Minima

Author

Listed:
  • Suliman Al-Homidan

    (King Fahd University of Petroleum and Minerals)

  • Nicolas Hadjisavvas

    (King Fahd University of Petroleum and Minerals
    University of the Aegean)

  • Loai Shaalan

    (King Fahd University of Petroleum and Minerals)

Abstract

Quasiconvex functions present some difficulties in global optimization, because their graph contains “flat parts”; thus, a local minimum is not necessarily the global minimum. In this paper, we show that any lower semicontinuous quasiconvex function may be written as a composition of two functions, one of which is nondecreasing, and the other is quasiconvex with the property that every local minimum is global minimum. Thus, finding the global minimum of any lower semicontinuous quasiconvex function is equivalent to finding the minimum of a quasiconvex function, which has no local minima other than its global minimum. The construction of the decomposition is based on the notion of “adjusted sublevel set.” In particular, we study the structure of the class of sublevel sets, and the continuity properties of the sublevel set operator and its corresponding normal operator.

Suggested Citation

  • Suliman Al-Homidan & Nicolas Hadjisavvas & Loai Shaalan, 2018. "Transformation of Quasiconvex Functions to Eliminate Local Minima," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 93-105, April.
    • Handle: RePEc:spr:joptap:v:177:y:2018:i:1:d:10.1007_s10957-018-1223-7
    DOI: 10.1007/s10957-018-1223-7
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    References listed on IDEAS

    as
    1. Alberto Cambini & Laura Martein, 2009. "Generalized Convexity and Optimization," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-70876-6, December.
    2. Jean-Pierre Crouzeix & Andrew Eberhard & Daniel Ralph, 2016. "(Convex) Level Sets Integration," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 865-886, December.
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    Citations

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    Cited by:

    1. Satoshi Suzuki, 2019. "Optimality Conditions and Constraint Qualifications for Quasiconvex Programming," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 963-976, December.
    2. Satoshi Suzuki, 2021. "Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential," Journal of Global Optimization, Springer, vol. 79(1), pages 191-202, January.
    3. Sorin-Mihai Grad & Felipe Lara, 2022. "An extension of the proximal point algorithm beyond convexity," Journal of Global Optimization, Springer, vol. 82(2), pages 313-329, February.
    4. 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.
    5. Frank Plastria, 2020. "On the Structure of the Weakly Efficient Set for Quasiconvex Vector Minimization," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 547-564, February.
    6. Felipe Lara, 2020. "Optimality Conditions for Nonconvex Nonsmooth Optimization via Global Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 134-150, April.

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