Journal of Applied Mathematics

Volume 2012 (2012), Article ID 168172, 12 pages

doi:10.1155/2012/168172

Research Article

A Class of -Semipreinvex Functions and Optimality

Xue Wen Liu,¹ Ke Quan Zhao,¹ and Zhe Chen1,2

¹Department of Mathematics, Chongqing Normal University, Chongqing 400047, China

²School of Economics and Management, Tsinghua University, Beijing 100084, China

Received 31 May 2012; Accepted 27 October 2012

Academic Editor: Xue-Xiang Huang

Copyright © 2012 Xue Wen Liu 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 -semipreinvex functions, which are some generalizations of the semipreinvex functions, and the -convex functions, is introduced. Examples are given to show their relations among -semipreinvex functions, semipreinvex functions and -convex functions. Some characterizations of -semipreinvex functions are also obtained, and some optimality results are given for a class of -semipreinvex functions. Ours results improve and generalize some known results.

1. Introduction

Generalized convexity has been playing a central role in mathematical programming and optimization theory. The research on characterizations of generalized convexity is one of most important parts in mathematical programming and optimization theory. Many papers have been published to study the problems of how to weaken the convex condition to guarantee the optimality results. Schaible and Ziemba [1] introduced -convex function which is a generalization of convex function and studied some characterizations of -convex functions. Hanson [2] introduced invexity which is an extension of differentiable convex function. Ben-Israel and Mond [3] considered the functions for which there exists such that, for any ,  ,

Weir et al. [4, 5] named such kinds of functions which satisfied the condition (1.1) as preinvex functions with respect to . Further study on characterizations and generalizations of convexity and preinvexity, including their applications in mathematical programming, has been done by many authors (see [6–18]). As a generalization of preinvexity, Yang and Chen [15] introduced semipreinvex functions and discussed the applications in prevariational inequality. Yang et al. [16] investigated some properties of semipreinvex functions. As a generalization of -convex functions and preinvex functions, Antczak [17] introduced -preinvex functions and obtained some optimality results for a class of constrained optimization problems. As a generalization of -vexity and semipreinvexity, Long and Peng [18] introduced the concept of semi--preinvex functions. Zhao et al. [19] introduced -semipreinvex functions and established some optimality results for a class of nonlinear programming problems.

Motivated by the results in [17–19], in this paper, we propose the concept of -semipreinvex functions and obtain some important characterizations of -semipreinvexity. At the same time, we study some optimality results under -semipreinvexity. Our results unify the concepts of -convexity, preinvexity, -preinvexity, semipreinvexity, and -semipreinvexity.

2. Preliminaries and Definitions

Definition 2.1 (see [1]). Let be a continuous real-valued strictly monotonic function defined on . A real-valued function defined on a convex set is said to be -convex if for any ,  ,

where is the inverse of , .

Remark 2.2. Every convex functions is -convex, but the converse is not necessarily true.

Example 2.3. Let ,  , be the range of real-valued function , and let be defined by

Then, we can verify that is a -convex function. But is not a convex function because the following inequality

holds for ,  ,  and  .

Weir et al. [4, 5] presented the concepts of invex sets and preinvex functions as follows.

Definition 2.4 (see [4, 5]). A set is said to be invex if there exists a vector-valued function such that for any ,  ,

Definition 2.5 (see [4, 5]). Let be invex with respect to vector-valued function . Function is said to be preinvex with respect to if for any ,  ,

Remark 2.6. Every convex function is a preinvex function with respect to , but the converse is not necessarily true.

Example 2.7. Let . be defined by

Then, we can verify that is a preinvex function with respect to , where

But is not convex a function in Example 2.3.

Antczak [17] introduced the concept of -preinvex functions as follows.

Definition 2.8 (see [17]). Let be a nonempty invex (with respect to ) subset of . A function is said to be (strictly) -preinvex at with respect to if there exists a continuous real-valued increasing function such that for all , ,

If (2.8) is satisfied for any , then is said to be (strictly) a -preinvex function on with respect to .

