Abstract

There are many generalizations of Ekeland's variational principle for vector optimization problems with fixed ordering structures, i.e., ordering cones. These variational principles are useful for deriving optimality conditions, ε-Kolmogorov conditions in approximation theory, and ε-maximum principles in optimal control. Here, we present several generalizations of Ekeland's variational principle for vector optimization problems with respect to variable ordering structures. For deriving these variational principles we use nonlinear scalarization techniques. Furthermore, we derive necessary conditions for approximate solutions of vector optimization problems with respect to variable ordering structures using these variational principles and the subdifferential calculus by Mordukhovich

Author information

Contact details are reproduced from the original publication and may be historical.

Truong Quang Bao

Dept. of Mathematics and Computer Science, Northern Michigan University, 1401 Presque Isle Avenue, 1135 NSF Marquette, MI 49855, U.S.A.

btruong@nmu.edu

Keywords

  • Nonconvex vector optimization
  • variable ordering structure
  • Ekeland’s variational principle
  • separation theorem
  • limiting (Mordukhovich) subdifferential

References

  1. T. Q. Bao, B. S. Mordukhovich: Relative Pareto minimizers in multiobjective optimization: existence and optimality conditions, Math. Progr. 122 (2010) 301–347.https://doi.org/10.1007/s10107-008-0249-2Google Scholar
  2. T. Q. Bao, B. S. Mordukhovich: Necessary nondomination conditions in set and vector optimization with variable ordering structures, J. Optim. Theory Appl. 162 (2014) 350–370.https://doi.org/10.1007/s10957-013-0332-6Google Scholar
  3. T. Q. Bao, B. S. Mordukhovich, A. Soubeyran: Variational analysis in Psychological modeling, J. Optim. Theory Appl. 164 (2015) 290–315.https://doi.org/10.1007/s10957-014-0569-8Google Scholar
  4. J. Y. Bello Cruz, G. Bouza Allende: A steepest descent-like method for variable order vector optimization problems, J. Optim. Theory Appl. 162 (2014) 371–391.https://doi.org/10.1007/s10957-013-0308-6Google Scholar
  5. G.-Y. Chen, X. Huang, X. Q. Yang: Vector Optimization, Set-Valued and Variational Analysis, Springer, Heidelberg (2005).Google Scholar
  6. G.-Y. Chen, X. Q. Yang: Characterizations of variable domination structures via nonlinear scalarization, J. Optim. Theory Appl. 112 (2002) 97–110.https://doi.org/10.1023/A:1013044529035Google Scholar
  7. G.-Y. Chen, X. Q. Yang, H. Yu: A nonlinear scalarization function and generalized quasi-vector equilibrium problems, J. Glob. Optim. 32 (2005) 451–466.https://doi.org/10.1007/s10898-003-2683-2Google Scholar
  8. L. C. Ceng, S. Huang: Existence theorems for generalized vector variational inequalities with a variable ordering relation, J. Glob. Optim. 46 (2010) 521–535.https://doi.org/10.1007/s10898-009-9436-9Google Scholar
  9. M. Durea, C. Tammer: Fuzzy necessary optimality conditions for vector optimization problems, Optimization 58 (2009) 449–467.https://doi.org/10.1080/02331930701761615Google Scholar
  10. G. Eichfelder: Optimal elements in vector optimization with a variable ordering structure, J. Optim. Theory Appl. 151 (2011) 217–240.https://doi.org/10.1007/s10957-011-9928-xGoogle Scholar
  11. G. Eichfelder: Variable ordering structures in vector optimization, in: Recent Developments in Vector Optimization, Q. H. Ansari et al. (ed.), Springer, Berlin (2012) 95–126.https://doi.org/10.1007/978-3-642-21114-0_4Google Scholar
  12. G. Eichfelder: Cone-valued maps in optimization, Appl. Anal. 91 (2012) 1831–1846.https://doi.org/10.1080/00036811.2011.616499Google Scholar
  13. G. Eichfelder: Variable Ordering Structures in Vector Optimization, Springer, Heidelberg (2014).https://doi.org/10.1007/978-3-642-54283-1Google Scholar
  14. G. Eichfelder, T. X. D. Ha: Optimality conditions for vector optimization problems with variable ordering structures, Optimization 62 (2013) 597–627.https://doi.org/10.1080/02331934.2011.575939Google Scholar
  15. I. Ekeland: Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979) 443–474.https://doi.org/10.1090/S0273-0979-1979-14595-6Google Scholar
