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Henig globally efficiency for set-valued optimization and vector variational inequality

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Abstract

This paper deals with Henig globally efficiency in vector optimization involving generalized cone-preinvex set-valued mapping. Some properties of generalized cone-preinvex set-valued map are derived. It also disclose the closed relationships between Henig globally efficiency of generalized conepreinvex set-valued optimization problem and Henig globally efficiency of a kind of vector variational inequality.

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References

  1. Weir T and Mond B, Preinvex functions in multiple-objective optimization, Journal of Mathematical Analysis and Applications, 1988, 136: 29–38.

    Article  MATH  MathSciNet  Google Scholar 

  2. Bhatiaa D and Mehra A, Lagrangian duality for preinvex set-valued functions, Journal of Mathematical Analysis and Applications, 1997, 214(2): 599–612.

    Article  MathSciNet  Google Scholar 

  3. Mishra S K, Giorgi G, and Wang S Y, Duality in vector optimization in Banach spaces with generalized convexity, Journal of Global Optimization, 2004, 29: 415–424.

    Article  MATH  MathSciNet  Google Scholar 

  4. Noor M A, On generalized preinvex functions and monotonicities, J. Inequal. Pure Appl. Math., 2004, 5(4): 1–9.

    Google Scholar 

  5. Jabarootian T and Zafarani J, Characterizations of preinvex and prequasiinvex set-valued maps, Taiwanese Journal of Mathematics, 2009, 113(3): 871–898.

    MathSciNet  Google Scholar 

  6. Yu G L and Liu S Y, Some vector optimization problems in Banach spaces with generalized convexity, Computers and Mathematics with Applications, 2007, 54: 1403–1410.

    Article  MATH  MathSciNet  Google Scholar 

  7. Yu G L, Directional derivatives and generalized preinvex set-valued optimization, Acta Mathematica Sinica, 2011, 54(5): 705–880.

    MathSciNet  Google Scholar 

  8. Qiu J H, Cone-directed contingent derivatives and generalized preinvex set-valued optimization, Acta Mathematica Scientia, 2007, 27B(1): 211–218.

    Article  Google Scholar 

  9. Borwein J M and Zhuang D M, Super efficiency in vector optimization, Trans. Amer. Math. Soc., 1993, 338: 105–122.

    Article  MATH  MathSciNet  Google Scholar 

  10. Benson H P, An improved definition of proper efficiency for vector maximization with respect to cones, Journal of Mathematical Analysis and Applications, 1979, 71: 232–241.

    Article  MATH  MathSciNet  Google Scholar 

  11. Henig M I, Proper efficiency with respect to cones, Journal of Optimization Theory and Applications, 1982, 36: 387–407.

    Article  MATH  MathSciNet  Google Scholar 

  12. Gong X H, Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior, Journal of Mathematical Analysis and Applications, 2005, 307: 12–31.

    Article  MATH  MathSciNet  Google Scholar 

  13. Gong X H, Dong H B, and Wang S Y, Optimality conditions for proper efficient solutions of vector set-valued optimization, Journal of Mathematical Analysis and Applications, 2003, 284: 332–350.

    Article  MATH  MathSciNet  Google Scholar 

  14. Yu G L and Liu S Y, Optimality conditions of globally proper efficient solutions for set-valued optimization problem, Acta Mathematica Applicatae Sinica (Chinese Series), 2010, 33(1): 150–160.

    MATH  Google Scholar 

  15. Yu G L and Liu S Y, Globally proper saddle point in ic-cone-convexlike set-valued optimization problems, Acta Mathematica Sinica (English Series), 2009, 25(11): 1921–1928.

    Article  MATH  MathSciNet  Google Scholar 

  16. Giannessi F, Variational Inequalities and Complementarity Problems, Wiley, New York, 1980.

    Google Scholar 

  17. Noor M A, Preinvex functions and variational inequalities, J. Nat. Geometry, 1996, 9: 63–76.

    MATH  Google Scholar 

  18. Khan M F, On generalized vector variational-like inequalities, Nonlinear Anal., 2004, 59: 879–889.

    Article  MATH  MathSciNet  Google Scholar 

  19. Ruiz-Garzon G, Osuna-Gomez R, and Rufian-Lizana A, Relationships between vector variationallike inequality and optimization problems, Eur. J. Oper. Res., 2004, 157: 113–119.

    Article  MATH  MathSciNet  Google Scholar 

  20. Fang Y P and Huang N J, Variational-like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory Appl., 2003, 118: 327–338.

    Article  MATH  MathSciNet  Google Scholar 

  21. Liu W and Gong X H, Proper efficiency for set-valued vector optimization problems and vector variational inequalites, Mathematical Methods of Operations Research, 2000, 51: 443–457.

    Article  MATH  MathSciNet  Google Scholar 

  22. Zeng J and Li S J, On vector variational-like inequalities and set-valued optimization problems, Optimization Letters, 2011, 5(1): 55–69.

    Article  MATH  MathSciNet  Google Scholar 

  23. Rezaie M and Zafarani J, Vector optimization and variational-like inequalities, Journal of Global Optimization, 2009, 43(1): 47–66.

    Article  MATH  MathSciNet  Google Scholar 

  24. Jahn J and Rauh R, Contingent epiderivatives and set-valued optimzation, Mathematical Methods of Operation Research, 1997, 46: 193–211.

    Article  MATH  MathSciNet  Google Scholar 

  25. Jahn J and Rauh R, The existence of contingent epiderivatives for set-valued maps, Applied Mathematics Letters, 2003, 16(8): 1179–1185.

    Article  MATH  MathSciNet  Google Scholar 

  26. Chen G Y and Jahn J, Optimality conditions for set-valued optimization problem, Mathematical Methods of Operation Research, 1998, 48: 187–200.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics, Beifang University of Nationalities, Yinchuan, 750021, China

    Guolin Yu

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  1. Guolin Yu
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Correspondence to Guolin Yu.

Additional information

This research was supported by the Natural Science Foundation of China under Grant No. 11361001; Ministry of Education Science and technology key projects under Grant No. 212204; the Natural Science Foundation of Ningxia under Grant No. NZ12207; and the Science and Technology key project of Ningxia institutions of higher learning under Grant No. NGY2012092.

This paper was recommended for publication by Editor DAI Yuhong.

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Yu, G. Henig globally efficiency for set-valued optimization and vector variational inequality. J Syst Sci Complex 27, 338–349 (2014). https://doi.org/10.1007/s11424-014-1215-0

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  • DOI: https://doi.org/10.1007/s11424-014-1215-0

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