#### Abstract

We mainly discuss the properties and applications of semi--preinvexity, which was introduced in Peng and Chang 2013. Examples are given to show the existence of semi--preinvex functions. Three important properties and characterizations of semi--preinvex functions are established. Furthermore, under semi--preinvexity, some optimality results are obtained in nonlinear programming problems with inequality constraint. Our results improve and generalize some known results.

#### 1. Introduction

It is well known that the research on convexity and generalized convexity is one of the most important aspects in mathematical programming and optimization theory; see, for example, [15]. In fact, there are a number of nonlinear programming problems whose objective and constraints functions are nonconvex. Therefore, in the recent years, attempts are made by several authors to define various nonconvex classes of functions and to study their optimality criteria in solving such types of problems; see, for example, [614]. One of such a generalization of convex function is invexity notion introduced in [13] and called in [15]. Over the years, many generalizations of this concept have been given in the literature. In 1972, Avriel [16] introduced the definition of -convex functions, which is a generalization of convex functions, and discussed some characterizations of -convex functions. Later, Ben-Israel and Mond [17] considered a class of nondifferentiable functions that were called preinvex in [18] as a generalization of convexity and studied how and where preinvexity can replace convexity in multiobjective programming problems. Then, the concept of preinvexity was generalized to -preinvexity [19] and semipreinvexity [20]. Using the definition of a weighted -mean (where is a real number) for a sequence of positive numbers, Antczak [21] introduced new classes of functions and called them -preinvex with respect to . Moreover, Antczak [22] considered a class of -preinvex functions, which is a generalization of -convex functions, and obtained some optimality results under appropriate -preinvexity conditions for constrained optimization problems. In 2011, Zhao et al. [23] introduced the definition of -semipreinvex functions and obtained some basic characterizations and optimality results in nonlinear programming.

On the other hand, Avriel et al. [24] introduced the definition of -convex functions, which is another generalization of convex functions, and obtained some characterizations and the relations between -convexity and convexity, where is a continuous real-valued increasing function. As a generalization of -convexity and invexity, Antczak [25] introduced the concept of -invexity and obtained some optimality conditions for constrained optimization problems under assumptions of -invexity. Recently, Antczak [26] introduced the definition of -preinvex functions, which is a generalization of -preinvex functions with [22] with respect to the same . Then, by using the -preinvexity, some optimality results for constrained optimization problems were derived. Very recently, Peng and Chang [27] introduced a class of nonconvex functions, semi--preinvex functions, which is a generalization of -preinvex functions [26], semipreinvex functions [20, 28], and -semipreinvex functions with [23] with respect to the same , and then they obtained some interesting properties of semi--preinvex functions.

But, to the best of our knowledge, there are few optimality results in nonlinear programming (especially, in mathematical programming with constant) that have been concerned under semi--preinvex functions in the literature. So, in this paper, we mainly discuss the characterizations and applications in nonlinear programming of semi--preinvexity.

The rest of the paper is organized as follows. In Section 2, we recall some concepts and give some examples to show the existence of semi--preinvex functions. In Section 3, we establish three important properties and characterizations of semi--preinvex functions, which include that the ratio of semi--preinvex functions is semi-preinvex function. In Section 4, under semi--preinvexity, we obtain some optimality results in nonlinear programming problems with inequality constraint. Some examples are given for illustration of our results. Our results extend and generalize the corresponding ones in the literature [20, 2628].

#### 2. Preliminaries

Throughout this paper, let be a nonempty subset of . Let be a real-valued function and a vector-valued function. And let be the range of , that is, the image of under , and let be the inverse of .

Now we recall some useful definitions.

Definition 1 (see [24]). Let be a continuous real-valued strictly monotonic function defined on , and , where is a nonempty convex set in . is said to be -convex on if and only if, for any , , where is the inverse of , .

Remark 2. All convex functions are -convex functions with , but the converse is not necessarily true.

Weir et al. [5, 18] presented the concepts of invex sets and preinvex functions which are generalization of convex sets and convex functions, respectively, as follows.

Definition 3. A set is said to be invex if there exists a vector-valued function such that

Definition 4. Let be an invex set with respect to and let be a mapping. We say that is preinvex if and only if
Antczak presented the concepts of -invex functions [25] and -preinvex functions [26] which are generalization of -convex functions and preinvex functions, respectively, as follows.

