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A Linear Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization

Author

Listed:
  • Erik Alex Papa Quiroz

    (Universidad Nacional Mayor de San Marcos and Universidad Privada del Norte)

  • Hellena Christina Fernandes Apolinário

    (Federal University of Tocantins)

  • Kely Diana Villacorta

    (Federal University of Paraíba)

  • Paulo Roberto Oliveira

    (Federal University of Rio de Janeiro)

Abstract

In this paper, we propose a linear scalarization proximal point algorithm for solving lower semicontinuous quasiconvex multiobjective minimization problems. Under some natural assumptions and, using the condition that the proximal parameters are bounded, we prove the convergence of the sequence generated by the algorithm and, when the objective functions are continuous, we prove the convergence to a generalized critical point of the problem. Furthermore, for the continuously differentiable case we introduce an inexact algorithm, which converges to a Pareto critical point.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:3:d:10.1007_s10957-019-01582-z
    DOI: 10.1007/s10957-019-01582-z
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    References listed on IDEAS

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    1. Glaydston Carvalho Bento & J.X. Cruz Neto & Antoine Soubeyran, 2014. "A Proximal Point-Type Method for Multicriteria Optimization," Post-Print hal-01463765, HAL.
    2. Villacorta, Kely D.V. & Oliveira, P. Roberto, 2011. "An interior proximal method in vector optimization," European Journal of Operational Research, Elsevier, vol. 214(3), pages 485-492, November.
    3. Ceng, Lu-Chuan & Yao, Jen-Chih, 2007. "Approximate proximal methods in vector optimization," European Journal of Operational Research, Elsevier, vol. 183(1), pages 1-19, November.
    4. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    5. 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.
    6. H. Apolinário & E. Papa Quiroz & P. Oliveira, 2016. "A scalarization proximal point method for quasiconvex multiobjective minimization," Journal of Global Optimization, Springer, vol. 64(1), pages 79-96, January.
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    Cited by:

    1. 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.
    2. Xiaopeng Zhao & Jen-Chih Yao, 2022. "Linear convergence of a nonmonotone projected gradient method for multiobjective optimization," Journal of Global Optimization, Springer, vol. 82(3), pages 577-594, March.
    3. Xiaopeng Zhao & Markus A. Köbis & Yonghong Yao & Jen-Chih Yao, 2021. "A Projected Subgradient Method for Nondifferentiable Quasiconvex Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 82-107, July.
    4. 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.

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