Second-Order Contingent Derivative of the Perturbation Map in Multiobjective Optimization

Abstract

Some relationships between the second-order contingent derivative of a set-valued map and its profile map are obtained. By virtue of the second-order contingent derivatives of set-valued maps, some results concerning sensitivity analysis are obtained in multiobjective optimization. Several examples are provided to show the results obtained.

1. Introduction

In this paper, we consider a family of parametrized multiobjective optimization problems

(1.1)

Here, is a -dimensional decision variable, is an -dimensional parameter vector, is a nonempty set-valued map from to , which specifies a feasible decision set, and is an objective map from to , where , , are positive integers. The norms of all finite dimensional spaces are denoted by . is a closed convex pointed cone with nonempty interior in . The cone induces a partial order on , that is, the relation is defined by

(1.2)

We use the following notion. For any ,

(1.3)

Based on these notations, we can define the following two sets for a set in :

(i) is a -minimal point of with respect to if there exists no , such that , ,

(ii) is a weakly -minimal point of with respect to if there exists no , such that .

The sets of -minimal point and weakly -minimal point of are denoted by and , respectively.

Let be a set-valued map from to defined by

(1.4)

  is considered as the feasible set map. In the vector optimization problem corresponding to each parameter valued , our aim is to find the set of -minimal point of the feasible set map . The set-valued map from to is defined by

(1.5)

for any , and call it the perturbation map for .

Sensitivity and stability analysis is not only theoretically interesting but also practically important in optimization theory. Usually, by sensitivity we mean the quantitative analysis, that is, the study of derivatives of the perturbation function. On the other hand, by stability we mean the qualitative analysis, that is, the study of various continuity properties of the perturbation (or marginal) function (or map) of a family of parametrized vector optimization problems.

Some interesting results have been proved for sensitivity and stability in optimization (see [1–16]). Tanino [5] obtained some results concerning sensitivity analysis in vector optimization by using the concept of contingent derivatives of set-valued maps introduced in [17], and Shi [8] and Kuk et al. [7, 11] extended some of Tanino's results. As for vector optimization with convexity assumptions, Tanino [6] studied some quantitative and qualitative results concerning the behavior of the perturbation map, and Shi [9] studied some quantitative results concerning the behavior of the perturbation map. Li [10] discussed the continuity of contingent derivatives for set-valued maps and also discussed the sensitivity, continuity, and closeness of the contingent derivative of the marginal map. By virtue of lower Studniarski derivatives, Sun and Li [14] obtained some quantitative results concerning the behavior of the weak perturbation map in parametrized vector optimization.

Higher order derivatives introduced by the higher order tangent sets are very important concepts in set-valued analysis. Since higher order tangent sets, in general, are not cones and convex sets, there are some difficulties in studying set-valued optimization problems by virtue of the higher order derivatives or epiderivatives introduced by the higher order tangent sets. To the best of our knowledge, second-order contingent derivatives of perturbation map in multiobjective optimization have not been studied until now. Motivated by the work reported in [5–11, 14], we discuss some second-order quantitative results concerning the behavior of the perturbation map for .

The rest of the paper is organized as follows. In Section 2, we collect some important concepts in this paper. In Section 3, we discuss some relationships between the second-order contingent derivative of a set-valued map and its profile map. In Section 4, by the second-order contingent derivative, we discuss the quantitative information on the behavior of the perturbation map for .

2. Preliminaries

In this section, we state several important concepts.

Let be nonempty set-valued maps. The efficient domain and graph of are defined by

(2.1)

respectively. The profile map of is defined by , for every , where is the order cone of .

Definition 2.1 (see [18]).

A base for is a nonempty convex subset of with , such that every , , has a unique representation of the form , where and .

Definition 2.2 (see [19]).

is said to be locally Lipschitz at if there exist a real number and a neighborhood of , such that

(2.2)

where denotes the closed unit ball of the origin in .

3. Second-Order Contingent Derivatives for Set-Valued Maps

In this section, let be a normed space supplied with a distance , and let be a subset of . We denote by the distance from to , where we set . Let be a real normed space, where the space is partially ordered by nontrivial pointed closed convex cone . Now, we recall the definitions in [20].

Definition 3.1 (see [20]).

Let be a nonempty subset , , and , where denotes the closure of .

(i)The second-order contingent set of at is defined as

(3.1)

(ii)The second-order adjacent set of at is defined as

(3.2)

Definition 3.2 (see [20]).

Let , be normed spaces and be a set-valued map, and let and .

(i)The set-valued map from to defined by

(3.3)

is called second-order contingent derivative of at .

(ii)The set-valued map from to defined by

(3.4)

is called second-order adjacent derivative of at .

Definition 3.3 (see [21]).

The -domination property is said to be held for a subset of if .

Proposition 3.4.

Let and , then

(3.5)

for any .

Proof.

The conclusion can be directly obtained similarly as the proof of [5, Proposition  2.1].

It follows from Proposition 3.4 that

(3.6)

Note that the inclusion of

(3.7)

may not hold. The following example explains the case.

Example 3.5.

Let , , and . Consider a set-valued map defined by

(3.8)

Let and , then, for any ,

(3.9)

Thus, one has

(3.10)

which shows that the inclusion of (3.7) does not hold here.

Proposition 3.6.

Let and . Suppose that has a compact base , then for any ,

(3.11)

Proof.

