- Research Article
- Open Access

# A Note on Essential Components and Essential Weakly Efficient Solutions for Multiobjective Optimization Problems

- Qun Luo
^{1}Email author

**2009**:928968

https://doi.org/10.1155/2009/928968

© Qun Luo. 2009

**Received:**19 June 2009**Accepted:**24 November 2009**Published:**30 December 2009

## Abstract

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.

## Keywords

- Banach Space
- Compact Subset
- Convex Subset
- Closed Subset
- Distinct Point

## 1. Introduction

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 [2] 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

Let be a nonempty compact subset of a Banach space ,

Definition 2.1.

Clearly, , but the reverse containment may not hold.

Example 2.2 (see [8]).

Then , .

It is easy to see that , and (VMP) is just a scalar optimization problem, in this case, we still denote by or the set of optimal solution.

Remark 2.3.

Let , for all , one has .

Clearly, is a complete metric space.

For any , by [2], it has been shown that is a nonempty compact subset, and the following Lemma 2.4 is due to [2].

Lemma 2.4.

The mapping is upper semicontinuous with nonempty compact values.

where is an index set, for any , is a nonempty connected compact and for any , .

Definition 2.5.

For , is called an essential component of if, for any open set containing , there exists such that for all with , .

The following Definition 2.6 is from [2].

Definition 2.6.

For , is said to be an essential weakly efficient solution to (VMP) if, for any open neighborhood of in , there exists an open neighborhood at in such that for all .

Remark 2.7.

For , if is an essential weakly efficient solution to (VMP), then the component which contains the point is an essential component.

Remark 2.8.

For , maybe, there is no essential component in , and no essential weakly efficient solution in .

Example 2.9.

Then, , and for all , . By [3, Theorem ], has no essential component.

Example 2.10.

Then, , and for all , . By [3, Theorem ], contains no essential weakly efficient solution.

Definition 2.11.

for all , , .

## 3. Essential Component and Essential Weakly Efficient Solution

Theorem 3.1.

For , , if there is such that , then is an essential weakly efficient solution of (VMP). Hence, the component that contains the point is an essential component.

Proof.

Suppose that , , and there exists such that . For any open neighborhood of in , there is an open neighborhood of in such that , where denotes the closure of

Then, , and hence , which implies . By Definition 2.6, is an essential weakly efficient solution to (VMP). Hence, the component that contains the point is an essential component.

Corollary 3.2 (see [2]).

When , for , if is a singleton, then is an essential optimum solution.

Lemma 3.3.

Let be a nonempty compact convex subset of a Banach space, the function continuous and strongly quasiconvex, then is a singleton.

Proof.

which is a contradiction, then is a singleton.

By Theorem 3.1 and Lemma 3.3, we have the following Theorem 3.4.

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.

Theorem 3.5.

For , if has only one component , then is an essential component.

Proof.

By Lemma 2.4, the mapping is upper semicontinuous at , hence, for any open set with , there exists such that for all with , one has , and hence, . By Definition 2.5, is an essential component.

Remark 3.6.

When , by [3], the optimum solution set (where ) has an essential component if and only if is connected. But, when , it is not true.

Example 3.7.

Example 3.8.

Thus ; therefore, is not an essential weakly efficient solution. Similarly, is not an essential weakly efficient solution.

## 4. Conclusions

When , by [3], 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.

## Declarations

### Acknowledgments

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.

## Authors’ Affiliations

## References

- Yu PL:
*Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions, Mathematical Concepts and Methods in Science and Engineering*.*Volume 30*. Plenum Press, New York, NY, USA; 1985:xiv+388.View ArticleGoogle Scholar - Yu J:
**Essential weak efficient solution in multiobjective optimization problems.***Journal of Mathematical Analysis and Applications*1992,**166**(1):230–235. 10.1016/0022-247X(92)90338-EMathSciNetView ArticleMATHGoogle Scholar - Luo Q:
**Essential component and essential optimum solution of optimization problems.***Journal of Optimization Theory and Applications*1999,**102**(2):433–438. 10.1023/A:1021740709876MathSciNetView ArticleMATHGoogle Scholar - Miglierina E, Molho E:
**Scalarization and stability in vector optimization.***Journal of Optimization Theory and Applications*2002,**114**(3):657–670. 10.1023/A:1016031214488MathSciNetView ArticleMATHGoogle Scholar - Huang XX:
**Stability in vector-valued and set-valued optimization.***Mathematical Methods of Operations Research*2000,**52**(2):185–193. 10.1007/s001860000085MathSciNetView ArticleMATHGoogle Scholar - Xiang SW, Yin WS:
**Stability results for efficient solutions of vector optimization problems.***Journal of Optimization Theory and Applications*2007,**134**(3):385–398. 10.1007/s10957-007-9214-0MathSciNetView ArticleMATHGoogle Scholar - Chuong TD, Huy NQ, Yao JC:
**Stability of semi-infinite vector optimization problems under functional perturbations.***Journal of Global Optimization*2009,**45**(4):583–595. 10.1007/s10898-008-9391-xMathSciNetView ArticleMATHGoogle Scholar - Warburton AR:
**Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives.***Journal of Optimization Theory and Applications*1983,**40**(4):537–557. 10.1007/BF00933970MathSciNetView ArticleMATHGoogle Scholar - Engelking R:
*General Topology, Sigma Series in Pure Mathematics*.*Volume 6*. 2nd edition. Heldermann, Berlin, Germany; 1989:viii+529.MATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.