A new ABB theorem in normed vector spaces

Optimization

Volumn 53, Issue 4, 2004, Pages 369-376

  Gopfert A ^c   Tammer, Chr ^a   Zalinescu C ^b  

^a MARTIN LUTHER UNIVERSITY HALLE WITTENBERG   (Germany)

^b ALEXANDRU IOAN CUZA UNIVERSITY OF IASI   (Romania)

^c NONE

Author keywords

Asymptotic cone; Asymptotically compact set; Density results; Henig efficient point; Maximal point; Properly efficient point

Indexed keywords

EID: 4444359686     PISSN: 02331934     EISSN: None     Source Type: Journal    
DOI: 10.1080/02331930412331282427     Document Type: Article

References (25)

1. K.J. Arrow, E.W. Barankin, D. Blackwell (1953). Admissible points of convex sets. In: H.W. Kuhn and A.W. Tucker (Eds.), Contributions to the Theory of Games 2. Ann. of Math. Studies, 28, pp. 87-91. Princeton Univ. Press, Princeton.
2. J.M. Borwein and D. Zhuang (1993). Super efficiency in vector optimization. Trans. Amer. Math. Soc., 338, 105-122.
3. A. Daniilidis (2000). Arrow-Barankin-Blackwell theorems and related results in cone duality: a survey. In: V.H. Nguyen, J.-J. Strodiot and P. Tossings (Eds.), Optimization, Lecture Notes in Econom. and Math. Systems, Vol. 481, pp. 119-131. Springer, Berlin Heidelberg.
4. Dedieu, J.-P. (1978). Critères de fermeture pour l'image d'un fermé non convexe par une multiapplication. C. R. Acad. Sci., Paris Ser. 1, 287, 941-943.
5. F. Ferro (1991). A generalization of the Arrow-Barankin-Blackwell theorem in normed spaces. J. Math. Anal. Appl., 158, 47-54.
6. F. Ferro (1999). A new ABB theorem in Banach spaces. Optimization, 46, 353-362.
7. W. Fu (1996). On the density of proper efficient points. Proc. Amer. Math. Soc., 124, 1213-1217.
8. R.J. Gallagher (1995). The Arrow-Barankin-Blackwell theorem in a dual space setting. J. Optim. Theory Appl., 84, 665-674.
9. R.J. Gallagher and O.A. Saleh (1993). Two generalizations of a theorem of Arrow, Barankin and Blackwell. SIAM J. Control Optim., 31, 247-256.
10. A. Göpfert, H. Riahi, Chr. Tammer and C. Zǎlinescu (2003). Variational Methods in Partially Ordered Spaces. Springer, New York.
11. X.H. Gong (1994). Connectedness of efficient solution sets for set-valued maps in normed spaces. J. Optim. Theory Appl., 83, 83-96.
12. X.H. Gong (1995). Density of the set of positive proper minimal points in the set of minimal points. J. Optim. Theory Appl., 86, 609-630.
13. M.I. Henig (1982). Proper efficiency with respect to cones. J. Optim. Theory Appl., 36, 387-407.
14. J. Jahn (1988). A generalization of a theorem of Arrow, Barankin and Blackwell. SIAM J. Control Optim., 26, 999-1005.
15. D.T. Luc (1989). Theory of Vector Optimization, Vol. 319, Lecture Notes in Econom. and Math. Systems, Springer, Berlin.
16. E.K. Makarov and N.N. Rachkowski (1996). Density theorems for generalized Henig proper efficiency. J. Optim. Theory Appl., 91, 419-437.
17. M. Petschke (1990). On a theorem of Arrow, Barankin, and Blackwell. SIAM J. Control Optim., 28, 395-401.
18. W. Song (1997). Generalization of the Arrow-Barankin-Blackwell theorem in a dual space setting. J. Optim. Theory Appl., 95, 225-230.
19. Y. Sonntag and C. Zǎlinescu (2000). Comparison of existence results for efficient points. J. Optim. Theory Appl., 105, 161-188.
20. A. Sterna-Karwat (1989). Approximating families of cones and proper efficiency of vector optimization. Optimization, 20, 799-808.
21. X.D.H. Truong (2001). Existence and density results for proper efficiency in cone compact sets. J. Optim. Theory Appl., 111, 173-194.
22. L.W. Woo and L.W. Goodrich (2003). Maximal points of convex sets in locally convex topological vector spaces: Generalization of the Arrow-Barankin-Blackwell theorem. J. Optim. Theory Appl., 116, 647-658.
23. C. Zǎlinescu (1989). Stability for a class of nonlinear optimization problems and applications. In: Nonsmooth Optimization and Related Topics (Erice, 1988), pp. 437-458. Plenum, New York.
24. D. Zhuang (1994). Density results for proper efficiency. SIAM J. Control Optim., 32, 51-58.
25. X.Y. Zheng (1998). Generalizations of a theorem of Arrow, Barankin, and Blackwell in topological vector spaces. J. Optim. Theory Appl., 96, 221-233.