Semistrict G-preinvexity and its application
© Peng; licensee Springer. 2012
Received: 4 April 2012
Accepted: 6 August 2012
Published: 7 September 2012
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  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].
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 ()
The set K is said to be invex with respect to η if K is invex at each .
Definition 2.2 ()
Remark 2.1 Any convex function is a preinvex function with . But the converse is not true.
Definition 2.3 ()
Remark 2.2 Any semistrictly (or strong) convex function is a semistrictly preinvex function with . But the converse is not true.
Definition 2.4 ()
is said to be strictly G-preinvex on K, if the inequality (2.1) is strict for all , and .
Definition 2.5 ()
Remark 2.3 It is clear that the semistrictly G-preinvex function is a generalization of semistrictly preinvex function.
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.
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 ()
is (strictly) increasing if and only if G is (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 2.2 If G is increasing and concave, then is convex.
3 Some properties of semistrictly G-preinvex functions
In this section, we derive some interesting properties of semistrictly G-preinvex functions.
Theorem 3.1 Let K be a nonempty invex set with respect to η, where η satisfies Condition C, and let () be a finite or infinite collection of both semistrictly G-preinvex and G-preinvex functions for the same η on K. Define , for every . Assume that for every , there exists an such that . Then f is both a semistrictly G-preinvex and G-preinvex function with respect to the same η on K.
- (i)If , then by the semistrict G-preinvexity of ,(3.3)
- (ii)If , then by the G-preinvexity of ,(3.4)
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.
This completes the proof. □
From Definitions 2.4 and 2.5, we can obtain the following lemma.
Lemma 3.1 Let K be a nonempty invex set with respect to η, where η satisfies Condition C. Let be continuous and semistrictly G-preinvex with respect to η on K, and satisfy (). Then f is a G-preinvex function on K.
- (ii)Since f is a semistrictly G-preinvex function on K, for with , it follows form the above results that(3.5)
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.
From now on, we suppose that is a vector-valued mapping, where X is an invex subset of endowed with the Euclidean norm .
for every ;
is the negation of .
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 .
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.
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 .
Let G be increasing and concave on ;
satisfies Condition C;
is a strictly local minimizer of order m for (MOP).
If , , are semistrictly G-preinvex on X with respect to η, then is a strictly global minimizer of order m for (MOP).
where Lemma 2.2 is used in the second inequality.
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.
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 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).
- Mangasarin OL: Nonlinear Programming. Mcgraw-Hill, New York; 1969.Google Scholar
- Bazaraa MS, Sherali HD, Shetty CM: Nonlinear Programming Theory and Algorithms. Wiley, New York; 1979.MATHGoogle Scholar
- Schaible S, Ziemba WT: Generalized Concavity in Optimization and Economics. Academic Press, London; 1981.MATHGoogle Scholar
- Ward DE: Characterizations of strict local minima and necessary conditions for weak sharp minima. J. Optim. Theory Appl. 1994, 80: 551–571. 10.1007/BF02207780MathSciNetView ArticleMATHGoogle Scholar
- Hanson MA: On sufficiency of Kuhn-Tucker conditions. J. Math. Anal. Appl. 1981, 80: 545–550. 10.1016/0022-247X(81)90123-2MathSciNetView ArticleMATHGoogle Scholar
- Weir T, Mond B: Pre-invex functions in multiple objective optimization. J. Optim. Theory Appl. 1988, 136: 29–38.MathSciNetMATHGoogle Scholar
- Weir T, Jeyakumar V: A class of nonconvex functions and mathematical programming. Bull. Aust. Math. Soc. 1988, 38: 177–189. 10.1017/S0004972700027441MathSciNetView ArticleMATHGoogle Scholar
- Yang XM, Li D: On properties of preinvex functions. J. Math. Anal. Appl. 2001, 256: 229–241. 10.1006/jmaa.2000.7310MathSciNetView ArticleMATHGoogle Scholar
- Yang XM, Li D: Semistrictly preinvex functions. J. Math. Anal. Appl. 2001, 258: 287–308. 10.1006/jmaa.2000.7382MathSciNetView ArticleMATHGoogle Scholar
- Yang XM, Yang XQ, Teo KL: Characterizations and applications of prequasi-invex functions. J. Optim. Theory Appl. 2001, 110: 645–668. 10.1023/A:1017544513305MathSciNetView ArticleMATHGoogle Scholar
- Peng JW, Yang XM: Two properties of strictly preinvex functions. Oper. Res. Trans. 2005, 9: 37–42.Google Scholar
- Avriel M, Diewert WE, Schaible S, Zang I: Generalized Concavity. Plenum, New York; 1975.MATHGoogle Scholar
- Antczak T: On G -invex multiobjective programming. I. Optimality. J. Glob. Optim. 2009, 43: 97–109. 10.1007/s10898-008-9299-5MathSciNetView ArticleMATHGoogle Scholar
- Antczak T: G -preinvex functions in mathematical programming. J. Comput. Appl. Math. 2008, 217: 212–226. 10.1016/j.cam.2007.06.026MathSciNetView ArticleMATHGoogle Scholar
- Antczak T: r -preinvexity and r -invexity in mathematical programming. Comput. Math. Appl. 2005, 50: 551–566. 10.1016/j.camwa.2005.01.024MathSciNetView ArticleMATHGoogle Scholar
- Luo HZ, Wu HX: On the relationships between G -preinvex functions and semistrictly G -preinvex functions. J. Comput. Appl. Math. 2008, 222: 372–380. 10.1016/j.cam.2007.11.006MathSciNetView ArticleMATHGoogle Scholar
- Bhatia G: Optimality and mixed saddle point criteria in multiobjective optimization. J. Math. Anal. Appl. 2008, 342: 135–145. 10.1016/j.jmaa.2007.11.042MathSciNetView ArticleMATHGoogle Scholar
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