Remark 2.9. Every preinvex function with respect to is -preinvex function with respect to the same , where . Every -convex function is -preinvex function with respect to . However, the converse is not necessarily true.

Example 2.10. Let . , be defined by

Then, we can verify that is a -preinvex function with respect to , where

But is not a preinvex function because the following inequality

holds for ,  , and  .

And is not a -convex function because the following inequality

holds for ,  , and  .

Definition 2.11 (see [15]). A set is said to be a semiconnected set if there exists a vector-valued function such that for any ,  ,

Definition 2.12 (see [15]). Let be a semiconnected set with respect to a vector-valued function . Function is said to be semipreinvex with respect to if for any ,  , ,

Next we present the definition of -semipreinvex functions as follows.

Definition 2.13. Let be semiconnected set with respect to vector-valued function . A function is said to be (strictly) -semipreinvex at with respect to if there exists a continuous real-valued strictly increasing function such that for all , , ,

If (2.15) is satisfied for any , then is said to be (strictly) -semipreinvex on with respect to .

Remark 2.14. Every semipreinvex function with respect to is a -semipreinvex function with respect to the same , where . However, the converse is not true.

Example 2.15. Let . Then is a semiconnected set with respect to and , where

Let , be defined by

Then, we can verify that is a -semipreinvex function with respect to . But is not a semipreinvex function with respect to because the following inequality

holds for , and  ,  .

Example 2.16. Let . Then is a semiconnected set with respect to and , where

Let ,   be defined by

Then, we can verify that is a -semipreinvex function with respect to classes of functions . But is not semipreinvex function with respect to because the following inequality

holds for ,  ,  and  .

Remark 2.17. Every a -convex function is -semipreinvex function with respect to . But the converse is not true.

Example 2.18. Let , it is easy to check that is a semiconnected set with respect to and , where

Let ,   be defined by

Then, we can verify that is a -semipreinvex function with respect to . But is not a -convex function, because the following inequality

holds for  ,  ,  and  .

3. Some Properties of -Semipreinvex Functions

In this section, we give some basic characterizations of -semipreinvex functions.

Theorem 3.1. Let be a -semipreinvex function with respect to on a nonempty semiconnected set with respect to , and let be a continuous strictly increasing function on . If the function is convex on the image under of the range of , then is also -semipreinvex function on with respect to the same function .

Proof. Let be a nonempty semiconnected subset of with respect to , and we assume that is -semipreinvex with respect to . Then, for any ,  ,

Let be a continuous strictly increasing function on . Then,

By the convexity of , it follows the following inequality

for all ,  . Therefore,

Thus, we have

Theorem 3.2. Let be a -semipreinvex function with respect to on a nonempty semiconnected set with respect to . If the function is concave on , then is semipreinvex function with respect to the same function .

Proof. Let , from the assumption is concave on , we have

Let

then

It follows that

Then,

This means that is convex. Let ,  , then is convex. Hence by Theorem 3.1, is -semipreinvex with respect to . But is the identity function; hence, is a semipreinvex function with respect to the same function .

Theorem 3.3. Let be a nonempty semiconnected set with respect to subset of and let , be finite collection of -semipreinvex function with respect to the same and on . Define , for every . Further, assume that for every , there exists , such that . Then is -semipreinvex function with respect to the same function .

Proof. Suppose that the result is not true, that is, is not -semipreinvex function with respect to on . Then, there exists ,   such that

We denote there exist , satisfying

Therefore, by (3.11),

By the condition, we obtain

From the definition of -semipreinvexity, is an increasing function on its domain. Then, is increasing. Since , then (3.14) gives

The inequality (3.15) above contradicts (3.13).

Theorem 3.4. Let be a -semipreinvex function with respect to on a nonempty semiconnected set with respect to . Then, the level set is a semiconnected set with respect to , for each .

Proof. Let , for any arbitrary real number . Then, . Hence, it follows that

Then, by the definition of level set we conclude that , for any , we conclude that is a semiconnected set with respect to .

