# Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

- XK Sun
^{1}Email author and - SJ Li
^{1}

**2010**:510838

https://doi.org/10.1155/2010/510838

© X. K. Sun and S. J. Li. 2010

**Received: **29 March 2010

**Accepted: **3 August 2010

**Published: **16 August 2010

## Abstract

By virtue of higher-order adjacent derivative of set-valued maps, relationships between higher-order adjacent derivative of a set-valued map and its profile map are discussed. Some results concerning stability analysis are obtained in parametrized vector optimization.

## Keywords

## 1. Introduction

Research on stability and sensitivity analysis is not only theoretically interesting but also practically important in optimization theory. A number of useful results have been obtained in scalar optimization (see [1, 2]). Usually, by stability, we mean the qualitative analysis, which is the study of various continuity properties of the perturbation or marginal function or map of a family of parametrized optimization problems. On the other hand, by sensitivity, we mean the quantitative analysis, which is the study of derivatives of the perturbation function.

Some authors have investigated the sensitivity of vector optimization problems. In [3], Tanino studied some results concerning the behavior of the perturbation map by using the concept of contingent derivative of set-valued maps for general multiobjective optimization problems. In [4], Shi introduced a weaker notion of set-valued derivative TP-derivative and investigated the behavior of contingent derivative for the set-valued perturbation maps in a nonconvex vector optimization problem. Later on, Shi also established sensitivity analysis for a convex vector optimization problem see [5] . In [6], Kuk et al. investigated the relationships between the contingent derivatives of the perturbation maps i.e., perturbation map, proper perturbation map, and weak perturbation map and those of feasible set map in the objective space by virtue of contingent derivative, TP-derivative and Dini derivative. Considering convex vector optimization problems, they also investigated the behavior of the above three kinds of perturbation maps under some convexity assumptions see [7]).

On the other hand, some interesting results have been proved for stability analysis in vector optimization problems. In [8], Tanino studied some qualitative results concerning the behavior of the perturbation map in convex vector optimization. In [9], Li investigated the continuity and the closedness of contingent derivative of the marginal map in multiobjective optimization. In [10], Xiang and Yin investigated some continuity properties of the mapping which associates the set of efficient solutions to the objective function by virtue of the additive weight method of vector optimization problems and the method of essential solutions.

To the best of our knowledge, there is no paper to deal with the stability of higher-order adjacent derivative for weak perturbation maps in vector optimization problems. Motivated by the work reported in [3–9], in this paper, by higher-order adjacent derivative of set-valued maps, we first discuss some relationships between higher-order adjacent derivative of a set-valued map and its profile map. Then, by virtue of the relationships, we investigate the stability of higher-order adjacent derivative of the perturbation maps.

The rest of this paper is organized as follows. In Section 2, we recall some basic definitions. In Section 3, after recalling the concept of higher-order adjacent derivative of set-valued maps, we provide some relationships between the higher-order adjacent derivative of a set-valued map and its profile map. In Section 4, we discuss some stability results of higher-order adjacent derivative for perturbation maps in parametrized vector optimization.

## 2. Preliminaries

Throughout this paper, let and be two finite dimensional spaces, and let be a pointed closed convex cone with a nonempty interior , where is said to be pointed if . Let be a set-valued map. The domain and the graph of are defined by and , respectively. The so-called profile map is defined by for all

At first, let us recall some important definitions.

Definition 2.1 (see [11]).

Let be a nonempty subset of . An elements is said to be a minimal point (resp. weakly minimal point) of if resp., . The set of all minimal points resp., weakly minimal point of is denoted by (resp., .

Definition 2.2 (see [12]).

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

Definition 2.3 (see [13]).

The weak domination property is said to hold for a subset of if

Definition 2.4 (see [14]).

Let be a set-valued map from to .

(i) is said to be lower semicontinuous (l.s.c) at if for any generalized sequence with and , there exists a generalized sequence with such that .

(ii) is said to be upper semicontinuous (u.s.c) at if for any neighborhood of , there exists a neighborhood of such that , for all .

(iii) is said to be closed at if for any generalized sequence , , it yields .

We say that is l.s.c (resp., u.s.c, closed) on if it is l.s.c (resp., u.s.c, closed) at each . is said to be continuous on if it is both l.s.c and u.s.c on .

Definition 2.5 (see [14]).

where denotes the closed unit ball of the origin in .

Definition 2.6 (see [14]).

is said to be uniformly compact near if there exists a neighborhood of such that is a compact set.

## 3. Higher-Order Adjacent Derivatives of Set-Valued Maps

In this section, we recall the concept of higher-order adjacent derivative of set-valued maps and provide some basic properties which are necessary in the following section. Throughout this paper, let be an integer number and .

Definition 3.1 (see [15]).

Definition 3.2 (see [15]).

Proposition 3.3.

Proof.

The proof follows on the lines of Proposition in [3] by replacing contingent derivative by th-order adjacent derivative.

Note that the converse inclusion of (3.3) may not hold. The following example explains the case where we only take .

Example 3.4.

Proposition 3.5.

where is a closed convex cone contained in .

Proof.

Since is a closed convex cone contained in , has a compact base. It is clear that is a compact base for , where is a compact base for . In this proposition, we assume that is a compact base of . Since , there exist and such that . Since is compact, we may assume without loss of generality that .

which contradicts (3.10). Therefore, and Thus, it follows from (3.11) that and the proof is complete.

Remark 3.6.

may not hold under the assumptions of Proposition 3.5. The following example explains the case where we only take

Example 3.7.

Proposition 3.8.

where is a closed convex cone contained in .

Proof.

