#### Abstract

We firstly construct a concrete semi-invex set which is not invex. Basing on concept of semi-invex set, we introduce some kinds of generalized convex functions, which include semi--preinvex functions, strictly semi--preinvex functions and explicitly semi--preinvex functions. Moreover, we establish relationships between our new generalized convexity and generalized convexity introduced in the literature. With these relationships and the well-known results pertaining to common generalized convexity, we obtain results for our new generalized convexities. We extend the existing results in the literature.

#### 1. Introduction

It is well known that convexity has been playing a key role in mathematical programming, engineering, and optimization theory. The generalization of convexity is one of the most important aspects in mathematical programming and optimization theory. There have been many attempts to weaken the convexity assumptions in the literature [1–17].

One of generalization of convexity, invexity, was introduced by Hanson in [5]. Further, he proved that invexity has a common property that Karush-Kuhn-Tucker conditions are sufficient for global optimality of nonlinear programming under the invexity assumptions. Ben-Israel and Mond [6] introduced the concept of preinvex functions, which is a special case of invexity. On the other hand, Avriel [1] introduced the definition of -convex functions which is another generalization of convex functions. He also discussed some characterizations and the relations between -convexity and other generalization of convexity. In [18], Antczak introduced the concept of a class of -preinvex functions which is a generalization of -convex functions and preinvex functions, obtained some optimality results under -preinvexity assumption for constrained optimization problems.

Recently, Antczak [19] extended invexity concept to -invexity for scalar differentiable functions. In the natural way, Antczak’s definition of -invexity was also extended to the differentiable vector-valued case in [20]. With vector -invexity, Antczak [21] proved new duality results for nonlinear differentiable multiobjective programming problems. To deal with programming which is not necessarily differential, Antczak [22] introduced the concept of -preinvexity, which unifies the concepts of nondifferentiable convexity, preinvexity, and -preinvexity. Antczak [23], Luo and Wu [24] also discussed relations between concepts of different preinvexity. Further, various concepts of D--properly prequasi-invex functions were introduced in [25].

Note that characterizing the generalized convex functions are important in mathematical programming and optimization theory. Many researchers have extensively studied the properties of different generalized convex functions. Yang et al. [26] presented characterizations for prequasiinvex functions, semistrictly prequasi-invex functions, and strictly prequasi-invex functions. In [16, 17], Yang and Li presented characterizations for preinvex functions and semistrictly preinvex functions. Next, Luo and Wu [27], Luo and Xu [28], Luo et al. [29] obtained the same results or even more general ones under weaker assumptions. Luo and Wu [27] also gave characterization for strictly preinvex functions under mild conditions. Yang et al. [30] proved that the explicit -preinvexity, together with the intermediate-point -preinvexity, implies -preinvexity, while the explicit -preinvexity, together with a lower semicontinuity, implies the -preinvexity. Characterizations of D--properly prequasi-invex functions were presented in [25, 31, 32].

Motivated by [10, 11, 14, 16, 17, 22–24, 31], we present some new kinds of generalized convex functions, which include semi--preinvex functions, strictly semi--preinvex functions and explicitly semi--preinvex functions. We have managed to characterize these new kinds of generalized convex functions. The rest of the paper is organized as follows. In Section 2, we firstly construct a concrete set which is not invex but semi-invex; basing on the semi-invex set, we define some new classes of generalized convex functions and discuss the relations with each other; we also establish relation theorems with common generalized convex functions introduced in the literature; moreover, we present the optimality properties for semi--preinvex functions and explicitly semi--preinvex functions. Section 3 obtains properties for these new kinds of generalized convexity. In Section 4, we discuss relations between -preinvexity and explicitly -preinvexity; we also obtain the characterizations of -preinvexity and explicitly -preinvexity. Section 5 gives some conclusions.

#### 2. Definitions and Preliminaries

In this section, we provide some definitions and some results which we will use throughout the paper.

*Definition 2.1. **Let *, . The set is said to be semi-invex at with respect to if for all , such that
is said to be semi-invex set with respect to if is semi-invex at each . If is independent with respect to the third argument , then semi-invex set is called invex with respect to .

*Remark 2.2. *If is an invex set with respect to , then is a semi-invex set with respect to . But the converse is not true. See the following example.

*Example 2.3. *Let be a subset in defined as follows:
Consider the point . Since the tangent line of the curve at point is the line . Then, for any , there exists such that
Therefore, there exists no vector-valued function such that
However, define for , then
Hence, is semi-invex at with respect to .

