# Optimality Conditions for Approximate Solutions in Multiobjective Optimization Problems

- Ying Gao
^{1}Email author, - Xinmin Yang
^{1}and - HeungWingJoseph Lee
^{2}

**2010**:620928

https://doi.org/10.1155/2010/620928

© Ying Gao et al. 2010

**Received: **18 July 2010

**Accepted: **25 October 2010

**Published: **27 October 2010

## Abstract

We study first- and second-order necessary and sufficient optimality conditions for approximate (weakly, properly) efficient solutions of multiobjective optimization problems. Here, tangent cone, -normal cone, cones of feasible directions, second-order tangent set, asymptotic second-order cone, and Hadamard upper (lower) directional derivatives are used in the characterizations. The results are first presented in convex cases and then generalized to nonconvex cases by employing local concepts.

## Keywords

## 1. Introduction

The investigation of the optimality conditions is one of the most attractive topics of optimization theory. For vector optimization, the optimality solutions can be characterized with the help of different geometrical concepts. Miettinen and Mäkelä [1] and Huang and Liu [2] derived several optimality conditions for efficient, weakly efficient, and properly efficient solutions of vector optimization problems with the help of several kinds of cones. Engau and Wiecek [3] derived the cone characterizations for approximate solutions of vector optimization problems by using translated cones. In [4], Aghezzaf and Hachimi obtained second-order optimality conditions by means of a second-order tangent set which can be considered an extension of the tangent cone; Cambini et al. [5] and Penot [6] introduced a new second-order tangent set called asymptotic second-order cone. Later, second-order optimality conditions for vector optimization problems have been widely studied by using these second-order tangent sets; see [7–9].

During the past decades, researchers and practitioners in optimization had a keen interest in approximate solutions of optimization problems. There are several important reasons for considering this kind of solutions. One of them is that an approximate solution of an optimization problem can be computed by using iterative algorithms or heuristic methods. In vector optimization, the notion of approximate solution has been defined in several ways. The first concept was introduced by Kutateladze [10] and has been used to establish vector variational principle, approximate Kuhn-Tucker-type conditions and approximate duality theorems, and so forth, (see [11–20]). Later, several authors have proposed other -efficiency concepts (see, e.g., White [21]; Helbig [22] and Tanaka [23]).

In this paper, we derive different characterizations for approximate solutions by treating convex case and nonconvex cases. Giving up convexity naturally means that we need local instead of global analysis. Some definitions and notations are given in Section 2. In Section 3, we derive some characterizations for global approximate solutions of multiobjective optimization problems by using tangent cone, the cone of feasible directions and -normal cone. Finally, in Section 3, we introduce some local approximate concepts and present some properties of these notions, and then, first and second-order sufficient conditions for local (properly) approximate efficient solutions of vector optimization problems are derived. These conditions are expressed by means of tangent cone, second-order tangent set and asymptotic second-order set. Finally, some sufficient conditions are given for local (weakly) approximate efficient solutions by using Hadamard upper (lower) directional derivatives.

## 2. Preliminaries

Let be the -dimensional Euclidean space and let be its nonnegative orthant. Let be a subset of , then, the cone generated by the set is defined as , and and referred to as the interior and the closure of the set , respectively. A set is said to be a cone if . We say that the cone is solid if , and pointed if . The cone is said to have a base if is convex, and . The positive polar cone and strict positive polar cone of are denoted by and , respectively.

We recall that is an efficient solution of (2.1) with respect to if . is a weakly efficient solution of (2.1) with respect to if (in this case, it is assumed that is solid). is a Benson properly efficient solution (see [24]) of (2.1) with respect to if . is a Henig' properly efficient solution (see [24]) of (2.1) with respect to if , for some convex cone with .

Definition 2.1 (see [18, 25]).

(i) is said to be a weakly -efficient solution of problem (2.1) if (in this case it is assumed that is solid).

(ii) is said to be a efficient -solution of problem (2.1) if .

(iii) is said to be a properly -efficient solution of problem (2.1), if .

