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Inexact subgradient methods for quasi-convex optimization problems

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  • Hu, Yaohua
  • Yang, Xiaoqi
  • Sim, Chee-Khian

Abstract

In this paper, we consider a generic inexact subgradient algorithm to solve a nondifferentiable quasi-convex constrained optimization problem. The inexactness stems from computation errors and noise, which come from practical considerations and applications. Assuming that the computational errors and noise are deterministic and bounded, we study the effect of the inexactness on the subgradient method when the constraint set is compact or the objective function has a set of generalized weak sharp minima. In both cases, using the constant and diminishing stepsize rules, we describe convergence results in both objective values and iterates, and finite convergence to approximate optimality. We also investigate efficiency estimates of iterates and apply the inexact subgradient algorithm to solve the Cobb–Douglas production efficiency problem. The numerical results verify our theoretical analysis and show the high efficiency of our proposed algorithm, especially for the large-scale problems.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ejores:v:240:y:2015:i:2:p:315-327
    DOI: 10.1016/j.ejor.2014.05.017
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    Cited by:

    1. Shinji Yamada & Akiko Takeda, 2018. "Successive Lagrangian relaxation algorithm for nonconvex quadratic optimization," Journal of Global Optimization, Springer, vol. 71(2), pages 313-339, June.
    2. 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.
    3. Regina S. Burachik & Yaohua Hu & Xiaoqi Yang, 2022. "Interior quasi-subgradient method with non-Euclidean distances for constrained quasi-convex optimization problems in hilbert spaces," Journal of Global Optimization, Springer, vol. 83(2), pages 249-271, June.
    4. 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.
    5. Hishinuma, Kazuhiro & Iiduka, Hideaki, 2020. "Fixed point quasiconvex subgradient method," European Journal of Operational Research, Elsevier, vol. 282(2), pages 428-437.
    6. 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.
    7. Yaohua Hu & Carisa Kwok Wai Yu & Xiaoqi Yang, 2019. "Incremental quasi-subgradient methods for minimizing the sum of quasi-convex functions," Journal of Global Optimization, Springer, vol. 75(4), pages 1003-1028, December.
    8. Yaohua Hu & Jiawen Li & Carisa Kwok Wai Yu, 2020. "Convergence rates of subgradient methods for quasi-convex optimization problems," Computational Optimization and Applications, Springer, vol. 77(1), pages 183-212, September.
    9. Jifeng Bao & Carisa Kwok Wai Yu & Jinhua Wang & Yaohua Hu & Jen-Chih Yao, 2019. "Modified inexact Levenberg–Marquardt methods for solving nonlinear least squares problems," Computational Optimization and Applications, Springer, vol. 74(2), pages 547-582, November.

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