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Vector Optimization Problems with Generalized Functional Constraints in Variable Ordering Structure Setting

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Abstract

This article aims at studying a special class of constrained vector optimization problems in the setting of a variable ordering structure, having a geometric constraint and finitely many generalized functional constraints. Based on a version of the extended extremal principle, some necessary optimality conditions for a minimal solution of the proposed problem are derived in terms of coderivatives and normal cones.

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References

  1. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  2. Borwein, J.M., Treiman, J.S., Zhu, Q.J.: Necessary conditions for constrained optimization problems with semicontinuous and continuous data. Trans. Am. Math. Soc. 350, 2409–2429 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Zheng, X.Y., Ng, K.F.: The Lagrange multiplier rule for multifunctions in Banach spaces. SIAM J. Optim. 17, 1154–1175 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  5. Yu, P.L.: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14, 319–377 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bergstresser, K., Yu, P.L.: Domination structures and multicriteria problems in N-person games. Theory Decis. 8, 5–48 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Yu, P.L.: Multiple-Criteria Decision Making: Concepts, Techniques and Extensions. Plenum Press, New York (1985)

    Book  Google Scholar 

  8. Bao, T.Q., Mordukhovich, B.S.: Necessary nondomination conditions in set and vector optimization with variable ordering structures. J. Optim. Theory Appl. 162, 350–370 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Durea, M., Strugariu, R., Tammer, C.: On set-valued optimization problems with variable ordering structure. J. Glob. Optim. 61, 745–767 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eichfelder, G.: Optimal elements in vector optimization with a variable ordering structure. J. Optim. Theory Appl. 151, 217–240 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eichfelder, G.: Variable Ordering Structures in Vector Optimization. Springer, Heidelberg (2014)

    Book  MATH  Google Scholar 

  12. Eichfelder, G., Ha, T.X.D.: Optimality conditions for vector optimization problems with variable ordering structures. Optimization 62, 597–627 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, G.Y., Yang, X.Q.: Characterizations of variable domination structures via nonlinear scalarization. J. Optim. Theory Appl. 112, 97–110 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, G.Y., Huang, X., Yang, X.: Vector Optimization, Set-Valued and Variational Analysis. Springer, Berlin (2005)

    MATH  Google Scholar 

  15. Penot, J.-P.: Cooperative behavior of functions, relations and sets. Math. Methods Oper. Res. 48, 229–246 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, S., Penot, J.-P., Xue, X.: Codifferential calculus. Set-Valued Var. Anal. 19, 505–536 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Durea, M., Huynh, V.N., Nguyen, H.T., Strugariu, R.: Metric regularity of composition set-valued mappings: metric setting and coderivative conditions. J. Math. Anal. Appl. 412, 41–62 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Florea, E.-A.: Coderivative necessary optimality conditions for sharp and robust efficiencies in vector optimization with variable ordering structure. Optimization 65, 1417–1435 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Durea, M., Strugariu, R.: On some Fermat rules for set-valued optimization problems. Optimization 60, 575–591 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mordukhovich, B.S., Treiman, J.S., Zhu, Q.J.: An extended extremal principle with applications to multiobjective optimization. SIAM J. Optim. 14, 359–379 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Ioffe, A.: Necessary conditions for nonsmooth optimization. Math. Oper. Res. 9, 59–189 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zheng, X.Y., Ng, K.F.: A unified separation theorem for closed sets in a Banach space and optimality conditions for vector optimization. SIAM J. Optim. 21, 886–911 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Soleimani, B., Tammer, C.: Optimality conditions for approximate solutions of vector optimization problems with variable ordering structures. Bull. Iran. Math. Soc. 42, 5–23 (2016)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is grateful for the comments and suggestions of the editor and the two anonymous reviewers. Also, the author expresses his particular gratitude to Prof. Franco Giannessi, whose remarks allowed to improve considerably the original submission of this paper. This research was supported by a grant of Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, Project Number PN-III-P4-ID-PCE-2016-0188, within PNCDI III.

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  1. Faculty of Mathematics, “Alexandru Ioan Cuza” University, Bd. Carol I, nr. 11, 700506, Iasi, Romania

    Elena-Andreea Florea

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  1. Elena-Andreea Florea
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Correspondence to Elena-Andreea Florea.

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Xiaoqi Yang.

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Florea, EA. Vector Optimization Problems with Generalized Functional Constraints in Variable Ordering Structure Setting. J Optim Theory Appl 178, 94–118 (2018). https://doi.org/10.1007/s10957-018-1269-6

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  • DOI: https://doi.org/10.1007/s10957-018-1269-6

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