The sets of -efficient solutions, weakly -efficient solutions, and properly -efficient solutions of problem (2.1) are denoted by , , and , respectively.

Remark 2.2.

If , then -efficient solution, weakly -efficient solution, and properly -efficient solution reduce to efficient solution, weakly efficient solution and properly efficient solution of problem (2.1).

Definition 2.3.

Lemma 2.4 (see [26]).

Let be closed convex cones such that . Suppose that is pointed and locally compact, or , then, .

## 3. Cone Characterizations of Approximate Solutions: Convex Case

In this section, we assume that is a convex set.

Theorem 3.1.

Proof.

Suppose, on the contrary, that , then, there exist and such that . That is, . Therefore, , which is a contradiction to . This completes the proof.

Theorem 3.2.

(ii)Let , and is solid set and . If , then .

Since is convex set, . Hence, . From Lemma 2.4, there exists such that , for all .

- (ii)

In fact, if there exists such that , then, from and , there exists such that . Hence, , which is a contradiction to the assumption.

By using the convex separation theorem, there exists such that , for all and , for all . It is easy to get that , for all . Hence, , for all .

Which implies . On the other hand, from , we have , which yields a contradiction. This completes the proof.

Remark 3.3.

If , then the conditions of Theorems 3.1 and 3.2 are also necessary(see [2]). But for , these are not necessary conditions, see the following example.

Example 3.4.

Let , , , , , and , then, and . But . Hence, and .

Theorem 3.5.

Let , , be a solid set and . If there exists such that and , then . Conversely, if , then there exists such that and .

Proof.

On the other hand, from , we have , which is a contradiction to the above inequality. Hence, .

Conversely, let , then, . Since is convex and is a convex cone, there exists such that , for all . Since , there exists such that and , for all . Therefore, , for all , which implies . This completes the proof.

Theorem 3.6.

Let and . If there exists such that and , then . Conversely, assume that is a locally compact set, if , then there exists such that and .

Proof.

and there exists , for all such that . From and , we have . Hence, there exists such that , for all . From , for all , there exist , , and such that , for all . Therefore, , for all , which combing with and yields , for all , which is a contradiction to . Hence, .

Since is a convex set, is a closed convex cone. From Lemma 2.4, there exists such that . Since , and are cone, there exists such that and .

Now, we prove that . That is, , for all .

Which implies . This completes the proof.

Example 3.7.

Let , , , , , and , then, and . Let , then and .

- (i)
If and , then Theorems 3.1 and 3.5 reduce to the corresponding results in [1].

- (ii)
In [1], the cone characterizations of Henig' properly efficient solution were derived. We know that Henig' properly efficient solution equivalent to Benson properly efficient solution, when is a closed convex pointed cone(see [24]). Therefore, if and , Theorems 3.2 and 3.6 reduce to the corresponding results in [1].

## 4. Cone Characterizations of Approximate Solutions: Nonconvex Case

In this section, is no longer assumed to be convex. In nonconvex case, the corresponding local concepts are defined as follows.

Definition 4.1.

(i) is said to be a local weakly -efficient solution of problem (2.1), if there exists a neighborhood of such that (in this case, it is assumed that is solid).

(ii) is said to be a local -efficient solution of problem (2.1), if there exists a neighborhood of such that .

(iii) is said to be a local properly -efficient solution of problem (2.1), if there exists a neighborhood of such that .

The sets of local -efficient solutions, local weakly -efficient solutions and local properly -efficient solutions of problem (2.1) are denoted by , and , respectively.

If , then, (i), (ii), and (iii) reduce to the definitions of local weakly efficient solution, local efficient solution and local properly efficient solution, respectively, and the sets of local (weakly, properly) efficient solutions of problem (2.1) are denoted by , , respectively.

In [4–9], some properties of second-order tangent sets have been derived, see the following Lemma.

Lemma 4.3.

(i) and are closed sets contained in , and is a cone.

(ii)If , then . If , then . If , then , and .

(iii)Let is convex. If and , then .

