Advertisement

Journal of Global Optimization

, Volume 70, Issue 4, pp 875–901 | Cite as

Approximate solutions of vector optimization problems via improvement sets in real linear spaces

  • C. Gutiérrez
  • L. Huerga
  • B. Jiménez
  • V. Novo
Article

Abstract

We deal with a constrained vector optimization problem between real linear spaces without assuming any topology and by considering an ordering defined through an improvement set E. We study E-optimal and weak E-optimal solutions and also proper E-optimal solutions in the senses of Benson and Henig. We relate these types of solutions and we characterize them through approximate solutions of scalar optimization problems via linear scalarizations and nearly E-subconvexlikeness assumptions. Moreover, in the particular case when the feasible set is defined by a cone-constraint, we obtain characterizations by means of Lagrange multiplier rules. The use of improvement sets allows us to unify and to extend several notions and results of the literature. Illustrative examples are also given.

Keywords

Vector optimization Improvement set Approximate weak efficiency Approximate proper efficiency Nearly E-subconvexlikeness Linear scalarization Lagrange multipliers algebraic interior Vector closure 

Mathematics Subject Classification

Primary 90C26 90C29 Secondary 90C46 90C48 49K27 

Notes

Acknowledgements

The authors are grateful to the anonymous referee and the Associated Editor for their useful suggestions and remarks.

References

  1. 1.
    Adán, M., Novo, V.: Optimality conditions for vector optimization problems with generalized convexity in real linear spaces. Optimization 51(1), 73–91 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adán, M., Novo, V.: Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness. Eur. J. Oper. Res. 149(3), 641–653 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adán, M., Novo, V.: Proper efficiency in vector optimization on real linear spaces. J. Optim. Theory Appl. 121(3), 515–540 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Adán, M., Novo, V.: Proper efficiency in vector optimization on real linear spaces. Errata corrige. J. Optim. Theory Appl. 124(3), 751 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bair, J., Fourneau, R.: Etude Géométrique des Espaces Vectoriels. Une Introduction. Lecture Notes in Mathematics 489. Springer, Berlin (1975)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chicco, M., Mignanego, F., Pusillo, L., Tijs, S.: Vector optimization problems via improvement sets. J. Optim. Theory. Appl. 150(3), 516–529 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Debreu, G.: Theory of Value: An Axiomatic Analysis of Economic Equilibrium. Wiley, New York (1959)zbMATHGoogle Scholar
  8. 8.
    Gutiérrez, C., Huerga, L., Novo, V.: Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems. J. Math. Anal. Appl. 389(2), 1046–1058 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gutiérrez, C., Jiménez, B., Novo, V.: Improvement sets and vector optimization. Eur. J. Oper. Res. 223(2), 304–311 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Proper approximate solutions and \(\varepsilon \)-subdifferentials in vector optimization: basic properties and limit behaviour. Nonlinear Anal. 79, 52–67 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Henig approximate proper efficiency and optimization problems with difference of vector mappings. J. Convex Anal. 23(3), 661–690 (2016)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hernández, E., Jiménez, B., Novo, V.: Weak and proper efficiency in set-valued optimization on real linear spaces. J. Convex Anal. 14(2), 275–296 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hiriart-Urruty, J.-B.: \(\varepsilon \)-Subdifferential calculus. In: Aubin, J.P., Vinter, R.B. (eds.) Convex Analysis and Optimization, Research Notes in Mathematics 57, pp. 43–92. Pitman, Boston (1982)Google Scholar
  14. 14.
    Jahn, J.: Vector Optimization. Theory, Applications, and Extensions. Springer, Berlin (2011)zbMATHGoogle Scholar
  15. 15.
    Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization. An Introduction with Applications. Springer, Berlin (2015)zbMATHGoogle Scholar
  16. 16.
    Kiyani, E., Soleimani-damaneh, M.: Approximate proper efficiency on real linear vector spaces. Pac. J. Optim. 10(4), 715–734 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kutateladze, S.S.: Convex \(\varepsilon \)-programming. Sov. Math. Dokl. 20, 391–393 (1979)zbMATHGoogle Scholar
  18. 18.
    Lalitha, C.S., Chatterjee, P.: Stability and scalarization in vector optimization using improvement sets. J. Optim. Theory Appl. 166(3), 825–843 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)CrossRefGoogle Scholar
  20. 20.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II. Applications. Springer, Berlin (2006)Google Scholar
  21. 21.
    Taa, A.: \(\varepsilon \)-Subdifferentials of set-valued maps and \(\varepsilon \)-weak Pareto optimality for multiobjective optimization. Math. Methods Oper. Res. 62(2), 187–209 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Xia, Y.M., Zhang, W.L., Zhao, K.Q.: Characterizations of improvement sets via quasi interior and applications in vector optimization. Optim. Lett. 10, 769–780 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhao, K.Q., Yang, X.M.: Characterizations of the E-Benson proper efficiency in vector optimization problems. Numer. Algebra Control Optim. 3(4), 643–653 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhao, K.Q., Yang, X.M., Peng, J.W.: Weak E-optimal solution in vector optimization. Taiwan. J. Math. 17(4), 1287–1302 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhou, Z.A., Peng, J.W.: Scalarization of set-valued optimization problems with generalized cone subconvexlikeness in real ordered linear spaces. J. Optim. Theory Appl. 154(3), 830–841 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhou, Z.A., Yang, X.M., Peng, J.W.: \(\varepsilon \)-Henig proper efficiency of set-valued optimization problems in real ordered linear spaces. Optim. Lett. 8(6), 1813–1827 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhou, Z.A., Yang, X.M., Peng, J.W.: \(\varepsilon \)-Optimality conditions of vector optimization problems with set-valued maps based on the algebraic interior in real linear spaces. Optim. Lett. 8(3), 1047–1061 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.IMUVA (Institute of Mathematics of University of Valladolid)ValladolidSpain
  2. 2.Departamento de Matemática Aplicada, E.T.S.I. IndustrialesUniversidad Nacional de Educación a Distancia (UNED)MadridSpain

Personalised recommendations