*Definition 2.4 (see [33]). * Let be a nonempty semi-invex subset of . A real-valued function is said to be semi--preinvex at with respect to if there exist vector-valued function and real functions such that for all
where . The real-valued function is said to be semi--preinvex on with respect to if is semi--preinvex at each with respect to ; is said to be strictly semi--preinvex on with respect to if strict inequality (2.6) holds for all such that ; is said to be explicitly semi--preinvex on with respect to if strict inequality (2.6) holds for all such that .

*Remark 2.5. *Note that semi--preinvexity is a special kind of convexity defined in [11, 12]. Furthermore, assume that is an invex subset. Then semi--preinvexity is -preinvexity [14]; explicitly semi--preinvexity is explicitly -preinvexity [30]; strictly semi--preinvexity is strictly -preinvexity [34]. Moreover, if be a convex set, then semi--preinvexity is -vexity defined in [8, 9].

*Definition 2.6. * Let be a nonempty semi-invex subset of . A real-valued function is said to be semi--preinvex at on with respect to if there exists a continuous real-valued function such that is a strictly increasing function on its domain, a vector-valued function , and real functions such that for all
If inequality (2.7) holds for any , then is semi--preinvex on with respect to ; is said to be strictly semi--preinvex on with respect to if strict inequality (2.7) holds for all such that ; is said to be explicitly semi--preinvex on with respect to if strict inequality (2.7) holds for all such that .

*Remark 2.7. *Let be an invex subset. Then semi--preinvexity, strictly semi--preinvexity, and explicitly semi--preinvexity are called -preinvexity, strictly -preinvexity, and explicitly -preinvexity, respectively.

*Remark 2.8. *Every -preinvex function with respect to introduced in [19, 22] is semi--preinvex function with respect to , where , , ; every semi--preinvex function with respect to introduced in [14] is semi--preinvex function with respect to , where , . The converse results are, in general, not true, see Example 2.10.

*Remark 2.9. * Every semistrictly -preinvex function with respect to introduced in [24] is explicitly -preinvex function with respect to , where , , ; every explicitly semi--preinvex function with respect to introduced in [30] is explicitly semi--preinvex function with respect to , where , . The converse results are, in general, not true. See Example 2.10 too.

*Example 2.10. *Let be the subset defined in Example 2.3, , . Define
where is a point on the line between and , which is different from , such that . Define
Then, it is easy to check that is both an explicitly semi--preinvex function and a semi--preinvex function on with respect to . However, is not a -preinvex function on with respect to and is also not a semistrictly -preinvex function on with respect to , because is not an invex set. Moreover, by letting , , , we have
Hence, is not an explicitly semi--preinvex function and is also not a semi--preinvex function on with respect to .

From Definition 2.6, the inverse of function must exist. Hence function must be a strictly increasing one. Thus, we can assume that function is a strictly increasing function on its domain. Now we give the following useful lemma.

Lemma 2.11. *Let . Then:*(i)* is semi--preinvex on with respect to if and only if is semi--preinvex on with respect to ;*(ii)* is strictly semi--preinvex on with respect to if and only if is strictly semi--preinvex on with respect to ;*(iii)* is explicitly semi--preinvex on with respect to if and only if is explicitly semi--preinvex on with respect to .*

*Proof. *(i) By the monotonicity of , we know that the inequality (2.7) is equivalent with
Therefore, by Definitions 2.6 and 2.4, is semi--preinvex on with respect to if and only if is semi--preinvex on with respect to .

Similar to part (i), we can prove (ii) and (iii). This completes the proof.

Theorems 2.12 and 2.13, present the optimality properties for semi--preinvex functions and explicitly semi--preinvex functions, respectively.

Theorem 2.12. *Let be a nonempty semi-invex set in with respect to , and be a semi--preinvex function on with respect to . If is a local minimum to the problem of minimizing subject to , then is a global one.*

*Proof. *Let be a semi--preinvex function on with respect to . Then, by Lemma 2.11(i), is a semi--preinvex function on with respect to . Since is increasing on its domain , then is a local minimum to the problem of minimizing subject to if and only if is a local minimum to the problem of minimizing subject to . Therefore, by Theorem 3.1 in [33], is a global one to the problem of minimizing subject to . Hence is a global one for the problem of minimizing subject to . This completes the proof.

