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.

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.

Consider the following multiobjective optimization problem:

where is an arbitrary nonempty set, . As usual, the preference relation defined in by a closed convex pointed cone is used, which models the preferences used by the decision-maker:

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]).

Let be a fixed element, and .

(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.

Let be a nonempty convex set.

The contingent cone of at is defined as

The cone of feasible directions of at is defined as

Let , the -normal set of at is defined as

Lemma 2.4 (see [26]).

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

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

Theorem 3.1.

Let and . If

then .

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.

Let .

(i)If , then .

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

Proof.

(i) Suppose, on the contrary, that , then, there exists such that . Hence, there exist , and such that . Since , there exists such that .

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

On the other hand, from , we have . Therefore, there exists such that , and so , which deduces a contradiction, and the proof is completed.

(ii) Now, we let . From , we have

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

Since is a convex set, . Hence,

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 .

Suppose, on the contrary, that , then, there exists such that

and there exist , for all such that . That is, there exist and , for all such that , for all . Since , there exists such that , for all . From , and , for all , we have , for all . Therefore,

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.

Assume that, there exists such that and . Suppose, on the contrary, that , then, there exist and such that . From and , we have . Hence,

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.

Assume that, there exists such that and . Suppose, on the contrary, that , then, there exists such that

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, .

Conversely, let , then,

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 .

From , we have

Since and , we have

Which implies . This completes the proof.

Example 3.7.

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

Remark 3.8.

(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].

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

Definition 4.1.

Let be a fixed element and .

(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.

Definition 4.2 (see [4, 5]).

Let and .

(i)The second-order tangent set to at is defined as

(ii)The asymptotic second-order tangent cone to at is defined as

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

Lemma 4.3.

Let and , then,

(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]).

Let and be a nonsmooth function. The Hadamard upper directional derivative and the Hadamard lower directional derivative derivative of at in the direction are given by

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.

Theorem 4.6.

(i) Let , then, for any fixed ,

Conversely, if , and there exists a neighborhood of such that , for all , that is, , for all , then .

(ii) For any fixed , . Conversely, if and there exists a neighborhood of such that for any fixed and , , then .

(iii) For any fixed , . Conversely, if and there exists a neighborhood of such that for any fixed and , is a closed set, and , then .

Proof.

(i) Let , then, there exists a neighborhood of such that . From , we have

Which implies .

Conversely, we assume that there exists a neighborhood of such that , for all . Suppose, on the contrary, that , then, for any neighborhood of . Take , then, there exist and such that . Therefore, if is sufficiently small, we have , which is a contradiction to , for all . This completes the proof.

(ii) It is easy to see that .

Conversely, we assume that there exists a neighborhood of such that for any fixed and , . Suppose, on the contrary, that , then, for any neighborhood of , we have . Take , then, there exist and such that . Take and , then, , which is a contradiction to the assumption. This completes the proof.

(iii) It is easy to see that .

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 .

(i)If , then .

(ii)If , and for each

then .

Proof.

(i) Let . Suppose, on the contrary, that , then, there exists and such that , for all . Since is a continuous function and is a pointed cone, , for all and . Therefore, .

On the other hand, for any , we have

Since and , there exists such that

Hence, , which is a contradiction to the assumption. This completes the proof.

(ii) Suppose, on the contrary, that . Similar to the proof of (i), we have there exists such that

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).

(ii)If , and for each

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

, and for each

since by Lemma 4.3(iii).

Theorem 4.10.

Let be continuous on , , and .

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

(ii)Assume that , and there exists such that for each the following conditions hold

then , where, denotes the closed unit ball of .

Proof.

(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

Suppose, on the contrary, that , then, for any neighborhood of 0, we have

Therefore,

That is, for any there exist , and so, for any there exists , and such that and . Since , there exists such that , for all . By , we have

Let for and , then,

Let , then, , since is a convex set, and so,

On the other hand, from , we have when and . Since is a continuous function, when and , which combining with the assumption yields there exist and such that

From , there exists such that , for all . Take , and let . Since is an arbitrary set, it follows that

Which is a contradiction to (4.13). This completes the proof.

(ii) Suppose, on the contrary, that , then, for any and , we have

Let . Similar to the proof of (i), we have for any there exist , , and such that and . It is obvious that . Otherwise, , which is a contradiction to the assumption that is a pointed cone. Since and , there exists such that and , for all . From , we have , when and . Since is a continuous function and , it is easy to see that . From , we have for sufficiently large

On the other hand, we have

for sufficiently large . In fact, for sufficiently large

Hence,

when and sufficiently large enough. Since is arbitrary,

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 .

(ii), and there exists such that for each

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.

Let , and .

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

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

Where, .

Proof.

(i) Suppose, on the contrary, that , then, there exists , and such that . Let and , then, , and . Hence,

Since , there exists such that , for all . Hence,

Therefore,

Which is a contradictions to the assumption. This completes the proof.

(ii) Similar to the proof of (i), we have there exists and such that . Hence, there exists such that , for all . It is easy to see that, if we take an appropriate subsequences and of and , respectively, then there exist an index , and such that

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

Remark 4.18.

The following necessary conditions for -local weakly (efficient) solutions may not be true.

See the following example.

Example 4.19.

Let ,

,, , . Consider the following problem:

It is easy to see that is an -efficient solution of (MP), but, . In fact,

, for all . It is obvious that . On the other hand, . Hence, .