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.
Keywords:G-preinvex function; semistrictly G-preinvex function; optimality; multiobjective optimization
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  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  and Weir and Jeyakumar  introduced preinvex functions, and they also studied how and where preinvex functions can replace convex functions in an optimization problem. Then, Yang and Li  obtained some properties of a preinvex function in 2001. Yang and Li  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.  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  introduced the concept of G-invex functions and derived some optimality conditions for constrained optimization problems under the assumption of G-invexity. Antczak  introduced a class of G-preinvex functions, which is a generalization of G-invex , preinvex functions [6,8] and r-preinvex functions . Then, Luo and Wu  introduced the concept of semistrictly G-preinvex functions, which includes semistrictly preinvex functions  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].
Now we recall some definitions.
Definition 2.1 ()
Definition 2.2 ()
Definition 2.3 ()
Definition 2.4 ()
Definition 2.5 ()
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
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 , 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 ) with respect to η.
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.
In the sequel, we will use the following lemma.
Lemma 2.1 ()
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.
3 Some properties of semistrictly G-preinvex functions
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 , 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
This fact together with (3.1) yields
which contradicts (3.2).
which contradicts (3.2). This completes the proof. □
Remark 3.1 Theorem 3.1 generalizes Theorem 3.8  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.
Therefore, we obtain
This completes the proof. □
From Definitions 2.4 and 2.5, we can obtain the following lemma.
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
It follows that
From Lemma 3.1 and Theorem 3.2, we get
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.
4 Semistrict G-preinvexity and optimality
In the section, we consider a class of multiobjective optimization problems and obtain an important optimality result under semistrict G-preinvexity.
We consider the following multiobjective optimization problem:
Remark 4.1 If , then Definitions 4.1-4.2 reduce to Definitions 4.2-4.3 introduced by Bhatia , 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.
Consider the multiobjective optimization problem,
Theorem 4.1LetXbe an invex set with respect toη. Suppose the following conditions are satisfied:
where Lemma 2.2 is used in the second inequality.
According to (4.2), (4.3) and Condition C, we have
Now, we give an example of an optimization problem to illustrate Theorem 4.1.
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).
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