Theorem 2.13. *Let be a nonempty semi-invex set in with respect to , and be an explicitly semi--preinvex function on with respect to . If is a local minimum to the problem of minimizing subject to , then is a global one.*

*Proof. *Similar to the proof of Theorem 2.12, from Theorem 3.1 in [17], we can establish the result.

From Example 2.10, Theorems 2.12 and 2.13, we can conclude that these new generalized convex functions constitutes an important class of generalized convex functions in mathematical programming.

#### 3. Properties of Semi--Preinvex Functions

In this section, we first discuss the relations between our new kinds of generalized convex functions. By definitions of strictly semi--preinvexity, explicitly semi--preinvexity, and semi--preinvexity, the following result is obviously true.

Theorem 3.1. * If is strictly semi--preinvex function on with respect to , then is both an explicitly semi--preinvex function and a semi--preinvex function on with respect to .*

The following example illustrates that semi--preinvexity does not imply strictly semi--preinvexity; also explicitly semi--preinvexity does not imply strictly semi--preinvexity.

*Example 3.2. *Let be the set defined in Example 2.3; let , , and be functions defined in Example 2.10. define
Then is both an explicitly semi--preinvex function and a semi--preinvex function on with respect to , but is not a strictly semi--preinvex function on with respect to , where , .

Note that -preinvex function is semi--preinvex, and explicitly -preinvex function is explicitly semi--preinvex, where , . Examples 2.1 and 2.2 in [30] can illustrate that semi--preinvexity does not imply explicitly semi--preinvexity, and also explicitly semi--preinvexity does not imply semi--preinvexity.

Next, we present properties of semi--preinvex functions and explicitly semi--preinvex functions.

Theorem 3.3. *Let be a nonempty semi-invex set in with respect to , be an explicitly semi--preinvex function on with respect to , and be both a convex function and an increasing function. Then is an explicitly semi--preinvex function on with respect to the same .*

*Proof. * If is an explicitly semi--preinvex function on with respect to . Then, by Lemma 2.11(i), is an explicitly semi--preinvex function on with respect to . Therefore, there exist such that, for any , , , the inequality
holds. Note the convexity and monotonicity of , we have
Hence, is an explicitly semi--preinvex function on with respect to . Again, by Lemma 2.11(i), is an explicitly semi--preinvex function on with respect to . This completes the proof.

Theorem 3.4. *Let be a nonempty semi-invex set in with respect to , be semi--preinvex function on with respect to the same , , , and . Moreover, is both a convex function and a concave function on . Then, for any , , the function is semi--preinvex on with respect to the same , , , and . Further, if there exists such that is explicitly semi--preinvex on with respect to the same , , , and , then is explicitly semi--preinvex on with respect to the same , , , and .*

*Proof. *If is semi--preinvex on with respect to the same , , , and , . Then, by Lemma 2.11(i), is a semi--preinvex function on with respect to the same , , and , . Therefore, for any , , the inequality
holds for . Since is both a convex function and a concave function on , then
Multiplying (3.4) by , we have
Hence, is a semi--preinvex function on with respect to , , and . Again, by Lemma 2.11(i), is a semi--preinvex function on with respect to , , , and .

Furthermore, if there exists such that is an explicitly semi--preinvex function on with respect to the same , , , and , then, the inequality
holds for any and . Hence, for any and ,
Therefore, is an explicitly semi--preinvex function on with respect to , , and . Again, by Lemma 2.11(i), is an explicitly semi--preinvex function on with respect to , , and . This completes the proof.

Theorem 3.5. *Let be a nonempty semi-invex set in with respect to , be semi--preinvex on with respect to the same , , and , , where is a finite or infinite index set. Then function is a semi--preinvex function on with respect to the same , , , and .*

*Proof. * If is a semi--preinvex function on with respect to the same , , and , . Then, by Lemma 2.11(i), is a semi--preinvex function on with respect to the same , and , . Therefore, for any , the inequality
holds for . Define . Then,
Therefore, we have
Hence, is a semi--preinvex function on with respect to . Again, by Lemma 2.11(i), is a semi--preinvex function on with respect to , , , and . This completes the proof.

We remark that explicitly semi--preinvexity does not possess an analogous property, see the following example.

*Example 3.6. *Let , and . Define
and define
It is obvious that and are explicitly semi--preinvex functions on . Further, it can be verified that
Taking , , , we have
On the other hand,
Hence, is not an explicitly semi--preinvex functions on .

But we have the following result.

