- Research Article
- Open access
- Published:

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

*Fixed Point Theory and Applications*
**volumeÂ 2011**, ArticleÂ number:Â 857520 (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.

## 1. Introduction

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

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

We use the following notion. For any ,

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

â€‰â€‰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

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 [1â€“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 [5â€“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

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

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

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

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

is called second-order contingent derivative of at .

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

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

for any .

Proof.

The conclusion can be directly obtained similarly as the proof of [5, Propositionâ€‰â€‰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.

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

Let and , then, for any ,

Thus, one has

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 ,

Proof.

Let . Ifâ€‰â€‰, then (3.11) holds trivially. So, we assume that , and let

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

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

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

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

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

Let and . For any ,

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 ,

Proof.

From Proposition 3.4, one has

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

and then

Thus, for any ,

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

then

Let , , then, for any ,

Obviously, does not satisfy the -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 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

where is some neighborhood of .

Remark 4.2.

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

Hence, if is -minicomplete by near , then

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 ,

Proof.

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

So, .

Suppose that , then there exists ,, such that

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

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

where is the closed ball of .

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

Naturally, there exists , such that

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

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

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

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 (â€‰â€‰is not a single-point set near ).

Let and be defined by

then

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

and then

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

then

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

and then

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

then

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

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

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

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

then,

Let , , then, for any ,

for any ,

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

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

Proof.

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

## References

Fiacco AV:

*Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Mathematics in Science and Engineering*.*Volume 165*. Academic Press, Orlando, Fla, USA; 1983:xii+367.Alt W:

**Local stability of solutions to differentiable optimization problems in Banach spaces.***Journal of Optimization Theory and Applications*1991,**70**(3):443â€“466. 10.1007/BF00941297Xiang SW, Yin WS:

**Stability results for efficient solutions of vector optimization problems.***Journal of Optimization Theory and Applications*2007,**134**(3):385â€“398. 10.1007/s10957-007-9214-0Zhao J:

**The lower semicontinuity of optimal solution sets.***Journal of Mathematical Analysis and Applications*1997,**207**(1):240â€“254. 10.1006/jmaa.1997.5288Tanino T:

**Sensitivity analysis in multiobjective optimization.***Journal of Optimization Theory and Applications*1988,**56**(3):479â€“499. 10.1007/BF00939554Tanino T:

**Stability and sensitivity analysis in convex vector optimization.***SIAM Journal on Control and Optimization*1988,**26**(3):521â€“536. 10.1137/0326031Kuk H, Tanino T, Tanaka M:

**Sensitivity analysis in vector optimization.***Journal of Optimization Theory and Applications*1996,**89**(3):713â€“730. 10.1007/BF02275356Shi DS:

**Contingent derivative of the perturbation map in multiobjective optimization.***Journal of Optimization Theory and Applications*1991,**70**(2):385â€“396. 10.1007/BF00940634Shi DS:

**Sensitivity analysis in convex vector optimization.***Journal of Optimization Theory and Applications*1993,**77**(1):145â€“159. 10.1007/BF00940783Li SJ:

**Sensitivity and stability for contingent derivative in multiobjective optimization.***Mathematica Applicata*1998,**11**(2):49â€“53.Kuk H, Tanino T, Tanaka M:

**Sensitivity analysis in parametrized convex vector optimization.***Journal of Mathematical Analysis and Applications*1996,**202**(2):511â€“522. 10.1006/jmaa.1996.0331Ferro F:

**An optimization result for set-valued mappings and a stability property in vector problems with constraints.***Journal of Optimization Theory and Applications*1996,**90**(1):63â€“77. 10.1007/BF02192246Goberna MA, LÃ³pes MA, Todorov MI:

**The stability of closed-convex-valued mappings and the associated boundaries.***Journal of Mathematical Analysis and Applications*2005,**306**(2):502â€“515. 10.1016/j.jmaa.2005.01.003Sun XK, Li SJ:

**Lower Studniarski derivative of the perturbation map in parametrized vector optimization.***Optimization Letters*. In pressChuong TD, Yao J-C:

**Coderivatives of efficient point multifunctions in parametric vector optimization.***Taiwanese Journal of Mathematics*2009,**13**(6A):1671â€“1693.Meng KW, Li SJ:

**Differential and sensitivity properties of gap functions for Minty vector variational inequalities.***Journal of Mathematical Analysis and Applications*2008,**337**(1):386â€“398. 10.1016/j.jmaa.2007.04.009Aubin J-P:

**Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions.**In*Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud.*.*Volume 7*. Edited by: Nachbin L. Academic Press, New York, NY, USA; 1981:159â€“229.Holmes RB:

*Geometric Functional Analysis and Its Applications, Graduate Texts in Mathematics, no. 2*. Springer, New York, NY, USA; 1975:x+246.Aubin J-P, Ekeland I:

*Applied Nonlinear Analysis, Pure and Applied Mathematics (New York)*. John Wiley & Sons, New York, NY, USA; 1984:xi+518.Aubin J-P, Frankowska H:

*Set-Valued Analysis, Systems & Control: Foundations & Applications*.*Volume 2*. BiekhÃ¤user, Boston, Mass, USA; 1990:xx+461.Luc DT:

*Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems*.*Volume 319*. Springer, Berlin, Germany; 1989:viii+173.

## 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).

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Wang, Q., Li, S. Second-Order Contingent Derivative of the Perturbation Map in Multiobjective Optimization.
*Fixed Point Theory Appl* **2011**, 857520 (2011). https://doi.org/10.1155/2011/857520

Received:

Accepted:

Published:

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