Abstract and Applied Analysis

Volume 2013 (2013), Article ID 103864, 7 pages

http://dx.doi.org/10.1155/2013/103864

Research Article

Semistrict -Preinvexity and Optimality in Nonlinear Programming

Z. Y. Peng,1,2 Z. Lin,¹ and X. B. Li¹

¹College of Science, Chongqing JiaoTong University, Chongqing 400074, China

²Department of Mathematics, Inner Mongolia University, Hohhot 010021, China

Received 26 October 2012; Accepted 10 April 2013

Academic Editor: Natig M. Atakishiyev

Copyright © 2013 Z. Y. Peng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A class of semistrictly -preinvex functions and optimality in nonlinear programming are further discussed. Firstly, the relationships between semistrictly -preinvex functions and -preinvex functions are further discussed. Then, two interesting properties of semistrictly -preinvexity are given. Finally, two optimality results for nonlinear programming problems are obtained under the assumption of semistrict -preinvexity. The obtained results are new and different from the corresponding ones in the literature. Some examples are given to illustrate our results.

1. Introduction

It is well known that convexity and generalized convexity have been playing a central role in mathematical programming, economics, engineering, and optimization theory. The research convexity and generalized convexity are one of the most important aspects in mathematical programming and optimization theory in [1–4]. Various kinds of generalized convexity have been introduced by many authors (see, e.g., [5–21] and the references therein). In 1981, Hanson [5] introduced the concept of invexity which is extension of differentiable convex functions and proved the sufficiency of Kuhn-Tucker condition. Later, Weir and Mond [6] considered functions (not necessarily differentiable) for which there exists a vector function such that, for all , , one hasthe following:

which has been named as preinvex functions with respect to vector-valued function . In 2001, Yang and Li [8] obtained some properties of preinvex function. At the same time, Yang and Li [9] introduced the concept of semistrictly preinvex functions and investigated the relationships between semistrictly preinvex functions and preinvex functions. It is worth mentioning that many properties and applications in mathematical programming for invex functions and preinvex functions are discussed by many authors (see, e.g., [6–11, 21] and the references therein).

On the other hand, Avriel et al. [12] introduced a class of -convex functions which is another generalization of convex functions and obtained some relations with other generalization of convex functions. In [13], Antczak introduced the concept of a class of -invex fuctions, which is a generalization of -convex functions and invex functions. Recently, Antczak [14] introduced a class of -preinvex functions, which is a generalization of -invex [13], preinvex functions [8] and derived some optimality results for constrained optimization problems under -preinvexity. Very recently, Luo and Wu introduced a new class of functions called semistrictly -preinvex functions in [15], which include semistrictly preinvex functions [9] as a special case. They investigated the relationships between semistrictly -preinvex functions and -preinvex functions and gave a criterion for semistrict -preinvexity. Moreover, they also proposed three open questions (just as they said: “an interesting topic for our future research is to”: (1) investigate -preinvex functions and semicontinuity; (2) explore some properties of semistrictly -preinvex functions; (3) research into some applications in optimization problems under semistrictly -preinvexity [15]).

However, as far as we know, there are few papers dealing with the properties and applications of the semistrictly -preinvex functions [16]. The questions above in [15] have not been solved, and one condition in [15] is not mild in researching of relationships between -preinvexity and semistrict -preinvexity. So, in this paper, we further discuss semistrictly -preinvex functions. The rest of the paper is organized as follows. Firstly, we investigate -preinvex functions and semicontinuity and obtain a criterion of -preinvexity under semicontinuity. Then, we give new relationships between -preinvexity and semistrictly -preinvexity, which are different from the recent ones in the literature [15]. Finally, we get two optimality results under semistrict -preinvexity for nonlinear programming. The obtained results in this paper improve and extend the existing ones in the literature (e.g., [8, 9, 11, 15, 16]).

2. Preliminaries and Definitions

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

Now we recall some definitions.

Definition 1 (see [5, 6]). A set is said to be invex if there exists a vector-valued function () such that

Definition 2 (see [6, 8]). Let be an invex set with respect to and let be a mapping. One says that is preinvex if

Remark 3. Any convex function is a preinvex function with .

Definition 4 (see [10]). Let be an invex set with respect to . Let . One says that is prequasi-invex if

Definition 5 (see [14]). Let be a nonempty invex (with respect to ) subset . A function is said to be -preinvex at on if there exists a continuous real-valued increasing function such that for all and (),

If (5) is satisfied for any then is -preinvex on , with respect to .

Definition 6 (see [15]). Let be a nonempty invex (with respect to ) subset . A function is said to be semistrictly -preinvex at on if there exists a continuous real-valued increasing function such that for all () and ,

If (6) is satisfied for any , then is semistrictly -preinvex on with respect to .

Remark 7. In order to define an analogous class of semistrictly -preincave functions with respect to , the direction of the inequality in Definition 6 should be changed to the opposite one.

Remark 8. Every semistrictly preinvex function [8, 9] is semistrictly -preinvex with respect to the same function , where is defined by .

In order to prove our main result, we need Condition C as follows.

Condition C (see [9]). The vector-valued function is said to satisfy Condition C if for any , and ,

Example 9. Let

It can be verified that satisfies the Condition C.

3. Relationships with Semistrictly -Preinvexity

In [15], Luo and Wu obtained a sufficient condition of -preinvex functions under the condition of intermediate-point -preinvexity. Now we investigate -preinvex functions and semicontinuity without the condition of intermediate-point -preinvexity.

Theorem 10. Let be a nonempty invex set with respect to , where satisfies Condition C. Let be lower semicontinuous and semistrictly -preinvex for the same on , and let   (for all ). Then is -preinvex function on .

