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A proximal point method for difference of convex functions in multi-objective optimization with application to group dynamic problems

Author

Listed:
  • Glaydston Carvalho Bento

    (Federal University of Goiás)

  • Sandro Dimy Barbosa Bitar

    (Federal University of Amazonas)

  • João Xavier Cruz Neto

    (Federal University of Piauí)

  • Antoine Soubeyran

    (Aix-Marseille University (Aix-Marseille School of Economics))

  • João Carlos Oliveira Souza

    (Federal University of Piauí)

Abstract

We consider the constrained multi-objective optimization problem of finding Pareto critical points of difference of convex functions. The new approach proposed by Bento et al. (SIAM J Optim 28:1104–1120, 2018) to study the convergence of the proximal point method is applied. Our method minimizes at each iteration a convex approximation instead of the (non-convex) objective function constrained to a possibly non-convex set which assures the vector improving process. The motivation comes from the famous Group Dynamic problem in Behavioral Sciences where, at each step, a group of (possible badly informed) agents tries to increase his joint payoff, in order to be able to increase the payoff of each of them. In this way, at each step, this ascent process guarantees the stability of the group. Some encouraging preliminary numerical results are reported.

Suggested Citation

  • Glaydston Carvalho Bento & Sandro Dimy Barbosa Bitar & João Xavier Cruz Neto & Antoine Soubeyran & João Carlos Oliveira Souza, 2020. "A proximal point method for difference of convex functions in multi-objective optimization with application to group dynamic problems," Computational Optimization and Applications, Springer, vol. 75(1), pages 263-290, January.
    • Handle: RePEc:spr:coopap:v:75:y:2020:i:1:d:10.1007_s10589-019-00139-0
    DOI: 10.1007/s10589-019-00139-0
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    1. Albert Ferrer & Adil Bagirov & Gleb Beliakov, 2015. "Solving DC programs using the cutting angle method," Journal of Global Optimization, Springer, vol. 61(1), pages 71-89, January.
    2. F. Flores-BAZÁN & W. Oettli, 2001. "Simplified Optimality Conditions for Minimizing the Difference of Vector-Valued Functions," Journal of Optimization Theory and Applications, Springer, vol. 108(3), pages 571-586, March.
    3. G. Bento & J. Cruz Neto & G. López & Antoine Soubeyran & J. Souza, 2018. "The Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization with Application to the Compromise Problem," Post-Print hal-01985333, HAL.
    4. L. C. Ceng & B. S. Mordukhovich & J. C. Yao, 2010. "Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 267-303, August.
    5. J. Cruz Neto & G. Silva & O. Ferreira & J. Lopes, 2013. "A subgradient method for multiobjective optimization," Computational Optimization and Applications, Springer, vol. 54(3), pages 461-472, April.
    6. Qu, Shaojian & Liu, Chen & Goh, Mark & Li, Yijun & Ji, Ying, 2014. "Nonsmooth multiobjective programming with quasi-Newton methods," European Journal of Operational Research, Elsevier, vol. 235(3), pages 503-510.
    7. X. L. Guo & S. J. Li, 2014. "Optimality Conditions for Vector Optimization Problems with Difference of Convex Maps," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 821-844, September.
    8. Glaydston de C. Bento & João Xavier Cruz Neto & Lucas V. Meireles, 2018. "Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization of Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 37-52, October.
    9. 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.
    10. Glaydston Carvalho Bento & J.X. Cruz Neto & Antoine Soubeyran, 2014. "A Proximal Point-Type Method for Multicriteria Optimization," Post-Print hal-01463765, HAL.
    11. Bento, G.C. & Cruz Neto, J.X. & Oliveira, P.R. & Soubeyran, A., 2014. "The self regulation problem as an inexact steepest descent method for multicriteria optimization," European Journal of Operational Research, Elsevier, vol. 235(3), pages 494-502.
    12. Brito, A.S. & Cruz Neto, J.X. & Santos, P.S.M. & Souza, S.S., 2017. "A relaxed projection method for solving multiobjective optimization problems," European Journal of Operational Research, Elsevier, vol. 256(1), pages 17-23.
    13. 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.
    14. 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.
    15. G. C. Bento & J. X. Cruz Neto, 2013. "A Subgradient Method for Multiobjective Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 125-137, October.
    16. Ying Ji & Mark Goh & Robert Souza, 2016. "Proximal Point Algorithms for Multi-criteria Optimization with the Difference of Convex Objective Functions," Journal of Optimization Theory and Applications, Springer, vol. 169(1), pages 280-289, April.
    17. G. C. Bento & A. Soubeyran, 2015. "Generalized Inexact Proximal Algorithms: Routine’s Formation with Resistance to Change, Following Worthwhile Changes," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 172-187, July.
    18. Terry Ross, G. & Soland, Richard M., 1980. "A multicriteria approach to the location of public facilities," European Journal of Operational Research, Elsevier, vol. 4(5), pages 307-321, May.
    19. 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.
    20. Bo Wen & Xiaojun Chen & Ting Kei Pong, 2018. "A proximal difference-of-convex algorithm with extrapolation," Computational Optimization and Applications, Springer, vol. 69(2), pages 297-324, March.
    21. N. Dinh & J. Strodiot & V. Nguyen, 2010. "Duality and optimality conditions for generalized equilibrium problems involving DC functions," Journal of Global Optimization, Springer, vol. 48(2), pages 183-208, October.
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