Definition 5. Let be a nonempty invex (with respect to ) subset of and let be a differentiable function defined on . Further, we assume that there exists a differentiable real-valued increasing function . Then is said to be -invex at on with respect to if there exists a vector-valued function such that, for all ,

Definition 6. Let be a nonempty invex (with respect to ) subset of ; is said to be (strictly) -preinvex at with respect to if and only if there exist a continuous real-valued increasing function and a vector-valued function such that, for all and any , If the inequality above is satisfied for any , then is (strictly) -preinvex on with respect to .

Remark 7. Every -preinvex function with respect to with introduced in [22] is -preinvex with respect to the same function , where is defined by . Every preinvex function with respect to is -preinvex function with respect to the same , where .

Definition 8 (see [20]). is said to be semiconnected set if there exists such that, for any , , .

Definition 9 (see [20, 28]). Let be semiconnected set with respect to ; is said to be semipreinvex with respect to if and only if, for any , , and .

In [27], Peng and Chang presented the definition of semi--preinvex functions as follows.

Definition 10. Let be semiconnected set with respect to ; is said to be (strictly) semi--preinvex at with respect to if and only if there exist a continuous real-valued increasing function and a vector-valued function such that, for all , , and If the inequality above is satisfied for any , then is (strictly) semi--preinvex on with respect to .

Remark 11. In order to define an analogous class of (strictly) semi--preincave functions with respect to , the direction of the inequality in the definition of these functions should be changed to the opposite one.

Remark 12. Every -semipreinvex function [23] () with respect to is semi--preinvex function with respect to the same function , where .

Remark 13. Every -preinvex function [26] with respect to is semi--preinvex function with respect to the same function , where . However, the converse is not true.

Remark 14. Every -convex function [24] is semi--preinvex function with respect to the same function , where . However, the converse is not true.

Remark 15. Every semipreinvex function [20] with respect to is semi--preinvex function with respect to the same , where . However, the converse is not true.

Example 16. This example illustrates that a semi--preinvex function is not necessarily a semipreinvex function with respect to the same .
Let ; it is easy to check that is a semiconnected set with respect to and , where Let be defined by . From example 2.4 of [27], we know that is a semi--preinvex function with respect to , where , but it is not a semipreinvex function with respect to the same .

Now, we give another example that illustrates that semi--preinvex functions exist.

Example 17. Let ; it is easy to check that is a semiconnected set with respect to and , where Let be defined by .
From the definition of semi--preinvexity, it is not difficult to conclude that is a semi--preinvex function with respect to , where .
In the sequel, we will use the following lemmas.

Lemma 18. Let be a continuous real-valued function; then is increasing if and only if is increasing.

Lemma 19. Let be a continuous real-valued function. If is increasing and concave, then is convex. If is increasing and convex, then is concave.

#### 3. Properties and Characterizations of Semi--Preinvex Function

In this section, we establish three interesting properties for semi--preinvex function, which extend and generalize the corresponding ones in the literature [27, 28].

Firstly, we show that the ratio of semi--preinvex functions is semi--preinvex, which extends earlier results by Craven and Mond [11] and Yang et al. [28]. Since the proof is obvious, it has been omitted in this paper.

Theorem 20. Let be semiconnected set with respect to and let and be real-valued functions defined on . If , , is a semi--preinvex function, is a semi--preinvex function with respect to on , where is concave and is convex. Then,(i) is a semipreinvex function with respect to ;(ii)if , we also have be an invex function with respect to .

Theorem 21. Let be semi-connected set with respect to ; is a semi--preinvex function with respect to the same if and only if the set is semiconnected with respect to , where for all .

Proof. In a similar way to the proof of Theorem 3.5 of [27], with suitable modifications, we can obtain the conclusion.

Theorem 22. Let and , for all ; is continuous real-valued increasing. If is a semiconnected set with respect to , for all , where . Then is a semi--preinvex function with respect to .

Proof. (i)  Let , and let . Since is a semiconnected set with respect to , then, for any , From it follows that Thus, by the definition of , we have Hence, is semi--preinvex function with respect to on .
(ii)  When , the conclusion is correct obviously.

#### 4. Semi--Preinvexity and Optimality in Nonlinear Programming

In this section, we consider nonlinear programming problems with inequality constraint and obtain some optimality results under semi--preinvexity. Meanwhile, two examples are given for illustration of the main results. Because semi--preinvexity is a generalization of -(pre-)invexity and (semi-)preinvexity, some of our results generalize the corresponding results of [20, 26].

We consider the mathematical programming problem with inequality constraint as follows: where is a nonempty subset of , , . Denote the set of all feasible solutions for by

Theorem 23. Let the set of all feasible solutions in problem be semiconnected set with respect to . If is a nonconstant semi--preincave function with respect to on , then no interior point of is an optimal solution of , or equivalently, any optimal solution in problem , if it exists, must be a boundary point of .

