Journal of Global Optimization

, Volume 64, Issue 1, pp 79–96 | Cite as

A scalarization proximal point method for quasiconvex multiobjective minimization

  • H. C. F. ApolinárioEmail author
  • E. A. Papa Quiroz
  • P. R. Oliveira


In this paper we propose a scalarization proximal point method to solve multiobjective unconstrained minimization problems with locally Lipschitz and quasiconvex vector functions. We prove, under natural assumptions, that the sequence generated by the method is well defined and converges globally to a Pareto-Clarke critical point. Our method may be seen as an extension, for nonconvex case, of the inexact proximal method for multiobjective convex minimization problems studied by Bonnel et al. (SIAM J Optim 15(4):953–970, 2005).


Multiobjective minimization Clarke subdifferential  Quasiconvex functions Proximal point methods Fejér convergence Pareto-Clarke critical point 



The research of H.C.F. Apolinário was partially supported by CAPES/Brazil. The research of P.R.Oliveira was partially supported by CNPQ/Brazil. The research of E.A.Papa Quiroz was partially supported by the Postdoctoral Scholarship CAPES-FAPERJ Edital PAPD-2011.


  1. 1.
    Aussel, D.: Subdifferential properties of quasiconvex and pseudoconvex functions: unified approach. J. Optim. Theory Appl. 97(1), 29–45 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, New York (2006)CrossRefGoogle Scholar
  3. 3.
    Bagirov, A., Karmitsa, N., Makela, M.: Introduction to Nonsmooth Optimization: Theory, Practice and Software. Springer, Switzerland (2014)CrossRefGoogle Scholar
  4. 4.
    Bello Cruz, J.Y., Lucambio Pérez, L.R., Melo, J.G.: Convergence of the projected gradient method for quasiconvex multiobjective optimization. Nonlinear Anal. Theory Methods Appl. 74, 5268–5273 (2011)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bento, G.C., da Cruz Neto, J.X., Soubeyran, A.: A proximal point-type method for multicriteria optimization. Set Valued Var. Anal. (2014). doi: 10.1007/s11228-014-0279-2
  6. 6.
    Bolte, J., Daniilidis, A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18, 556–572 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Brito, A.S., da Cruz Neto, J.X., Lopes, J.O., Oliveira, P.R.: Interior proximal algorithm for quasiconvex programming and variational inequalities with linear constraints. J. Optim. Theory Appl. 154, 217–234 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Ceng, L., Yao, J.: Approximate proximal methods in vector optimization. Eur. J. Oper. Res. 183, 1–19 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Clarke, H.F.: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics. SIAM, New York (1983)Google Scholar
  11. 11.
    Custodio, A.L., Madeira, J.F.A., Vaz, A.I.F., Vicente, L.N.: Direct multisearch for multiobjective optimization. SIAM J. Optim. 21, 1109–1140 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    da Cruz Neto, J.X., Da Silva, G.J.P., Ferreira, O.P., Lopes, J.O.: A subgradient method for multiobjective optimization. Comput. Optim. Appl. 54, 461–472 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Gregório, R., Oliveira, P.R.: A Logarithmic-quadratic proximal point scalarization method for multiobjective programming. J. Global Optim. 49, 361–378 (2010)Google Scholar
  14. 14.
    Gromicho, J.: Quasiconvex Optimization and Location Theory. Kluwer Academic Publishers, Dordrecht, The Netherlands (1998)zbMATHCrossRefGoogle Scholar
  15. 15.
    Güler, O.: New proximal point proximal algorithms for convex minimization. SIAM J. Control Optim. 2, 649–664 (1992)zbMATHCrossRefGoogle Scholar
  16. 16.
    Hadjisavvas, N., Komlosi, S., Shaible, S.: Handbook of Generalized Convexity and Generalized Monotonicity. Nonconvex Optimization and its Applications, vol. 76. Springer, New York (2005)CrossRefGoogle Scholar
  17. 17.
    Huang, X.X., Yang, X.Q.: Duality for multiobjective optimization via nonlinear Lagrangian functions. J. Optim. Theory Appl. 120, 111–127 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kiwiel, K.C.: Convergence and efficience of subgradient methods for quasiconvex minimization. Math. Program. 90, 1–25 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Langenberg, N., Tichatschke, R.: Interior proximal methods for quasiconvex optimization. J. Global Optim. 52, 641–661 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Luc, T.D.: Theory of vector optimization. Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (1989)CrossRefGoogle Scholar
  21. 21.
    Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969)zbMATHGoogle Scholar
  22. 22.
    Martinet, B.: Regularization d’inequations variationelles par approximations sucessives. Révue Française d’informatique et Recherche Opérationelle. 4, 54–159 (1970)MathSciNetGoogle Scholar
  23. 23.
    Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, New York, NY, USA (1995)zbMATHGoogle Scholar
  24. 24.
    Miettinen, K.M.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston (1999)zbMATHGoogle Scholar
  25. 25.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Grundlehren Series [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2006)Google Scholar
  26. 26.
    Papa Quiroz, E.A., Oliveira, P.R.: Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal. 16, 49–69 (2009)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Papa Quiroz, E.A., Oliveira, P.R.: Full Convergence of the proximal point method for quasiconvex functions on Hadamard manifolds. ESAIM Control Optim. Calc. Var. 18, 483–500 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Papa Quiroz, E.A., Oliveira, P.R.: An extension of proximal methods for quasiconvex minimization on the nonnegative orthant. Eur. J. Oper. Res. 216, 26–32 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Papa Quiroz, E.A., Oliveira, P.R.: Proximal point method for minimizing quasiconvex locally Lipschitz functions on Hadamard manifolds. Nonlinear Anal. Theory Methods Appl. 75, 5924–5932 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Papa Quiroz, E.A., Mallma Ramirez, L., Oliveira, P.R.: An inexact proximal method for quasiconvex minimization. Eur. J. Oper. Res. 246, 721–729 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)zbMATHCrossRefGoogle Scholar
  32. 32.
    Schott, D.: Basic properties of Fejer monotone sequences. Rostocker Math. Kolloqu. 49, 57–74 (1995)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Villacorta, K.D.V., Oliveira, P.R.: An interior proximal method in vector optimization. Eur. J. Oper. Res. 214, 485–492 (2011)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Undergraduate Computation Sciences CourseFederal University of TocantinsPalmasBrazil
  2. 2.Department of Ciencias MatemáticasMayor de San Marcos National UniversityLimaPeru
  3. 3.Computing and Systems Engineering DepartmentFederal University of Rio de JaneiroRio de JaneiroBrazil

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