Scalarization of \(\epsilon \)-Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces

1. Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)

   Article  MATH  MathSciNet  Google Scholar 

2. Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979)

   Article  MATH  MathSciNet  Google Scholar 

3. Borwein, J.M.: Proper efficient points for maximizations with respect to cones. SIAM J. Control. Optim. 15, 57–63 (1977)

   Article  MATH  Google Scholar 

4. Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982)

   Article  MATH  MathSciNet  Google Scholar 

5. Borwein, J.M., Zhuang, D.M.: Super efficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993)

   Article  MATH  MathSciNet  Google Scholar 

6. Zheng, X.Y.: Proper efficiency in locally convex topological vector spaces. J. Optim. Theory Appl. 94, 469–486 (1997)

   Article  MATH  MathSciNet  Google Scholar 

7. Zhou, Z.A., Yang, X.M., Peng, J.W.: \(\epsilon \)-Henig proper efficiency of set-valued optimization problems in real ordered linear spaces, Optim. Lett. (2013). doi:10.1007/s11590-013-0667-9.

8. Li, Z.M.: The optimality conditions for vector optimization of set-valued maps. J. Math. Anal. Appl. 237, 413–424 (1999)

   Article  MATH  MathSciNet  Google Scholar 

9. Adán, M., Novo, V.: Proper efficiency in vector optimization on real linear spaces. J. Optim. Theory Appl. 121, 515–540 (2004)

   Article  MATH  MathSciNet  Google Scholar 

10. Hernández, E., Jiménez, B., Novo, V.: Weak and proper efficiency in set-valued optimization on real linear spaces. J. Convex. Anal. 14, 275–296 (2007)

    MATH  MathSciNet  Google Scholar 

11. Rong, W.D., Wu, Y.N.: \(\epsilon \)-Weak minimal solutions of vector optimization problems with set-valued maps. J. Optim. Theory Appl. 106, 569–579 (2000)

    Article  MATH  MathSciNet  Google Scholar 

12. Li, T.Y., Xu, Y.H., Zhu, C.X.: \(\varepsilon \)-Strictly efficient solutions of vector optimization problems with set-valued maps. Asia Pac. J. Oper. Res. 24, 841–854 (2007)

    Article  MATH  MathSciNet  Google Scholar 

13. Tuan, L.A.: \(\varepsilon \)-Optimality conditions for vector optimization problems with set-valued maps. Numer. Func. Anal. Optim. 31, 78–95 (2010)

    Article  MATH  MathSciNet  Google Scholar 

14. 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, 830–841 (2012)

    Article  MATH  MathSciNet  Google Scholar 

15. Zhou, Z.A., Yang, X.M., Peng, J.W.: \(\epsilon \)-optimality conditions of vector optimization problems with set-valued maps based on the algebraic interior in real linear spaces. Optim. Lett. 8, 1047–1061 (2014)

16. Tiel, J.V.: Convex Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

17. Adán, M., Novo, V.: Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness. Eur. J. Oper. Res. 149, 641–653 (2003)

    Article  MATH  Google Scholar 

18. Shi, S.Z.: Convex Analysis. Shanghai Science and Technology Press, Shanghai (1990)

    Google Scholar 

19. Xu, D.Z., Yao, Q.L.: Topological Linear Spaces. Lanzhou University Press, Lanzhou (1987)

    Google Scholar 

20. Huang, Y.W., Li, Z.M.: Optimality condition and Lagrangian multipliers of vector optimization with set-valued maps in linear spaces. Oper. Res. Trans. 5, 63–69 (2001)

    Google Scholar 

21. Xia, L.Y., Qiu, J.H.: Superefficiency in vector optimization with nearly subconvexlike set-valued maps. J. Optim. Theory Appl. 136, 125–137 (2008)

    Article  MATH  MathSciNet  Google Scholar 

22. Gong, X.H., Fu, W.T., Liu, W.: Super efficiency for a vector equilibrium in locally convex topological vector spaces. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 233–252. Kluwer Academic, Netherlands (2000)

    Chapter  Google Scholar 

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