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Quasi-convex feasibility problems: Subgradient methods and convergence rates

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  • Hu, Yaohua
  • Li, Gongnong
  • Yu, Carisa Kwok Wai
  • Yip, Tsz Leung

Abstract

The feasibility problem is at the core of the modeling of many problems in various areas, and the quasi-convex function usually provides a precise representation of reality in many fields such as economics, finance and management science. In this paper, we consider the quasi-convex feasibility problem (QFP), that is to find a common point of a family of sublevel sets of quasi-convex functions, and propose a unified framework of subgradient methods for solving the QFP. This paper is contributed to establish the quantitative convergence theory, including the iteration complexity and the convergence rates, of subgradient methods with the constant/dynamic stepsize rules and several general control schemes, including the α-most violated constraints control, the s-intermittent control and the stochastic control. An interesting finding is disclosed by iteration complexity results that the stochastic control enjoys both advantages of low computational cost requirement and low iteration complexity. More importantly, we introduce a notion of Hölder-type error bound property for the QFP, and use it to establish the linear (or sublinear) convergence rates for subgradient methods to a feasible solution of the QFP. Preliminary numerical results to the multiple Cobb-Douglas productions efficiency problem indicate the powerful modeling capability of the QFP and show the high efficiency and stability of subgradient methods for solving the QFP.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ejores:v:298:y:2022:i:1:p:45-58
    DOI: 10.1016/j.ejor.2021.09.029
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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. 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.
    3. Erik Alex Papa Quiroz & Hellena Christina Fernandes Apolinário & Kely Diana Villacorta & Paulo Roberto Oliveira, 2019. "A Linear Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 1028-1052, December.
    4. 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.
    5. X. X. Huang & X. Q. Yang, 2003. "A Unified Augmented Lagrangian Approach to Duality and Exact Penalization," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 533-552, August.
    6. Papa Quiroz, E.A. & Mallma Ramirez, L. & Oliveira, P.R., 2015. "An inexact proximal method for quasiconvex minimization," European Journal of Operational Research, Elsevier, vol. 246(3), pages 721-729.
    7. Alexander J. Zaslavski, 2016. "Approximate Solutions of Common Fixed-Point Problems," Springer Optimization and Its Applications, Springer, number 978-3-319-33255-0, June.
    8. Hishinuma, Kazuhiro & Iiduka, Hideaki, 2020. "Fixed point quasiconvex subgradient method," European Journal of Operational Research, Elsevier, vol. 282(2), pages 428-437.
    9. Papa Quiroz, E.A. & Roberto Oliveira, P., 2012. "An extension of proximal methods for quasiconvex minimization on the nonnegative orthant," European Journal of Operational Research, Elsevier, vol. 216(1), pages 26-32.
    10. 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.
    11. NESTEROV, Yurii, 2015. "Universal gradient methods for convex optimization problems," LIDAM Reprints CORE 2701, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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