A Note on Essential Components and Essential Weakly Efficient Solutions for Multiobjective Optimization Problems
© Qun Luo. 2009
Received: 19 June 2009
Accepted: 24 November 2009
Published: 30 December 2009
The concept of essential component of weakly efficient solution set is introduced first. Then we obtain some sufficient conditions for the existence of an essential component or an essential weakly efficient solution in the weakly efficient solution sets for multiobjective optimization problems.
In [1–7] and the references therein, the authors have studied the existence and the stability of (vector-valued, set-valued, semi-infinite vector) optimization and multiobjective optimization problems, while the author in  shows that most of the weakly efficient solution sets of multiobjective optimization problems (in the sense of Baire category) are stable. By the fact that there are still quite a few weakly efficient solution sets of multiobjective optimization problems which are not stable, in this paper, we discusses the stability of solution set for multiobjective optimization problems from the perspective of essential components.
2. Definitions and Lemmas
Example 2.2 (see ).
The following Definition 2.6 is from .
Then, , and for all , . By [3, Theorem ], has no essential component.
Then, , and for all , . By [3, Theorem ], contains no essential weakly efficient solution.
3. Essential Component and Essential Weakly Efficient Solution
Corollary 3.2 (see ).
By Theorem 3.1 and Lemma 3.3, we have the following Theorem 3.4.
Let be a nonempty compact convex subset of a Banach space, for , if there exists such that is strongly quasiconvex, then has an essential weakly efficient solution, consequently, has an essential component.
When , by , the optimum solution set (where ) has an essential component if and only if is connected. But, when , it is not true.
When , by , for , has an essential component if and only if is connected, and has an essential optimum solution if and only if is a singleton.
When , we obtain some sufficient conditions for the existence of an essential component or an essential weakly efficient solution in . Example 3.7 shows that disconnected, but has an essential component and is not a singleton, but has an essential weakly efficient solution. Example 3.8 shows that, if is not a singleton, for some , then for any , is not an essential weakly efficient solution.
The author expresses her sincerely thanks to anonymous referees for their comments and suggestions leading to the present version of this paper. This research was supported by the Natural Science Foundation of Guangdong Province (9251064101000015), China.
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