  16. C. Gert (Tammer), P. Weidner: Nonconvex separation theores and some applications in vector optimization, J. Optim. Theory Appl. 67 (1990) 297–320.https://doi.org/10.1007/BF00940478Google Scholar
  17. A. Göpfert, H. Riahi, C. Tammer, C. Zălinescu: Variational Methods in Partially Ordered Spaces, Springer, New York (2003).Google Scholar
  18. G. Göpfert, T. Riedrich, C. Tammer: Angewandte Funktionalanalysis, Vieweg Teubner, Wiesbaden (2009).https://doi.org/10.1007/978-3-8348-9572-1Google Scholar
  19. C. Hirsch, P. K. Shukla, H. Schmeck: Variable preference modeling using multiobjective evolutionary algorithms, in: Evolutionary Multi-criterion Optimization (Ouro Preto, 2011), R. H. C. Takahashi et al. (ed.), Lecture Notes in Computer Science 6576, Springer, Heidelberg (2011) 91–105.https://doi.org/10.1007/978-3-642-19893-9_7Google Scholar
  20. N. J. Huang, X. Q. Yang, W. K. Chan: Vector complementarity problems with a variable ordering relation, Eur. J. Oper. Res. 176 (2007) 15–26.https://doi.org/10.1016/j.ejor.2005.06.046Google Scholar
  21. J. Jahn: Vector Optimization. Theory, Applications and Extensions, 2nd Ed., Springer, Heidelberg (2011).https://doi.org/10.1007/978-3-642-17005-8Google Scholar
  22. E. Köbis: Variable ordering structures in set optimization, Preprint Univ. Erlangen-Nürnberg 378 (2014).Google Scholar
  23. P. Loridan: ε-solutions in vector minimization problems, J. Optim. Theory Appl. 43 (1984) 265–276.https://doi.org/10.1007/BF00936165Google Scholar
  24. D. T. Luc, A. Soubeyran: Variable preference relations: existence of maximal elements, J. Math. Econ. 49 (2013) 251–262.https://doi.org/10.1016/j.jmateco.2013.05.004Google Scholar
  25. G. Mastroeni: On the image space analysis for vector quasi-equilibrium problems with a variable ordering relation, J. Glob. Optim. 53 (2012) 203–214.https://doi.org/10.1007/s10898-011-9674-5Google Scholar
  26. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. I: Basic Theory, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  27. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. II: Applications, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  28. A. Pascoletti, P. Serafini: Scalarizing vector optimization problems, J. Optim. Theory Appl. 42 (1984) 499–524.https://doi.org/10.1007/BF00934564Google Scholar
  29. P. K. Shukla, M. A. Braun: Indicator based search in variable orderings: theory and algorithms, in: Evolutionary Multi-Criterion Optimization (Sheffield, 2013), R. C. Purshouse et al. (ed.), Lecture Notes in Computer Science 7811, Springer, Heidelberg (2013) 66–80.https://doi.org/10.1007/978-3-642-37140-0_9Google Scholar
  30. B. Soleimani: Characterization of approximate solutions of vector optimization problems with a variable order structure, J. Optim. Theory Appl. 162 (2014) 605–632.https://doi.org/10.1007/s10957-014-0535-5Google Scholar
  31. B. Soleimani, C. Tammer: Approximate solutions of vector optimization problem with variable ordering structure, AIP Conf. Proc. 1479 (2012) 2363–2366.https://doi.org/10.1063/1.4756669Google Scholar
  32. B. Soleimani, C. Tammer: Concepts for approximate solutions of vector optimization problem with variable order structure, Vietnam J. Math. 42(4) (2014) 543–566.https://doi.org/10.1007/s10013-014-0103-1Google Scholar
  33. C. Tammer: A generalization of Ekeland’s variational principle, Optimization 5 (1992) 129–141.https://doi.org/10.1080/02331939208843815Google Scholar
  34. C. Tammer, C. Zălinescu: Vector variational principles for set-valued functions, in: Recent Developments in Vector Optimization, Q. H. Ansari et al. (ed.), Springer, Berlin (2012) 367–415.https://doi.org/10.1007/978-3-642-21114-0_11Google Scholar
  35. P. L. Yu: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974) 319–377.https://doi.org/10.1007/BF00932614Google Scholar

Suggested citation

T. Q. Bao, G. Eichfelder, B. Soleimani, C. Tammer. “Ekeland's Variational Principle for Vector Optimization with Variable Ordering Structure.” Journal of Convex Analysis 24 (2017), No. 2, 393–415. https://doi.org/10.68381/jca24027

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.