Let . If  , then (3.11) holds trivially. So, we assume that , and let

(3.12)

Since , there exist sequences with , with , and with , such that

(3.13)

It follows from and has a compact base that there exist some and , such that, for any , one has . Since is compact, we may assume without loss of generality that .

We now show . Suppose that , then for some , we may assume without loss of generality that , for all , by taking a subsequence if necessary. Let , then, for any , and

(3.14)

Since , for all . Thus, . It follows from (3.14) that

(3.15)

which contradicts (3.12), since . Thus, and . Then, it follows from (3.13) that . So,

(3.16)

and the proof of the proposition is complete.

Note that the inclusion of

(3.17)

may not hold under the assumptions of Proposition 3.6. The following example explains the case.

Example 3.7.

Let , , and . Obviously, has a compact base. Consider a set-valued map defined by

(3.18)

Let and . For any ,

(3.19)

Then, for any , . So, the inclusion of (3.17) does not hold here.

Proposition 3.8.

Let and . Suppose that has a compact base and satisfies the -domination property for all , then for any ,

(3.20)

Proof.

From Proposition 3.4, one has

(3.21)

It follows from the -domination property of and Proposition 3.6 that

(3.22)

and then

(3.23)

Thus, for any ,

(3.24)

and the proof of the proposition is complete.

The following example shows that the -domination property of in Proposition 3.8 is essential.

Example 3.9 ( does not satisfy the -domination property).

Let , , and , and let be defined by

(3.25)

then

(3.26)

Let , , then, for any ,

(3.27)

Obviously, does not satisfy the -domination property and

(3.28)

4. Second-Order Contingent Derivative of the Perturbation Maps

The purpose of this section is to investigate the quantitative information on the behavior of the perturbation map for by using second-order contingent derivative. Hereafter in this paper, let , , and , and let be the order cone of .

Definition 4.1.

We say that is -minicomplete by near if

(4.1)

where is some neighborhood of .

Remark 4.2.

Let be a convex cone. Since , the -minicompleteness of by near implies that

(4.2)

Hence, if is -minicomplete by near , then

(4.3)

Theorem 4.3.

Suppose that the following conditions are satisfied:

(i) is locally Lipschitz at ;

(ii);

(iii) is -minicomplete by near ;

(iv)there exists a neighborhood of , such that for any , is a single point set,

then, for all ,

(4.4)

Proof.

Let . If , then (4.4) holds trivially. Thus, we assume that . Let , then there exist sequences with and with , such that

(4.5)

So, .

Suppose that , then there exists ,, such that

(4.6)

Since , for the preceding sequence , there exists a sequence with , such that

(4.7)

It follows from the locally Lipschitz continuity of that there exist and a neighborhood of , such that

(4.8)

where is the closed ball of .

From assumption (iii), there exists a neighborhood of , such that

(4.9)

Naturally, there exists , such that

(4.10)

Therefore, it follows from (4.7) and (4.8) that for any , there exists , such that

(4.11)

Thus, from (4.5), (4.9), and assumption (iv), one has

(4.12)

and then it follows from and is a closed convex cone that

(4.13)

which contradicts (4.6). Thus, and the proof of the theorem is complete.

The following two examples show that the assumption (iv) in Theorem 4.3 is essential.

Example 4.4 (  is not a single-point set near ).

Let and be defined by

(4.14)

then

(4.15)

Let , , then is not a single-point set near , and it is easy to check that other assumptions of Theorem 4.3 are satisfied.

For any , one has

(4.16)

and then

(4.17)

Thus, for any , the inclusion of (4.4) does not hold here.

Example 4.5 ( is not a single-point set near ).

Let and be defined by

(4.18)

then

(4.19)

Let , , and , then is not a single-point set near , and it is easy to check that other assumptions of Theorem 4.3 are satisfied.

For any , one has

(4.20)

and then

(4.21)

Thus, for , the inclusion of (4.4) does not hold here.

Now, we give an example to illustrate Theorem 4.3.

Example 4.6.

Let and be defined by

(4.22)

then

(4.23)

Let , . By directly calculating, for all , one has

(4.24)

Then, it is easy to check that assumptions of Theorem 4.3 are satisfied, and the inclusion of (4.4) holds.

Theorem 4.7.

If fulfills the -domination property for all and is -minicomplete by near , then

(4.25)

Proof.

Since , has a compact base. Then, it follows from Propositions 3.6 and 3.8 and Remark 4.2 that for any , one has

(4.26)

Then, the conclusion is obtained and the proof is complete.

Remark 4.8.

If the -domination property of is not satisfied in Theorem 4.7, then Theorem 4.7 may not hold. The following example explains the case.

Example 4.9 ( does not satisfy the -domination property for ).

Let and be defined by

(4.27)

then,

(4.28)

Let , , then, for any ,

(4.29)

for any ,

(4.30)

Hence, does not satisfy the -domination property, and . Then,

(4.31)

Theorem 4.10.

Suppose that the following conditions are satisfied:

(i) is locally Lipschitz at ;

(ii);

(iii) is -minicomplete by near ;

(iv)there exists a neighborhood of , such that for any , is a single-point set;

(v)for any , fulfills the -domination property;

then

(4.32)

Proof.

It follows from Theorems 4.3 and 4.7 that (4.32) holds. The proof of the theorem is complete.