Theorem 3.7. *Let be a nonempty semi-invex set in with respect to , be both a semi--preinvex function and an explicitly semi--preinvex function on with respect to the same , , , and , , where is a finite or infinite index set. Define function , for every . Assume that for every , there exists an , such that . Then function is both a semi--preinvex function and an explicitly semi--preinvex function on with respect to the same , , and .*

*Proof. *By Theorem 3.5, we know that is a semi--preinvex function on with respect to . It suffices to show that is an explicitly semi--preinvex function on with respect to . Assume that is not an explicitly semi--preinvex function on . Then, there exist , , such that
By the semi--preinvexity of , we have
Hence
Denote . From the assumptions of the theorem, there exist , and , satisfying
Then, (3.19) implies that

(i) If , by the explicitly semi--preinvexity of , we have
From , and (3.22), we have
which contradicts (3.19).

(ii) If , by the semi--preinvexity of , we have

Since , at least one of the inequalities and has to be a strict inequality. From (3.24), we obtain
which contradicts (3.19). This completes the proof.

#### 4. Characterizations of -Preinvexity

In this section, we consider -preinvexity and explicitly -preinvexity, which are special cases of semi--preinvexity and explicitly semi--preinvexity, respectively. We obtain two sufficient conditions or characterizations for -preinvexity under the Condition C, which was introduced by Mohan and Neogy in [13]. We say that the function satisfies the Condition C if the following identities hold for any and for any .

The upper and lower semicontinuity of a real function is defined as follows.

*Definition 4.1. * Let be a nonempty subset of . The function is said to be upper semicontinuous at , if for every , there exists a such that for all , if , then
If is upper semicontinuous at , then is said to be lower semicontinuous at .

Theorem 4.2. *Let be a nonempty invex set in with respect to , where satisfies the Condition C, and be an explicitly -preinvex function on with respect to . If there exists a such that for every the following inequality holds
**
Then is -preinvex on with respect to the same , , , and .*

*Proof. *Since is an explicitly -preinvex function on with respect to , , , and . Then, by Lemma 2.11(iii), is an explicitly -preinvex function on with respect to , , and . Therefore, from Theorem 4.1 in [30], we deduce that is a -preinvex function on with respect to , , and . Again, from Lemma 2.11(iii), is a -preinvex function on with respect to the same , , , and .

By Theorem 4.2, we get the following corollary, which is Theorem 2 in [24].

Corollary 4.3. *Let be a nonempty invex set in with respect to , where satisfies the Condition C, and be a semistrictly -preinvex function on with respect to and . If there exists a such that for every the following inequality holds
**
Then is a -preinvex function on with respect to the same and .*

Theorem 4.4. *Let be a nonempty invex set in with respect to , where satisfies the Condition C and be an explicitly -preinvex function on with respect to , , , and . Assume that is a lower semicontinuous function and is a continuous one on . Then is a -preinvex function on with respect to the same , , , and .*

*Proof. *Since is a lower semicontinuous function, and is a continuous one, then is a lower semicontinuous one. By the assumption of theorem, is an explicitly -preinvex function on with respect to , , and . Therefore, from Theorem 4.2 in [30], we deduce that is a -preinvex function on with respect to , , and . From Lemma 2.11(iii), is a -preinvex function on with respect to , , , and .

As an anonymous reviewer pointed out, an interesting question is to investigate under what conditions, the -preinvex function is also a explicitly -preinvex function. Until now, we have no definite answer to this question. However, we have Theorem 4.5 which is Theorem 1 in [24] for a special case and .

Theorem 4.5. *Let be a nonempty invex set in with respect to , where satisfies the Condition C, and be a -preinvex function on with respect to . If there exists a such that for every , , the inequality
**
holds, then is explicitly -preinvex on with respect to the same .*

#### 5. Conclusions

In this paper, we firstly construct a concrete set which is not invex but semi-invex; basing on the semi-invex set, we have introduced some new kinds of generalized convex functions, which include semi--preinvex functions, strictly semi--preinvex functions and explicitly semi--preinvex functions. From Example 2.10, Theorems 2.12 and 2.13, we can conclude that these new generalized convex functions constitutes an important class of generalized convex functions in mathematical programming. Moreover, we have established the relationships between the new kinds of generalized convex functions defined in this paper and the corresponding common kinds of generalized convex one introduced in the literature. Basing on these relationships and using the well-known results pertaining to common generalized convex functions, we have obtained results for these new kinds of generalized convex functions.