Proof. If problem has no solution, the theorem is trivially true. Let be an optimal solution in problem . By assumption, is nonconstant on . Then, there exists a feasible point such that Let be an interior point of . By assumption, is semiconnected set with respect to . Then there exists such that, for some , Since is a semi--preincave function with respect to at any optimal solution on , then From the inequality above, we know that does not attain its minimum at an interior point . This completes the proof of the theorem.

Let denote a neighborhood of of radius .

Theorem 24. Let the set of all feasible solutions in problem be semiconnected set with respect to and let be a local minimum in problem . Suppose that for every and for every there exists such that . If is strictly semi--preinvex with respect to at on , then is a strictly global minimum in .

Proof. By assumption, is strictly semi--preinvex with respect to at on . Then holds for all and any . Since is a local minimum in problem , then there exists such that the inequality holds for all .
Now, let , such that . Since is semiconnected set with respect to , then for all and any . By assumption, for every and for every , there exists such that . From (20), there exists such that By virtue of (19) and (22), we obtain If , from (23), we can get , which is impossible. Thus, ; it follows from (23) that the following inequality holds. By the arbitrariness of , we get the conclusion of the theorem; that is, is a strictly global minimum in .

Theorem 25. Let the objective function be the semi--preinvex with respect to on , and let the constraint functions be semi--preinvex with respect to the same function on . Then, the set of all optimal solutions is semiconnected set with respect to the same .

Proof. Let and be optimal solutions in such that . Then . By assumption, is semi--preinvex with respect to on . Since , from Definition 10, we have From , one has To prove that for any , it is sufficient to show that for any . By assumption, , are semi--preinvex with respect to the same function on . Therefore, for , From the definition of semi--preinvexity, is an increasing function on its domain. Using together with Lemma 18, for any , Thus, for any , So the set of all optimal solutions for is semiconnected set with respect to .

Now, we illustrate the theorem above by a nonconvex optimization problem.

Example 26. We consider the following optimization problem: Note that the set of all feasible solutions and the set of all optimal solutions .
It can be proved by Definition 10 that all functions constituting the considered optimization problem are semi--preinvex on with respect to the same function , where
Indeed, the objective function is semi--preinvex on with respect to and the constraint function is semi--preinvex on with respect to . Since all hypotheses of Theorem 25 are fulfilled, then the set of all optimal solutions in the considered optimization problem is semiconnected with respect to the function .
Indeed, it is not difficult to show by Definition 8 that the set of all optimal solutions in the considered optimization problem is semiconnected with respect to the same . However, the set of all optimal solutions is not convex.

Theorem 27. Let be optimal in . Moreover, suppose that is strictly semi--preinvex with respect to at on and the constraint functions , are semi--preinvex with respect to the same function on . Then, is a unique optimal solution in problem .

Proof. By contradiction, we assume that there exists , being another optimal solution in problem . By assumption, the constraint functions , are semi--preinvex with respect to the same function on . It follows from Definition 10 and , for all and any ; we have Then, for all and any , one has Since is strictly semi--preinvex with respect to at on , from Definition 10, the following inequality holds for any and all . So, we can get for .
Since , we obtain that the inequality above (35) is a contradiction to the optimality of in problem . The proof is complete.

Now, we give an example to illustrate Theorem 27 as follows.

Example 28. Consider the following optimization problem: Note that the set of all feasible solutions and the feasible solution are optimal in the considered optimization problem.
From Definition 10, we can verify that the objective function is strictly semi--preinvex on with respect to and and the constraint function is semi--preinvex on with respect to and , where the function is defined by All assumptions of Theorem 27 are satisfied; by virtue of Theorem 27, the optimal solution is unique in the considered optimization problem. While, in Example 26, we notice that the objective function is only semi--preinvex on , but it is not strictly semi--preinvex. Therefore, the considered nonconvex optimization problem has no unique optimal solution; what is more, the set of all optimal solutions is not bounded in Example 26.

Remark 29. From Theorems 2325, Theorem 27, and Examples 26 and 28, we can know that this class of semi--preinvexity plays an important role in optimization theory and applications.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the Natural Science Foundation of China (nos. 11301571, 71201126, and 11271389), the Science Foundation of Ministry of Education of China (no. 12XJC910001), the Natural Science Foundation Project of Chongqing (CSTC,2012jjA00016), and the Education Committee Research Foundation of Chongqing (KJ130428).