A Note on Essential Components and Essential Weakly Efficient Solutions 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 [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.

Let be a nonempty compact subset of a Banach space ,  

(2.1)

Consider the following multiobjective optimization problem:

(VMP)

Definition 2.1.

  A point is called a weakly efficient solution to (VMP), if there is no such that

(2.2)

  A point is called an efficient solution to (VMP), if there is no ( ) such that

(2.3)

or is used to denote the weakly efficient solution set of (VMP), and or is used to denote the efficient solution set of (VMP).

Clearly, , but the reverse containment may not hold.

Example 2.2 (see [8]).

Let , define

(2.4)

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 .

Denote

(2.5)

For any , , define

(2.6)

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.

For any , the component of a point is the union of all connected subsets of which contain the point . From [9, page 356], one knows that components are connected closed subsets of and thus they are also compact as is compact. It is easy to see that the components of two distinct points of either coincide or are disjoint, so that all components constitute a decomposition of into connected pairwise disjoint compact subsets, that is,

(2.7)

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.

Let , define

(2.8)

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

Example 2.10.

Let , define

(2.9)

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

Definition 2.11.

Let be a nonempty convex subset of a Banach space, and , the function is said to be strongly quasiconvex on , if

(2.10)

for all ,   ,   .

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

Since , and , then

(3.1)

Take such that

(3.2)

Then for any with , one has

(3.3)

Then

(3.4)

by (3.2) and (3.4), one has

(3.5)

therefore

(3.6)

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.

Suppose , and . By Definition 2.11,

(3.7)

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.

Let , define

(3.8)

is disconnected; however, is an essential weakly efficient solution, is an essential component, and is an essential component.

Example 3.8.

Let , define

(3.9)

Then ,   . By Theorem 3.1, is an essential weakly efficient solution. But, and are not essential weakly efficient solutions. In fact, for , take , , for all , take as the following:

(3.10)

Then , and

(3.11)

But . In fact, for all , if ,   , take , we have

(3.12)

If , take , we have

(3.13)

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

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.