Semistrict G-preinvexity and its application

In this paper, a class of semistrictly G-preinvex functions introduced by Luo and Wu (J. Comput. Appl. Math. 222:372-380, 2008) is further considered. Some properties of semistrictly G-preinvex functions are obtained, especially those containing an interesting gradient property. Then, some optimality results, which extend the corresponding results in the literature (Yang and Li in J. Math. Anal. Appl. 256:229-241, 2001; Yang and Li in J. Math. Anal. Appl. 258:287-308, 2001; Antczak in J. Glob. Optim. 43:97-109, 2009; Luo and Wu in J. Comput. Appl. Math. 222:372-380, 2008), are derived in multiobjective optimization problems.

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 on characterizations and generalizations of convexity and generalized convexity is 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. In 1981, Hanson [5] introduced the concept of invexity which is an extension of differentiable convex functions and proved the sufficiency of Kuhn-Tucker condition. Later, Weir and Mond [6] and Weir and Jeyakumar [7] introduced preinvex functions, and they also studied how and where preinvex functions can replace convex functions in an optimization problem. Then, Yang and Li [8] obtained some properties of a preinvex function in 2001. Yang and Li [9] also 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 of invex functions and (semistrictly) preinvex functions and their applications in mathematical programming are discussed in some existing literature (see [6-11]).

On the other hand, Avriel et al. [12] introduced the definition of G-convex functions, which is another generalization of convex functions, where G is a continuous real-valued increasing function. As a generalization of G-convex functions and invex functions, Antczak [13] introduced the concept of G-invex functions and derived some optimality conditions for constrained optimization problems under the assumption of G-invexity. Antczak [14] introduced a class of G-preinvex functions, which is a generalization of G-invex [13], preinvex functions [6,8] and r-preinvex functions [15]. Then, Luo and Wu [16] introduced the concept of semistrictly G-preinvex functions, which includes semistrictly preinvex functions [9] as a special case, and investigated the relations between semistrictly G-preinvex functions and G-preinvex functions.

However, to the best of our knowledge, it appears that there are no results on the properties and applications of semistrictly G-preinvex functions in literature. So, in this paper we study some properties of semistrictly G-preinvex functions and applications in a multiobjective optimization problem. The rest of the paper is organized as follows. In Section 2, we recall some definitions and give some examples to show that semistrictly G-preinvex functions are different from preinvex functions, G-invex functions, G-preinvex functions and strictly G-preinvex functions. In Section 3, we obtain some properties of semistrictly G-preinvex functions, especially those containing an interesting gradient property. Finally, optimality results for multiobjective optimization problems are obtained in Section 4. Our results extend and generalize the corresponding ones in [8,9,13,14,16].

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

Now we recall some definitions.

Definition 2.1 ([5])

A set K is said to be invex at y with respect to η if for all , such that

The set K is said to be invex with respect to η if K is invex at each .

Definition 2.2 ([6])

Let be an invex set with respect to η. The function f is said to be preinvex on K with respect to η iff

Remark 2.1 Any convex function is a preinvex function with . But the converse is not true.

Definition 2.3 ([9])

Let be an invex set with respect to η. The function f is said to be semistrictly preinvex on K with respect to η if, for all , , we have

Remark 2.2 Any semistrictly (or strong) convex function is a semistrictly preinvex function with . But the converse is not true.

Definition 2.4 ([14])

Let be an invex set with respect to η. The function f is said to be G-preinvex on K with respect to η if there exists a continuous real-valued increasing function such that for all and , we have

is said to be strictly G-preinvex on K, if the inequality (2.1) is strict for all , and .

Definition 2.5 ([16])

Let be an invex set with respect to η. The function f is said to be semistrictly G-preinvex on K with respect to η if there exists a continuous real-valued increasing function such that for all , and ,

Remark 2.3 It is clear that the semistrictly G-preinvex function is a generalization of semistrictly preinvex function.

Example 2.1 This example illustrates that a semistrictly G-preinvex function is not necessarily a (strictly) G-preinvex function with respect to the same η. Let

Then, we can verify that f is a semistrictly G-preinvex function with respect to η, where . However, by letting , (), , we have

Thus, f is not a G-preinvex function with respect to the same η, and it is also not a strictly G-preinvex function with respect to the same η.

Example 2.2 This example illustrates that a semistrictly G-preinvex function is not necessarily a G-invex function. Let

Then, by [16], Example 2], f is a semistrictly G-preinvex function with respect to η, where . It can be easily noticed that f is not differentiable at . Thus, f is not a G-invex function (see [13]) with respect to η.

Example 2.3 Seeing the function f and η in Example 2.1, it is obvious that f is a semistrictly G-preinvex function with respect to η, where . However, by letting , , , we have

Thus, f is not a preinvex function with respect to the same η.

