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Fixed Point Theory and Applications

, 2011:857520 | Cite as

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

  • QL Wang
  • SJ Li
Open Access
Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

In this paper, we consider a family of parametrized multiobjective optimization problems
Here, Open image in new window is a Open image in new window -dimensional decision variable, Open image in new window is an Open image in new window -dimensional parameter vector, Open image in new window is a nonempty set-valued map from Open image in new window to Open image in new window , which specifies a feasible decision set, and Open image in new window is an objective map from Open image in new window to Open image in new window , where Open image in new window , Open image in new window , Open image in new window are positive integers. The norms of all finite dimensional spaces are denoted by Open image in new window . Open image in new window is a closed convex pointed cone with nonempty interior in Open image in new window . The cone Open image in new window induces a partial order Open image in new window on Open image in new window , that is, the relation Open image in new window is defined by
We use the following notion. For any Open image in new window ,

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

(i) Open image in new window is a Open image in new window -minimal point of Open image in new window with respect to Open image in new window if there exists no Open image in new window , such that Open image in new window , Open image in new window ,

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

The sets of Open image in new window -minimal point and weakly Open image in new window -minimal point of Open image in new window are denoted by Open image in new window and Open image in new window , respectively.

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

for any Open image in new window , and call it the perturbation map for Open image in new window .

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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 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, 6, 7, 8, 9, 10, 11, 14], we discuss some second-order quantitative results concerning the behavior of the perturbation map for Open image in new window .

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 Open image in new window .

2. Preliminaries

In this section, we state several important concepts.

Let Open image in new window be nonempty set-valued maps. The efficient domain and graph of Open image in new window are defined by

respectively. The profile map Open image in new window of Open image in new window is defined by Open image in new window , for every Open image in new window , where Open image in new window is the order cone of Open image in new window .

Definition 2.1 (see [18]).

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

Definition 2.2 (see [19]).

Open image in new window is said to be locally Lipschitz at Open image in new window if there exist a real number Open image in new window and a neighborhood Open image in new window of Open image in new window , such that

where Open image in new window denotes the closed unit ball of the origin in Open image in new window .

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

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

Definition 3.1 (see [20]).

Let Open image in new window be a nonempty subset Open image in new window , Open image in new window , and Open image in new window , where Open image in new window denotes the closure of Open image in new window .

(i)The second-order contingent set Open image in new window of Open image in new window at Open image in new window is defined as

Definition 3.2 (see [20]).

Let Open image in new window , Open image in new window be normed spaces and Open image in new window be a set-valued map, and let Open image in new window and Open image in new window .

is called second-order contingent derivative of Open image in new window at Open image in new window .

is called second-order adjacent derivative of Open image in new window at Open image in new window .

Definition 3.3 (see [21]).

The Open image in new window -domination property is said to be held for a subset Open image in new window of Open image in new window if Open image in new window .

Proposition 3.4.

for any Open image in new window .

Proof.

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

It follows from Proposition 3.4 that
Note that the inclusion of

may not hold. The following example explains the case.

Example 3.5.

Thus, one has

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

Proposition 3.6.

Proof.

Let Open image in new window . If   Open image in new window , then (3.11) holds trivially. So, we assume that Open image in new window , and let

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

We now show Open image in new window . Suppose that Open image in new window , then for some Open image in new window , we may assume without loss of generality that Open image in new window , for all Open image in new window , by taking a subsequence if necessary. Let Open image in new window , then, for any Open image in new window , Open image in new window and
Since Open image in new window , for all Open image in new window . Thus, Open image in new window . It follows from (3.14) that
which contradicts (3.12), since Open image in new window . Thus, Open image in new window and Open image in new window . Then, it follows from (3.13) that Open image in new window . So,

and the proof of the proposition is complete.

Note that the inclusion of

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

Example 3.7.

Then, for any Open image in new window , Open image in new window . So, the inclusion of (3.17) does not hold here.

Proposition 3.8.

Proof.

From Proposition 3.4, one has
It follows from the Open image in new window -domination property of Open image in new window and Proposition 3.6 that
and then

and the proof of the proposition is complete.

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

Example 3.9 ( Open image in new window does not satisfy the Open image in new window -domination property).

Obviously, Open image in new window does not satisfy the Open image in new window -domination property and

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 Open image in new window by using second-order contingent derivative. Hereafter in this paper, let Open image in new window , Open image in new window , and Open image in new window , and let Open image in new window be the order cone of Open image in new window .

Definition 4.1.

where Open image in new window is some neighborhood of Open image in new window .

Remark 4.2.

Theorem 4.3.

Suppose that the following conditions are satisfied:

(i) Open image in new window is locally Lipschitz at Open image in new window ;

(ii) Open image in new window ;

(iii) Open image in new window is Open image in new window -minicomplete by Open image in new window near Open image in new window ;

(iv)there exists a neighborhood Open image in new window of Open image in new window , such that for any Open image in new window , Open image in new window is a single point set,

Proof.

So, Open image in new window .

It follows from the locally Lipschitz continuity of Open image in new window that there exist Open image in new window and a neighborhood Open image in new window of Open image in new window , such that

where Open image in new window is the closed ball of Open image in new window .

From assumption (iii), there exists a neighborhood Open image in new window of Open image in new window , such that
Naturally, there exists Open image in new window , such that
Therefore, it follows from (4.7) and (4.8) that for any Open image in new window , there exists Open image in new window , such that
Thus, from (4.5), (4.9), and assumption (iv), one has
and then it follows from Open image in new window and Open image in new window is a closed convex cone that

which contradicts (4.6). Thus, Open image in new window 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 ( Open image in new window   is not a single-point set near Open image in new window ).

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

and then

Thus, for any Open image in new window , the inclusion of (4.4) does not hold here.

Example 4.5 ( Open image in new window is not a single-point set near Open image in new window ).

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

and then

Thus, for Open image in new window , 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.

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

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

Remark 4.8.

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

Example 4.9 ( Open image in new window does not satisfy the Open image in new window -domination property for Open image in new window ).

Hence, Open image in new window does not satisfy the Open image in new window -domination property, and Open image in new window . Then,

Theorem 4.10.

Suppose that the following conditions are satisfied:

(i) Open image in new window is locally Lipschitz at Open image in new window ;

(ii) Open image in new window ;

(iii) Open image in new window is Open image in new window -minicomplete by Open image in new window near Open image in new window ;

(iv)there exists a neighborhood Open image in new window of Open image in new window , such that for any Open image in new window , Open image in new window is a single-point set;

(v)for any Open image in new window , Open image in new window fulfills the Open image in new window -domination property;

Proof.

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

Notes

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).

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Copyright information

© Q. L. Wang and S. J. Li. 2011

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.

Authors and Affiliations

  1. 1.College of SciencesChongqing Jiaotong UniversityChongqingChina
  2. 2.College of Mathematics and StatisticsChongqing UniversityChongqingChina

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