Abstract
In this paper, we study Henig weakly efficient solutions for set-valued optimization problems. The connectedness of the Henig weakly efficient solution set is proved under the condition that the objective function be a cone-arcwise connected set-valued mapping. As an application of the result, we establish the connectedness of the set of super efficient solutions.
Similar content being viewed by others
References
Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)
Gong, X.H.: Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior. J. Math. Anal. Appl. 307, 12–31 (2005)
Gong, X.H.: Connectedness of super efficient solution sets for set-valued maps in Banach spaces. Math. Methods Oper. Res. 44, 135–145 (1996)
Gong, X.H.: Connectedness of efficient solution sets for set-valued maps in normed spaces. J. Optim. Theory Appl. 83, 83–96 (1994)
Li, Z.F., Wang, S.Y.: Connectedness of super efficient sets in vector optimization of set-valued maps. Math. Methods Oper. Res. 48, 207–217 (1998)
Lalitha, C.S., Dutta, J., Govil, M.G.: Optimality criteria in set-valued optimization. J. Aust. Math. Soc. 75, 221–231 (2003)
Mehra, A.: Super efficiency in vector optimization with nearly convexlike set-valued maps. J. Math. Anal. Appl. 276, 815–832 (2002)
Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110, 413–427 (2001)
Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Pardalos, P.M. et al. (eds.) Nonlinear Analysis and Variational Problems, pp. 305–324. Springer, Berlin (2010)
Borwein, J.M., Zhuang, D.M.: Super efficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993)
Borwein, J.M., Zhuang, D.M.: Super efficiency in convex vector optimization. Math. Methods Oper. Res. 35, 75–84 (1991)
Dauer, J.P., Gallagher, R.J.: Positive proper efficient points and related cone results in vector optimization theory. SIAM J. Control Optim. 28, 158–172 (1990)
Jahn, J.: Mathematical Vector Optimization in Partially Order Linear Spaces. Peter Lang, Frankfurt (1986)
Zheng, X.Y.: Proper efficiency in locally convex topological vector spaces. J. Optim. Theory Appl. 94, 469–486 (1997)
Qiu, Q.S.: On Henig proper efficiency. J. Syst. Sci. Math. Sci. 31(4), 482–488 (2011)
Hiriart-Urruty, J.B.: Images of connected sets by semicontinuous multifunctions. J. Math. Anal. Appl. 111, 407–422 (1985)
Avriel, M., Zang, I.: Generalized arcwise connected functions and characterizations of local-global minimum properties. J. Optim. Theory Appl. 32, 407–425 (1980)
Robertson, A.P., Robertson, W.J.: Topological Vector Space. Cambridge University Press, London (1964)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jafar Zafarani.
The authors express their sincere gratitude to Professor J. Zafarani and the referees for comments and valuable suggestions. Q.S. Qiu was supported by the National Natural Science Foundation of China (Grant 11061023, 10831009), X.M. Yang was supported by the National Natural Science Foundation of China (Grant 10831009).
Rights and permissions
About this article
Cite this article
Qiu, Q.S., Yang, X.M. Connectedness of Henig Weakly Efficient Solution Set for Set-Valued Optimization Problems. J Optim Theory Appl 152, 439–449 (2012). https://doi.org/10.1007/s10957-011-9906-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-011-9906-3