Remark 2.4 From Examples 2.1-2.3, we know that semistrictly G-preinvex functions are different from G-preinvex functions, strictly G-preinvex functions, G-invex functions and preinvex functions with respect to the same η.

In order to discuss the properties of semistrictly G-preinvex functions, we recall the definition of Condition C as follows.

Condition C ([8,11])

The vector-valued function is said to satisfy Condition C if for any and ,

In the sequel, we will use the following lemma.

Lemma 2.1 ([14])

is (strictly) increasing if and only ifGis (strictly) increasing.

The next lemma can be easily proved by Lemma 2.1 and the definitions of a concave function and a convex function, so we omit it.

Lemma 2.2IfGis increasing and concave, thenis convex.

In this section, we derive some interesting properties of semistrictly G-preinvex functions.

Theorem 3.1LetKbe a nonempty invex set with respect toη, whereηsatisfies Condition C, and let () be a finite or infinite collection of both semistrictlyG-preinvex andG-preinvex functions for the sameηonK. Define, for every. Assume that for every, there exists ansuch that. Thenfis both a semistrictlyG-preinvex andG-preinvex function with respect to the sameηonK.

Proof By Proposition 12 in [14], we know that f is G-preinvex on K. We need to show that f is a semistrictly G-preinvex function on K. Assume that f is not a semistrictly G-preinvex function. Then, there exist with and such that

By the G-preinvexity of f, we have

It follows that

Let . By the assumptions, there exist , , , satisfying

This fact together with (3.1) yields

(i) If , then by the semistrict G-preinvexity of ,

From , and (3.3), we obtain

which contradicts (3.2).

(ii) If , then by the G-preinvexity of ,

Since , at least one of the inequalities and has to be a strict inequality. From (3.4) and the continuity and increasing property of G, we obtain

which contradicts (3.2). This completes the proof. □

Remark 3.1 Theorem 3.1 generalizes Theorem 3.8 [9] from a semistrictly preinvex case to a semistrictly G-preinvex case.

Next, we will establish an important gradient property of semistrictly G-preinvex functions. Before showing the property in Theorem 3.3, we first derive a result of G-preinvex functions.

Theorem 3.2LetKbe a nonempty invex set inwith respect to, andfbe aG-preinvex function with respect to the sameηonK. Assume thatηsatisfies Condition C. For anyand, let. Then

or equivalently,

Proof For , let , , . By Condition C,

We have

Therefore, we obtain

This completes the proof. □

From Definitions 2.4 and 2.5, we can obtain the following lemma.

Lemma 3.1LetKbe a nonempty invex set with respect toη, whereηsatisfies Condition C. Letbe continuous and semistrictlyG-preinvex with respect toηonK, and satisfy (). Thenfis aG-preinvex function onK.

Theorem 3.3LetKbe a nonempty invex set with respect toη, whereηsatisfies Condition C. Assume thatis differentiable and semistrictlyG-preinvex with respect toηonK, and satisfies (), whereGis a differentiable function. Then for anywith, we have

(i) ,

(ii) .

Proof (i) Suppose that f is a semistrictly G-preinvex function on K. By Definition 2.4, for any with , we have

which implies

It follows that

From Lemma 3.1 and Theorem 3.2, we get

that is

(ii) Since f is a semistrictly G-preinvex function on K, for with , it follows form the above results that

From (3.5) and (3.6), we can obtain

This completes the proof. □

Remark 3.2 As a matter of fact, the assumption of continuity for f can be extended to lower semicontinuity in Lemma 3.1.

In the section, we consider a class of multiobjective optimization problems and obtain an important optimality result under semistrict G-preinvexity.

From now on, we suppose that is a vector-valued mapping, where X is an invex subset of endowed with the Euclidean norm .

We consider the following multiobjective optimization problem:

In the sequel, we use the following notations. For

(i) for every ;

(ii) is the negation of .

Definition 4.1 Let be an integer. A point is said to be a strictly local minimizer of order m for (MOP) if there exist an and a vector such that

where is an ε neighbor of .

Now, we give the notion of a strict minimizer in the global sense if the neighbor is replaced by the whole space .

Definition 4.2 Let be an integer. A point is said to be a strictly global minimizer of order m for (MOP) if there exists a vector such that

Remark 4.1 If , then Definitions 4.1-4.2 reduce to Definitions 4.2-4.3 introduced by Bhatia [17], respectively.

Remark 4.2 From Definitions 4.1 and 4.2, we know that the concepts of a strictly local minimizer of order m and a strictly global minimizer of order m for (MOP) are stronger than the concepts of a strictly local minimizer and a strictly global minimizer for (MOP), respectively.

It is clear that any strictly global minimizer of order m is a strictly global minimizer. But the converse may not be true. We can see the case in the following example.

