Journal of Optimization Theory and Applications

, Volume 162, Issue 2, pp 371–391 | Cite as

A Steepest Descent-Like Method for Variable Order Vector Optimization Problems

  • J. Y. Bello Cruz
  • G. Bouza Allende


In some applications, the comparison between two elements may depend on the point leading to the so called variable order structure. Optimality concepts may be extended to this more general framework. In this paper, we extend the steepest descent-like method for smooth unconstrained vector optimization problems under a variable order structure. Roughly speaking, we see that every accumulation point of the generated sequence satisfies a necessary first order condition. We discuss the consequence of this fact in the convex case.


Convexity Gradient-like method Variable order Vector optimization Weakly efficient points 



The authors would like to thank the anonymous referees, whose suggestions helped us to improve the presentation of this paper. We also very grateful to Prof. Dr. Ole Peter Smith for correcting the manuscript.

This research was supported by Project CAPES-MES-CUBA 226/2012 “Modelos de Otimização e Aplicações”. First author was partially supported by PROCAD-nf-UFG/UnB/IMPA research and PRONEX-CNPq-FAPERJ Optimization research.


  1. 1.
    Eichfelder, G., Duc Ha, T.X.: Optimality conditions for vector optimization problems with variable ordering structures. Optimization (2011). doi: 10.1080/02331934.2011.575939 zbMATHGoogle Scholar
  2. 2.
    Eichfelder, G.: Optimal elements in vector optimization with variable ordering structure. J. Optim. Theory Appl. 151, 217–240 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Baatar, D., Wiecek, M.M.: Advancing equitability in multiobjective programming. Comput. Math. Appl. 2, 225–234 (2006) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Engau, A.: Variable preference modeling with ideal-symmetric convex cones. J. Global Optim. 42, 295–311 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Wiecek, M.M.: Advances in cone-based preference modeling for decision making with multiple criteria. Decis. Mak. Manuf. Serv. 1, 153–173 (2007) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15, 953–970 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Fliege, J., Graña Drummond, L.M., Svaiter, B.F.: Newton’s method for multiobjective optimization. SIAM J. Optim. 20, 602–626 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Graña Drummond, L.M., Maculan, N., Svaiter, B.F.: On the choice of parameters for the weighting method in vector optimization. Math. Program. 111, 201–216 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Jahn, J.: Scalarization in vector optimization. Math. Program. 29, 203–218 (1984) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Luc, D.T.: Scalarization of vector optimization problems. J. Optim. Theory Appl. 55, 85–102 (1987) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Bello Cruz, J.Y., Lucambio Pérez, L.R.: Convergence of a projected gradient method variant for quasiconvex objectives. Nonlinear Anal. 9, 2917–2922 (2010) CrossRefGoogle Scholar
  12. 12.
    Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51, 479–494 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Graña Drummond, L.M., Svaiter, B.F.: A steepest descent method for vector optimization. J. Comput. Appl. Math. 175, 395–414 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Bello Cruz, J.Y., Lucambio Pérez, L.R., Melo, J.G.: Convergence of the projected gradient method for quasiconvex multiobjective optimization. Nonlinear Anal. 74, 5268–5273 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Browder, F.E.: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 100, 201–225 (1967) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Iusem, A.N., Svaiter, B.F., Teboulle, M.: Entropy-like proximal methods in convex programming. Math. Oper. Res. 19, 790–814 (1994) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Burachik, R., Graña Drummond, L.M., Iusem, A.N., Svaiter, B.F.: Full convergence of the steepest descent method with inexact line searches. Optimization 32, 137–146 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Luc, D.T., Tan, N.X., Tinh, P.N.: Convex vector functions and their subdifferential. Acta Math. Vietnam. 23, 107–127 (1998) zbMATHMathSciNetGoogle Scholar
  19. 19.
    Fukuda, E.H., Graña Drummond, L.M.: On the convergence of the projected gradient method for vector optimization. Optimization 60, 1009–1021 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Graña Drummond, L.M., Iusem, A.N.: A projected gradient method for vector optimization problems. Comput. Optim. Appl. 28, 5–30 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989) Google Scholar
  22. 22.
    Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Meth. Oper. Res. 48, 187–200 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Luc, D.T.: Pareto Optimality, Game Theory and Equilibria. In: Pareto Optimality. Springer Optimization and its Applications, vol. 17, pp. 481–515 (2008) Google Scholar
  24. 24.
    Jahn, J.: Vector Optimization—Theory, Applications and Extensions. Springer, Berlin (2004) CrossRefzbMATHGoogle Scholar
  25. 25.
    Jahn, J.: Mathematical Vector Optimization in Partially Ordered Linear Spaces. Peter Lang, Frankfurt (1986) zbMATHGoogle Scholar
  26. 26.
    Isac, G., Tammer, C.: Application of a vector-valued Ekeland-type variational principle for deriving optimality conditions. In: Nonlinear Analysis and Variational Problems: In Honor of George Isac. Springer Optimization and Applications, vol. 35, pp. 343–365 (2010) CrossRefGoogle Scholar
  27. 27.
    Peressini, A.L.: Ordered Topological Vector Space. Harper and Row, New York (1967) Google Scholar
  28. 28.
    Kiwiel, K.C.: The efficiency of subgradient projection methods for convex optimization I. General level methods. SIAM J. Control Optim. 34, 660–676 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Kogan, J.: Introduction to Clustering Large and High-dimensional Data. Cambridge University Press, Cambridge (2007) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Universidade Federal de GoiásGoiâniaBrazil
  2. 2.Facultad de Matemática y ComputaciónUniversidad de La HabanaHavanaCuba

Personalised recommendations