Optimization Letters

, Volume 12, Issue 2, pp 311–320 | Cite as

Proximal point method for a special class of nonconvex multiobjective optimization functions

  • G. C. Bento
  • O. P. Ferreira
  • V. L. Sousa Junior
Original Paper


The proximal point method for a special class of nonconvex multiobjective functions is studied in this paper. We show that the method is well defined and that the accumulation points of any generated sequence, if any, are Pareto–Clarke critical points. Moreover, under additional assumptions, we show the full convergence of the generated sequence.


Multiobjective Pareto–Clarke optimality Nonconvex optimization 



The work was supported by CAPES and CNPq.


  1. 1.
    Apolinário, H.C.F., Papa Quiroz, E.A., Oliveira, P.R.: A scalarization proximal point method for quasiconvex multiobjective minimization. J. Glob. Optim. 64(1), 79–96 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers in multiobjective optimization: existence and optimality conditions. Math. Program. 122, 301–347 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bao, T.Q., Mordukhovich, B.S.: Necessary conditions for super minimizers in constrained multiobjective optimization. J. Glob. Optim. 43, 533–552 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bento, G.C., Cruz Neto, J.X., Soubeyran, A.: A proximal point-type method for multicriteria optimization. Set-Valued Var. Anal. 22(3), 557–573 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bertsekas, D.P.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)zbMATHGoogle Scholar
  6. 6.
    Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ceng, L.C., Mordukhovich, B.S., Yao, J.C.: Hybrid approximate proximal method with auxiliary variational inequality for vector optimization. J. Optim. Theory Appl. 146(2), 267–303 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ceng, L.C., Yao, J.C.: Approximate proximal methods in vector optimization. Eur. J. Oper. Res. 183(1), 1–19 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chuong, T.D., Mordukhovich, B.S., Yao, J.C.: Hybrid approximate proximal algorithms for efficient solutions in vector optimization. J. Nonlinear Convex Anal. 12(2), 257–286 (2011)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Clarke, F.H.: Optimization and Nonsmooth Analysis, Volume 5 of Classics in Applied Mathematics, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1990)CrossRefGoogle Scholar
  11. 11.
    Custódio, A.L., Madeira, J.F.A., Vaz, A.I.F., Vicente, L.N.: Direct multisearch for multiobjective optimization. SIAM J. Optim. 21(3), 1109–1140 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fukushima, M., Mine, H.: A generalized proximal point algorithm for certain nonconvex minimization problems. Int. J. Syst. Sci. 12(8), 989–1000 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gal, T., Hanne, T.: On the development and future aspects of vector optimization and MCDM. A tutorial. In: Multicriteria Analysis (Coimbra, 1994), pp. 130–145. Springer, Berlin (1997)Google Scholar
  14. 14.
    Grad, S.M., Pop, E.L.: Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone. Optimization 63(1), 21–37 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaplan, A., Tichatschke, R.: Proximal point methods and nonconvex optimization. J. Glob. Optim. 13(4), 389–406 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Luc, D.T.: Theory of Vector Optimization, volume 319 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (1989)Google Scholar
  17. 17.
    Mäkelä, M.M., Neittaanmäki, P.: Nonsmooth Optimization. World Scientific Publishing Co., Inc., River Edge (1992)CrossRefzbMATHGoogle Scholar
  18. 18.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Grundlehren Series in Fundamental Principles of Mathematical Sciences, vol. 331. Springer, Berlin (2006)Google Scholar
  19. 19.
    Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton (1970)Google Scholar
  20. 20.
    Villacorta, K.D.V., Oliveira, P.R.: An interior proximal method in vector optimization. Eur. J. Oper. Res. 214(3), 485–492 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • G. C. Bento
    • 1
  • O. P. Ferreira
    • 1
  • V. L. Sousa Junior
    • 1
  1. 1.Universidade Federal de GoiásGoiâniaBrazil

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