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

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.

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]).

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

(2.1)

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]).

Let and be elements of . The set is called the th-order adjacent set of at , if and only if, for any , for any sequence with , there exists a sequence with such that

(3.1)

Definition 3.2 (see [15]).

Let and , . The th-order adjacent derivative of at for vectors is the set-valued map from to defined by

(3.2)

Proposition 3.3.

Let and , . Then, for any ,

(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.

Let and , let be defined by

(3.4)

Let and . For any , we have

(3.5)

Thus, for any , we have

(3.6)

Proposition 3.5.

Let and , . Assume that has a compact base. Then, for any ,

(3.7)

where is a closed convex cone contained in .

Proof.

If the inclusion holds trivially. Thus, we suppose that Let Then,

(3.8)

Since ,

(3.9)

then it follows that

(3.10)

From (3.8) and the definition of th-order adjacent derivative, we have that for any sequence with , there exist sequences with and such that

(3.11)

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 .

Now, we show that Suppose that Then, for some , we may assume, without loss of generality, that , for all . Let . Then, we have

(3.12)

By (3.11) and (3.12), we obtain that

(3.13)

From (3.13) and we have

(3.14)

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

Remark 3.6.

The inclusion of

(3.15)

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

Example 3.7.

Let and , let and be defined by

(3.16)

Suppose that , . Then, for any ,

(3.17)

Naturally, we have

(3.18)

Thus, for any ,

(3.19)

Proposition 3.8.

Let , and , , and let has a compact base. Suppose that fulfills the weak domination property for any . Then, for any ,

(3.20)

where is a closed convex cone contained in .

Proof.

Let Then,

(3.21)

By Proposition 3.5, we also have

Suppose that Then, there exists such that

(3.22)

From and Proposition 3.3, we have

(3.23)

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

Conversely, let . Then,

(3.24)

Suppose that . Then, there exists such that

(3.25)

Since fulfills the weak domination property for any , there exists such that

(3.26)

From (3.25) and (3.26), we have

(3.27)

It follows from Proposition 3.5 and (3.27) that

(3.28)

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.

From Example 3.7, the equality of

(3.29)

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.

Since is Lipschitz at , there exist a real number and a neighborhood of such that

(3.30)

First, we prove that is l.s.c. at . Indeed, for any . From the definition of th-order adjacent derivative, we have that for any sequence with , there exists a sequence with such that

(3.31)

Take any and . Obviously, , for any sufficiently large. Therefore, by (3.30), we have

(3.32)

So, with (3.31), there exists such that

(3.33)

We may assume, without loss of generality, that . Thus, by (3.33),

(3.34)

It follows from (3.34) that for any sequence with , , there exists a sequence with

(3.35)

Obviously, . Hence, is l.s.c. on

We will prove that is u.s.c. on In fact, for any , we consider the neighborhood of . Let and . From the definition of , we have that for any sequence with , there exists a sequence with such that

(3.36)

Take any . Obviously, , for any sufficiently large. Therefore, by (3.30), we have

(3.37)

Similar to the proof of l.s.c., there exists such that

(3.38)

Thus, Hence, is u.s.c. on and the proof is complete.

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

In this section, we consider a family of parametrized vector optimization problems. Let be a set-valued map from to , where is the Banach space of perturbation parameter vectors, is the objective space, and is considered as the feasible set map in the objective space. In the optimization problem corresponding to each parameter valued , our aim is to find the set of weakly minimal points of the feasible objective valued set . Hence, we define another set-valued map from to by

(4.1)

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.

Since , the -minicompleteness of by near implies that

(4.2)

Hence, if is -minicomplete by near , then, for any

(4.3)

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 ;

(ii) is Lipschitz at ;

(iii) is -minicomplete by near .

Then, for any ,

(4.4)

Proof.

We first prove that

(4.5)

In fact, from Proposition 3.5, Proposition 3.8, and the -minicompleteness of by near , we have

(4.6)

Thus, result (4.5) holds.

Now, we prove that

(4.7)

In fact, assume that . Then, for any sequence with , there exists a sequence with such that

(4.8)

Suppose that Then, there exists such that Thus, for the preceding sequence there exists a sequence with such that

(4.9)

Obviously, for any sufficiently large. Therefore, by (ii), we have

(4.10)

So, with (4.9), there exists such that

(4.11)

Since and , , for sufficiently large. Then, we have

(4.12)

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 ).

Let , and , and let be defined by

(4.13)

Then, for any ,

(4.14)

Suppose that , . Then, is Lipschitz at , and for any ,

(4.15)

fulfills the weak domination property. We also have

(4.16)

On the other hand,

(4.17)

Thus, for any ,

(4.18)

Theorem 4.5.

Let and ,. Then, is closed on

Proof.

From the definition of , we have that

(4.19)

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.

It follows from Lemma 4.3 and Theorem 4.5 that is closed. By Proposition 3.10, we have that is u.s.c. at . Since is a compact set, it follows from Theorem of Chapter in [14] that

(4.20)

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.

Let , , and , and let be defined by

(4.21)

Then, for any ,

(4.22)

Suppose that , , and . Obviously, has a compact base, is Lipschitz at and is -minicomplete by near . By direct calculating, for any ,

(4.23)

fulfill the weak domination property, which is a strong property for a set-valued map. We also have

(4.24)

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.

By Lemma 4.3, it suffices to prove that is l.s.c. at . Let and

(4.25)

By Proposition 3.10, we have that is l.s.c. at . Then, there exists a sequence with such that . Since ,

(4.26)

Then, for any sequence with , we have

(4.27)

then it follows that

(4.28)

Since is uniformly compact near , we may assume, without loss of generality, that . It follows from the closedness of that

(4.29)

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.