By Proposition 3.5, we also have

So, by (3.21), (3.22), and (3.23), which leads to a contradiction. Thus, .

which contradicts . Thus, , and the proof is complete.

Obviously, Example 3.4 can also show that the weak domination property of is essential for Proposition 3.8.

Remark 3.9.

may still not hold under the assumptions of Proposition 3.8.

Proposition 3.10.

Let and , . Suppose that is Lipschitz at . Then, is continuous on

Proof.

## 4. Continuity of Higher-Order Adjacent Derivative for Weak Perturbation Map

The set-valued map is called the weak perturbation map. Throughout this section, we suppose that is a closed convex cone contained in .

Definition 4.1 (see [11]).

is said to be -minicomplete by near if , for any where is a neighborhood of .

Remark 4.2.

The following lemma palys a crucial role in this paper.

Lemma 4.3.

Let and , , and let have a compact base. Suppose that the following conditions are satisfied:

(i) fulfills the weak domination property for any ;

(iii) is -minicomplete by near .

Proof.

Thus, result (4.5) holds.

which contradicts (4.8). Then, This completes the proof.

The following example shows that the -minicompleteness of is essential in Lemma 4.3, where we only take

Example 4.4 ( is not -minicomplete by near ).

Theorem 4.5.

Let and , . Then, is closed on

Proof.

Since is closed set, is closed on , and the proof is complete.

Theorem 4.6.

Let and . If is a compact space, then is u.s.c. on

Proof.

Since is a compact space, it follows from Corollary of Chapter in [14] and Theorem 4.5 that is u.s.c. on Thus, the proof is complete.

Theorem 4.7.

Let . Suppose that is a compact set and the assumptions of Lemma 4.3 are satisfied. Then, is u.s.c. at .

Proof.

is u.s.c. at , and the proof is complete.

Now, we give an example to illustrate Theorem 4.7, where we also take

Example 4.8.

Thus, the conditions of Theorem 4.7 are satisfied. Obviously, both and are u.s.c at .

Theorem 4.9.

Let . Suppose that is uniformly compact near and the assumptions of Lemma 4.3 are satisfied. Then, is l.s.c. at .

Proof.

From (4.28) and is closed, we have . Then, it follows from (4.25) and (4.29) that . Thus, is l.s.c. at , and the proof is complete.

It is easy to see that Example 4.8 can also illustrate Theorem 4.9.

## Declarations

### Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (Grant no. 10871216) and Chongqing University Postgraduates Science and Innovation Fund (Grant no. 201005B1A0010338).

## Authors’ Affiliations

## References

- Bonnans JF, Shapiro A:
*Perturbation Analysis of Optimization Problems, Springer Series in Operations Research*. Springer, New York, NY, USA; 2000:xviii+601.View ArticleMATHGoogle Scholar - 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.Google Scholar - Tanino T: Sensitivity analysis in multiobjective optimization.
*Journal of Optimization Theory and Applications*1988, 56(3):479–499. 10.1007/BF00939554MathSciNetView ArticleMATHGoogle Scholar - Shi DS: Contingent derivative of the perturbation map in multiobjective optimization.
*Journal of Optimization Theory and Applications*1991, 70(2):385–396. 10.1007/BF00940634MathSciNetView ArticleMATHGoogle Scholar - Shi DS: Sensitivity analysis in convex vector optimization.
*Journal of Optimization Theory and Applications*1993, 77(1):145–159. 10.1007/BF00940783MathSciNetView ArticleMATHGoogle Scholar - Kuk H, Tanino T, Tanaka M: Sensitivity analysis in vector optimization.
*Journal of Optimization Theory and Applications*1996, 89(3):713–730. 10.1007/BF02275356MathSciNetView ArticleMATHGoogle Scholar - Kuk H, Tanino T, Tanaka M: Sensitivity analysis in parametrized convex vector optimization.
*Journal of Mathematical Analysis and Applications*1996, 202(2):511–522. 10.1006/jmaa.1996.0331MathSciNetView ArticleMATHGoogle Scholar - Tanino T: Stability and sensitivity analysis in convex vector optimization.
*SIAM Journal on Control and Optimization*1988, 26(3):521–536. 10.1137/0326031MathSciNetView ArticleMATHGoogle Scholar - Li Shengjie: Sensitivity and stability for contingent derivative in multiobjective optimization.
*Mathematica Applicata*1998, 11(2):49–53.MathSciNetMATHGoogle Scholar - Xiang SW, Yin WS: Stability results for efficient solutions of vector optimization problems.
*Journal of Optimization Theory and Applications*2007, 134(3):385–398. 10.1007/s10957-007-9214-0MathSciNetView ArticleMATHGoogle Scholar - Sawaragi Y, Nakayama H, Tanino T:
*Theory of Multiobjective Optimization, Mathematics in Science and Engineering*.*Volume 176*. Academic Press, Orlando, Fla, USA; 1985:xiii+296.Google Scholar - Holmes RB:
*Geometric Functional Analysis and Its Applications*. Springer, New York, NY, USA; 1975:x+246.View ArticleGoogle Scholar - Luc DT:
*Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems*.*Volume 319*. Springer, Berlin, Germany; 1989:viii+173.Google Scholar - Aubin J-P, Ekeland I:
*Applied Nonlinear Analysis, Pure and Applied Mathematics*. John Wiley & Sons, New York, NY, USA; 1984:xi+518.MATHGoogle Scholar - Aubin J-P, Frankowska H:
*Set-Valued Analysis, Systems & Control: Foundations & Applications*.*Volume 2*. Birkhäuser, Boston, Mass, USA; 1990:xx+461.MATHGoogle Scholar

## Copyright

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.