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A Proximal Point Algorithm with Quasi-distance in Multi-objective Optimization

Author

Listed:
  • Rogério A. Rocha

    (Federal University of Tocantins)

  • Paulo R. Oliveira

    (Federal University of Rio de Janeiro)

  • Ronaldo M. Gregório

    (Rural Federal University of Rio de Janeiro)

  • Michael Souza

    (Federal University of Ceará)

Abstract

In this paper, we present a generalized vector-valued proximal point algorithm for convex and unconstrained multi-objective optimization problems. Our main contribution is the introduction of quasi-distance mappings in the regularized subproblems, which has important applications in the computer theory and economics, among others. By considering a certain class of quasi-distances, that are Lipschitz continuous and coercive in any of their arguments, we show that any sequence generated by our algorithm is bounded and its accumulation points are weak Pareto solutions.

Suggested Citation

  • Rogério A. Rocha & Paulo R. Oliveira & Ronaldo M. Gregório & Michael Souza, 2016. "A Proximal Point Algorithm with Quasi-distance in Multi-objective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 964-979, December.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:3:d:10.1007_s10957-016-1005-z
    DOI: 10.1007/s10957-016-1005-z
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    References listed on IDEAS

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    1. Rocha, Rogério Azevedo & Oliveira, Paulo Roberto & Gregório, Ronaldo Malheiros & Souza, Michael, 2016. "Logarithmic quasi-distance proximal point scalarization method for multi-objective programming," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 856-867.
    2. X. X. Huang & X. Q. Yang, 2004. "Duality for Multiobjective Optimization via Nonlinear Lagrangian Functions," Journal of Optimization Theory and Applications, Springer, vol. 120(1), pages 111-127, January.
    3. Dinh The Luc, 2016. "Multiobjective Linear Programming," Springer Books, Springer, edition 1, number 978-3-319-21091-9, July.
    4. 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.
    5. Jörg Fliege & Benar Fux Svaiter, 2000. "Steepest descent methods for multicriteria optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(3), pages 479-494, August.
    6. 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.
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