Example 4.1 Let be defined as

Consider the multiobjective optimization problem,

is a strictly global minimizer but is not a strictly global minimizer of order m, because for any , and sufficiently small, we have , where or .

Theorem 4.1LetXbe an invex set with respect toη. Suppose the following conditions are satisfied:

(i) LetGbe increasing and concave on;

(ii) satisfies Condition C;

(iii) is a strictly local minimizer of ordermfor (MOP).

If, , are semistrictlyG-preinvex onXwith respect toη, thenis a strictly global minimizer of ordermfor (MOP).

Proof Let be a strictly local minimizer of order m for (MOP). Then there exist an ε neighborhood of and a vector such that

Hence, there exists no such that

where .

Suppose by contradiction that is not a strictly global minimizer of order m for (VP), then there exists with such that

for any . Since X is an invex set with respect to η,

Because , , are semistrictly G-preinvex on X with respect to η, G is increasing on and is convex, it follows that for any

where Lemma 2.2 is used in the second inequality.

According to (4.2), (4.3) and Condition C, we have

where . For a sufficiently small , we obtain

Let . Since is arbitrary, is also arbitrary. Therefore,

or

which implies that is not a strictly local minimizer of order m. It is a contradiction. Hence, is a strict minimizer of order m for (MOP). □

Now, we give an example of an optimization problem to illustrate Theorem 4.1.

Example 4.2 Let be defined as

where

From Definition 2.5, we can verify that () are semistrictly G-preinvex functions with respect to η, where . is a strictly local minimizer of order m for (MOP). From Theorem 4.1, we can get is also a strictly global minimizer of order m for (MOP), and is a global minimal value of (MOP).

The author declares that they have no competing interests.

The author is very grateful to the three anonymous referees for valuable comments and suggestions which helped to improve the paper. This work was supported by the Natural Science Foundation of China (No. 11271389, 11201509, 71271226), the Natural Science Foundation Project of Chongqing (No. CSTC, 2011AC6104.2012jjA00016) and the Education Committee Project Research Foundation of Chongqing (No. KJ100711).

1. Mangasarin, OL: Nonlinear Programming, Mcgraw-Hill, New York (1969)

2. Bazaraa, MS, Sherali, HD, Shetty, CM: Nonlinear Programming Theory and Algorithms, Wiley, New York (1979)

3. Schaible, S, Ziemba, WT: Generalized Concavity in Optimization and Economics, Academic Press, London (1981)

4. Ward, DE: Characterizations of strict local minima and necessary conditions for weak sharp minima. J. Optim. Theory Appl.. 80, 551–571 (1994). Publisher Full Text

5. Hanson, MA: On sufficiency of Kuhn-Tucker conditions. J. Math. Anal. Appl.. 80, 545–550 (1981). Publisher Full Text

6. Weir, T, Mond, B: Pre-invex functions in multiple objective optimization. J. Optim. Theory Appl.. 136, 29–38 (1988)

7. Weir, T, Jeyakumar, V: A class of nonconvex functions and mathematical programming. Bull. Aust. Math. Soc.. 38, 177–189 (1988). Publisher Full Text

8. Yang, XM, Li, D: On properties of preinvex functions. J. Math. Anal. Appl.. 256, 229–241 (2001). Publisher Full Text

9. Yang, XM, Li, D: Semistrictly preinvex functions. J. Math. Anal. Appl.. 258, 287–308 (2001). Publisher Full Text

10. Yang, XM, Yang, XQ, Teo, KL: Characterizations and applications of prequasi-invex functions. J. Optim. Theory Appl.. 110, 645–668 (2001). Publisher Full Text

11. Peng, JW, Yang, XM: Two properties of strictly preinvex functions. Oper. Res. Trans.. 9, 37–42 (2005)

12. Avriel, M, Diewert, WE, Schaible, S, Zang, I: Generalized Concavity, Plenum, New York (1975)

13. Antczak, T: On G-invex multiobjective programming. I. Optimality. J. Glob. Optim.. 43, 97–109 (2009). Publisher Full Text

14. Antczak, T: G-preinvex functions in mathematical programming. J. Comput. Appl. Math.. 217, 212–226 (2008). Publisher Full Text

15. Antczak, T: r-preinvexity and r-invexity in mathematical programming. Comput. Math. Appl.. 50, 551–566 (2005). Publisher Full Text

16. Luo, HZ, Wu, HX: On the relationships between G-preinvex functions and semistrictly G-preinvex functions. J. Comput. Appl. Math.. 222, 372–380 (2008). Publisher Full Text

17. Bhatia, G: Optimality and mixed saddle point criteria in multiobjective optimization. J. Math. Anal. Appl.. 342, 135–145 (2008). Publisher Full Text