- Research Article
- Open Access

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

- QL Wang
^{1}Email author and - SJ Li
^{2}

**2011**:857520

https://doi.org/10.1155/2011/857520

© Q. L. Wang and S. J. Li. 2011

**Received:**14 October 2010**Accepted:**24 January 2011**Published:**13 February 2011

## 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.

## Keywords

- Multiobjective Optimization
- Vector Optimization
- Vector Optimization Problem
- Multiobjective Optimization Problem
- Closed Convex Cone

## 1. Introduction

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.

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.

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]).

## 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 .

Definition 3.2 (see [20]).

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

is called second-order contingent derivative of at .

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.

Proof.

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

may not hold. The following example explains the case.

Example 3.5.

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

Proposition 3.6.

Proof.

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 .

and the proof of the proposition is complete.

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

Example 3.7.

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

Proposition 3.8.

Proof.

and the proof of the proposition is complete.

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

## 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.

where is some neighborhood of .

Remark 4.2.

Theorem 4.3.

Suppose that the following conditions are satisfied:

(iii) is -minicomplete by near ;

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

Proof.

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

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

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

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.

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

Now, we give an example to illustrate Theorem 4.3.

Example 4.6.

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

Theorem 4.7.

Proof.

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 ).

Theorem 4.10.

Suppose that the following conditions are satisfied:

(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;

Proof.

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

## Declarations

### Acknowledgments

This research was partially supported by the National Natural Science Foundation of China (no. 10871216 and no. 11071267), Natural Science Foundation Project of CQ CSTC and Science and Technology Research Project of Chong Qing Municipal Education Commission (KJ100419).

## Authors’ Affiliations

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