Skip to main content

Advertisement

Springer Nature Link
Log in

Optimality Conditions for Quasi-Solutions of Vector Optimization Problems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we deal with quasi-solutions of constrained vector optimization problems. These solutions are a kind of approximate minimal solutions and they are motivated by the Ekeland variational principle. We introduce several notions of quasi-minimality based on free disposal sets and we characterize these solutions through scalarization and Lagrange multiplier rules. When the problem fulfills certain convexity assumptions, these results are obtained by using linear separation and the Fenchel subdifferential. In the nonconvex case, they are stated by using the so-called Gerstewitz (Tammer) nonlinear separation functional and the Mordukhovich subdifferential.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bednarczuk, E.M., Zagrodny, D.: Vector variational principle. Arch. Math. 93, 577–586 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen, G.Y., Huang, X., Yang, X.: In: Vector Optimization. Set-Valued and Variational Analysis. Lecture Notes in Econom. and Math. Systems, vol. 541 (2005)

    Google Scholar 

  3. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. (NS) 1, 443–474 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  5. Flores-Bazán, F., Gutiérrez, C., Novo, V.: A Brézis-Browder principle on partially ordered spaces and related ordering theorems. J. Math. Anal. Appl. 375, 245–260 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gutiérrez, C., Jiménez, B., Novo, V.: A set-valued Ekeland’s variational principle in vector optimization. SIAM J. Control Optim. 47, 883–903 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Liu, C.G., Ng, K.F.: Ekeland’s variational principle for set-valued functions. SIAM J. Optim. 21, 41–56 (2011)

    Article  MathSciNet  Google Scholar 

  8. Tammer, C., Zălinescu, C.: Vector variational principles for set-valued functions. Optimization 60, 839–857 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Mordukhovich, B.S., Wang, B.: Necessary suboptimality and optimality conditions via variational principles. SIAM J. Control Optim. 41, 623–640 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program., Ser. A 122, 301–347 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ha, T.X.D.: The Ekeland variational principle for Henig proper minimizers and super minimizers. J. Math. Anal. Appl. 364, 156–170 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Durea, M., Dutta, J., Tammer, C.: Lagrange multipliers for ε-Pareto solutions in vector optimization with nonsolid cones in Banach spaces. J. Optim. Theory Appl. 145, 196–211 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gao, Y., Hou, S.H., Yang, X.M.: Existence and optimality conditions for approximate solutions to vector optimization problems. J. Optim. Theory Appl. 152, 97–120 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gutiérrez, C., López, R., Novo, V.: Generalized ε-quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions. Nonlinear Anal. 72, 4331–4346 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, C., Lee, H.: Lagrange multiplier rules for approximate solutions in vector optimization. J. Ind. Manag. Optim. 8, 749–764 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gerstewitz, C., Iwanow, E.: Dualität für nichtkonvexe vektoroptimierungsprobleme. Wiss. Z. - Tech. Hochsch. Ilmenau 31, 61–81 (1985)

    MATH  MathSciNet  Google Scholar 

  17. Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hiriart-Urruty, J.-B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Basic Theory. Springer, Berlin (2006)

    Google Scholar 

  20. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II. Applications. Springer, Berlin (2006)

    Google Scholar 

  21. Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55, 501–558 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ciligot-Travain, M.: On Lagrange-Kuhn-Tucker multipliers for Pareto optimization problems. Numer. Funct. Anal. Optim. 15, 689–693 (1994)

    Article  MathSciNet  Google Scholar 

  23. Durea, M., Tammer, C.: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449–467 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Chicco, M., Mignanego, F., Pusillo, L., Tijs, S.: Vector optimization problems via improvement sets. J. Optim. Theory Appl. 150, 516–529 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Gutiérrez, C., Jiménez, B., Novo, V.: Improvement sets and vector optimization. Eur. J. Oper. Res. 223, 304–311 (2012)

    Article  Google Scholar 

  26. Tammer, C., Zălinescu, C.: Lipschitz properties of the scalarization function and applications. Optimization 59, 305–319 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  27. Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)

    MATH  Google Scholar 

  28. Flores-Bazán, F., Hernández, E.: A unified vector optimization problem: complete scalarizations and applications. Optimization 60, 1399–1419 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Jahn, J.: Vector Optimization. Theory, Applications and Extensions. Springer, Berlin (2011)

    MATH  Google Scholar 

  30. Luc, D.T.: In: Theory of Vector Optimization. Lecture Notes in Econom. and Math. Systems, vol. 319 (1989)

    Chapter  Google Scholar 

  31. Gutiérrez, C., Jiménez, B., Novo, V.: A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM J. Optim. 17, 688–710 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Gutiérrez, C., Jiménez, B., Novo, V.: Optimality conditions via scalarization for a new ε-efficiency concept in vector optimization problems. Eur. J. Oper. Res. 201, 11–22 (2010)

    Article  MATH  Google Scholar 

  34. Giorgi, G., Guerraggio, A., Thierfelder, J.: Mathematics of Optimization: Smooth and Nonsmooth Case. Elsevier, Amsterdam (2004)

    Google Scholar 

  35. Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110, 413–427 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, New Jersey (2002)

    Book  MATH  Google Scholar 

  37. Gupta, D., Mehra, A.: A new notion of quasi efficiency in vector optimization. Pac. J. Optim. 8, 217–230 (2012)

    MATH  MathSciNet  Google Scholar 

  38. Hamel, A.: An ε-Lagrange multiplier rule for a mathematical programming problem on Banach spaces. Optimization 49, 137–149 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  39. Gupta, D., Mehra, A.: Two types of approximate saddle points. Numer. Funct. Anal. Optim. 29, 532–550 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was partially supported by the Ministerio de Economía y Competitividad (Spain) under project MTM2012-30942.

The authors are grateful to the editor and the anonymous referees for their helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, E.T.S. de Ingenieros de Telecomunicación, Universidad de Valladolid, Paseo de Belén 15, Campus Miguel Delibes, 47011, Valladolid, Spain

    C. Gutiérrez

  2. Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, Ciudad Universitaria, 28040, Madrid, Spain

    B. Jiménez & V. Novo

Authors
  1. C. Gutiérrez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Jiménez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. Novo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Novo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gutiérrez, C., Jiménez, B. & Novo, V. Optimality Conditions for Quasi-Solutions of Vector Optimization Problems. J Optim Theory Appl 167, 796–820 (2015). https://doi.org/10.1007/s10957-013-0393-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-013-0393-6

Keywords