Definition 4.4 (see [27]).

Lemma 4.5 (see [7]).

Let be a finite-dimensional space and . If the sequence converges to , then there exists a subsequence (denoted the same) such that converges to some nonnull vector , where , and either converges to some vector or there exists a sequence such that and converges to some vector , where denotes the orthogonal subspace to .

In the following theorem, we derive several properties of local (weakly, properly) approximate efficient solutions.

- (ii)

- (iii)

Conversely, we assume that there exists a neighborhood of such that for any fixed and , is a closed set, and . Suppose, on the contrary, that , then, for any neighborhood of , we have . Take , then, there exist , , and such that . Take and , similar to the proof of (ii) we can complete the proof.

Theorem 4.7.

Let be a continuous function on , , and .

Let and , for all . Similar to the proof of Lemma 4.3, we have there exists such that or , which is a contradiction to the assumptions. This completes the proof.

Corollary 4.8.

Let be a continuous function on , and .

(i)If , then is a local efficient solution of problem (2.1).

then is a local efficient solution of problem (2.1).

Proof.

The proof is similar to Theorem 4.7.

Remark 4.9.

If is convex, then the condition (ii) of Theorem 4.7 is equivalent to the following condition

Theorem 4.10.

Let be continuous on , , and .

(i)Assume that has a compact base , for and , and there exists such that . If , then .

then , where, denotes the closed unit ball of .

- (i)Let , then, , for all . The assumptions and the separation result [28, page 9] implies that for any there exists a neighborhood of 0 such that

- (ii)

Let and . Similar to the proof of Lemma 4.3, we have there exists such that or , which is a contradiction to the assumptions. This completes the proof.

Remark 4.11.

If is convex, then the conditions (i) and (ii) of Theorem 4.10 are equivalent to and , respectively.

has a compact base , for some , , and .

Remark 4.12.

The conditions of Theorem 4.7, Corollary 4.8 and Theorem 4.10 are not necessary conditions, see Examples 4.14 and 4.15.

Now, we give some examples to verify the results of Theorem 4.7, Theorem 4.10 and Corollary 4.8.

Example 4.13.

Let , , , , , and . We consider . It is easy to see that and . That is, the condition (i) of Theorem 4.10 is valid, and , for all .

If we let and , then, . But the condition (ii) of Theorem 4.10 is valid. Hence, .

Let , then, . But for all , the condition (ii) of Corollary 4.8 satisfies (see Example 3.7 in [7]), and is an efficient solution of this problem, since is a convex set. But for any , it is easy to check that there exists such that . In fact, for any , take , then, and . Hence, the condition (ii) of Theorem 4.10 is false, and is not a properly efficient solution of this problem.

Example 4.14.

Let , , , and . Take , then, it is easy to see that there exists such that and , for all . Hence, , for all . But is not a global properly efficient solution, where, is closed unit ball of .

We let and , then, , for all , and (ii) in Theorem 4.10 is false. In fact, for any and , , since . But . This implies that the conditions of Theorem 4.10 are not necessary.

Example 4.15.

Let , , , , and . We consider . It is easy to see that . But , and the condition (ii) of Theorem 4.7 is false. In fact, if we take , then, , and . Therefore, .

Example 4.16.

Let , , , , , and . We consider . It is easy to see that and . That is, the condition (i) of Theorem 4.7 is valid, and , for all .

Theorem 4.17.

(i)If , for any unit vector , then .

(ii)If , for any unit vector , then .

- (i)

- (ii)

Therefore, , for all , and , which is a contradiction to the assumption. This completes the proof.

Remark 4.18.

See the following example.

Example 4.19.

## Declarations

### Acknowledgments

This work was partially supported by the National Science Foundation of China (no. 10771228 and 10831009), the Research Committee of The Hong Kong Polytechnic University, the Doctoral Foundation of Chongqing Normal University (no.10XLB015) and the Natural Science Foundation project of CQ CSTC (no. CSTC. 2010BB2090).

## Authors’ Affiliations

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