Proof. Let . From the assumption of , when , we can know that

Then, there are two cases to be considered.

(i)If , then by the semistrict -preinvexity of , we have the following:

(ii)If , to show that is a -preinvex function, we need to show that

By contradiction, suppose that there exists an such that

Let . Since is lower semicontinuous, there exists , such that

From Condition C,

By the semistrict -preinvexity of and (12), we have the following:

On the other hand, from Condition C, one can obtain the following:

According to (13) and the semistrictly -preinvexity of , we have the following:

which contradicts (15). This completes the proof.

Remark 11. In Theorem 10, we investigate -preinvex functions and lower semicontinuity, and establish a new sufficient condition for -preinvexity without the condition of intermediate-point -preinvexity. Therefore, we also answer the open question (1) which proposed in [15] (“(1) investigate -preinvex functions and semicontinuity” [15]).

Now, we give a new sufficient condition for semistrictly -preinvexity.

Theorem 12. Let be a nonempty invex set with respect to , where satisfies Condition C. Let be a -preinvex function for the same on . Suppose that for any with , there exists an such that

Then, is semistrictly -preinvex function on .

Proof. For any with and , by assumption, we have the following:

(i) Let . From Condition C, we can obtain the following:

Using (18) and the -preinvexity of , we have the following:

(ii) Let ; that is,

From Condition C, we have the following:

According to (18) and the -preinvexity of , we get the following:

(21) and (24) imply that is a semistrictly -preinvex function on .

Remark 13. Theorem 12 extends and improves Theorem 1 in [15]. In Theorem 1 of [15], a uniform is needed; while in assumption (18) of Theorem 12, this condition has been weakened, where a uniform is not necessary. Moreover, our proof is also different from the corresponding result of [15].

The following example illustrates that assumption (18) in Theorem 12 is essential.

Example 14. Let

It is obvious that is a -preinvex function with respect to , where , .

However, from Definition 6, we can verify that is a semistrictly -preinvex function. The reason is that the assumption (18) is violated. Indeed, there exist , (with ), for any , we have the following:

Therefore, (18) is essential.

Lemma 15 (see [16]). Let be a nonempty invex set with respect to . Suppose that is a semistrictly -preinvex function with respect to and is a continuous and strictly increasing function on . If is convex on the image under of the range of , then is also semistrictly -preinvex function with respect to the same on .

Theorem 16. Let K be an invex set with respect to and be a semistrictly -preinvex function on . If is concave on , then is a preinvex function with respect to the same on .

Proof. Let and be two points in . Because is concave on , the following inequality

holds for any . Let and . Then, for each pair of points and in image of , that is, and , we have the following:

It follows from (28) and the increasing property of that

Thus,

This means that is convex. Letting , , then is convex. Hence, by virtue of Lemma 15, is a semistrictly -preinvex function with respect to . Because is identity function, is a preinvex function with respect to the same on .

From Theorem 16 and Definitions 2-4, we can obtain the following Corollary easily.

Corollary 17. Let K be an invex set with respect to and is a semistrictly -preinvex function on . If is concave on then be a prequasi-invex function with respect to the same on .

4. Semistrictly -Preinvexity and Optimality

In order to solve the open question (3) proposed in [15] (see, the part of Introduction), in this section, we consider nonlinear programming problems with constraint and obtain two optimality results under semistrict -preinvexity.

We consider the following nonlinear programming Problem with inequality constraint:

where , , , and is a nonempty subset of . We denote the set of all feasible solutions in by the following:

Theorem 18. Suppose the set of all feasible solutions of problem is an invex set with respect to , and at least contains two points with nonempty interior. Let be a nonconstant semistrictly -preincave function with respect to on . Then no interior of is an optimal solution of , or equivalently, any optimal solution in problem , if 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 a nonconstant on . Then, there exists a feasible point such that

Let () be an interior point of . By assumption, is an invex set with respect to . It follows from the definition of invex set that there exists such that for some ,

By assumption, is semistrictly -preincave with respect to on . Then, we have the following:

where is an interior point of .

From the inequality above, we conclude that no interior of is an optimal solution of , that is, any optimal solution in problem , if exists, must be a boundary point of . This completes the proof.

Theorem 19. Let be local optimal in problem . Moreover, we assume that is semistrictly -preinvex with respect to at on and the constraint functions , , are -preinvex with respect to at on . Then, is a global optimal solution in problem .

Proof. Assume that is a local optimal solution in . Hence, there is an neighborhood around such that

Suppose to the contrary, is not a global minimum in , there exists an such that

By assumption, the constraint functions , , are -preinvex with respect to the same at on . By using Definition 5 together with , , then for all and any , we have the following:

Thus, for all and any ,

By assumption, is semistrictly -preinvex with respect to the same at on . Therefore, by Definition 6, we have the following:

From (37) and (40), we have the following:

For a sufficiently small , it follows that

which contradicts that is local optimal in problem . This completes the proof.

Now, we give an example to illustrate Theorem 19.

Example 20. Let be defined as follows:

and let

From Definitions 5-6, we can get that is semistrictly -preinvex with respect to and the constraint functions , , are -preinvex with respect to , respectively, where , . Obviously, is a local optimal solution of . Then, all the conditions in Theorem 19 are satisfied. By virtue of Theorem 19, is a global optimal solution in problem .

Acknowledgments

The authors would like to express their thanks to Professor X. M. Yang and the anonymous referees for their valuable comments and suggestions which helped to improve the paper. This work was supported by the Natural Science Foundation of China (nos. 11271389, 11201509, and 71271226), the Natural Science Foundation Project of ChongQing (nos. CSTC, 2012jjA00016, and 2011AC6104), and the Research Grant of Chongqing Key Laboratory of Operations and System Engineering.

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