Let is a -semipreinvex function with respect to , its epigraph is said to be -semiconnected set with respect to if for any ,  ,

Theorem 3.5. Let with respect to be a nonempty semiconnected set, and let be a real-valued function defined on . Then, is a -semipreinvex function with respect to if and only if its epigraph is a -semiconnected set with respect to .

Proof. Let , then . Thus, for any ,

By the definition of an epigraph of , this means that

Thus, we conclude that is a semiconnected set with respect to .

Conversely, let be a semiconnected set. Then, for any , we have ,  . By the definition of an epigraph of , the following inequality

holds for any . This implies that is a -semipreinvex function on with respect to .

The following results characterize the class of -semipreinvex functions.

Theorem 3.6. Let be a semiconnected set with respect to ; is a -semipreinvex function with respect to the same if and only if for all ,  , and ,

Proof. Let be -semipreinvex functions with respect to , and let . From the definition of -semipreinvexity, we have

Conversely, let . For any ,

By the assumption of theorem, we can get that for ,

Since is a continuous real-valued increasing function, and can be arbitrarily small, let , it follows that

4. -Semipreinvexity and Optimality

In this section, we will give some optimality results for a class of -semipreinvex functions.

Theorem 4.1. Let be a -semipreinvex function with respect to , and we assume that satisfies the following condition: , when . Then, each local minimum point of the function is its point of global minimum.

Proof. Assume that is a local minimum point of which is not a global minimum point. Hence, there exists a point such that . By the -semipreinvexity of with respect to , we have

Then, for ,

Thus, we have

This is a contradiction with the assumption.

Theorem 4.2. The set of points which are global minimum of is a semiconnected set with respect to .

Proof. Denote by the set of points of global minimum of , and let . Since is -semipreinvex with respect to , then

is satisfied. Since , we have

So, for any ,

Since are points of a global minimum of , it follows that, for any , the following relation:

is satisfied. Then, is a semiconnected set with respect to .

Acknowledgments

This work is supported by the National Science Foundation of China (Grants nos. 10831009, 11271391, and 11001289) and Research Grant of Chongqing Key Laboratory of Operations Research and System Engineering. The authors are thankful to Professor Xinmin Yang, Chongqing Normal University, for his valuable comments on the original version of this paper.

References

S. Schaible and W. T. Ziemba, Generalized Concavity in Optimization and Economics, Academic Press, London, UK, 1981.

M. A. Hanson, “On sufficiency of the Kuhn-Tucker conditions,” Journal of Mathematical Analysis and Applications, vol. 80, no. 2, pp. 545–550, 1981.

A. Ben-Israel and B. Mond, “What is invexity?” Bulletin of Australian Mathematical Society B, vol. 28, no. 1, pp. 1–9, 1986.

T. Weir and B. Mond, “Pre-invex functions in multiple objective optimization,” Journal of Mathematical Analysis and Applications, vol. 136, no. 1, pp. 29–38, 1988.

T. Weir and V. Jeyakumar, “A class of nonconvex functions and mathematical programming,” Bulletin of the Australian Mathematical Society, vol. 38, no. 2, pp. 177–189, 1988.

R. Pini, “Invexity and generalized convexity,” Optimization, vol. 22, no. 4, pp. 513–525, 1991.

D. Bhatia and P. Kumar, “Multiobjective control problem with generalized invexity,” Journal of Mathematical Analysis and Applications, vol. 189, no. 3, pp. 676–692, 1995.

S. R. Mohan and S. K. Neogy, “On invex sets and preinvex functions,” Journal of Mathematical Analysis and Applications, vol. 189, no. 3, pp. 901–908, 1995.

L. V. Reddy and R. N. Mukherjee, “Some results on mathematical programming with generalized ratio invexity,” Journal of Mathematical Analysis and Applications, vol. 240, no. 2, pp. 299–310, 1999.

X. M. Yang and D. Li, “On properties of preinvex functions,” Journal of Mathematical Analysis and Applications, vol. 256, no. 1, pp. 229–241, 2001.

D. S. Kim and S. Schaible, “Optimality and duality for invex nonsmooth multiobjective programming problems,” Optimization, vol. 53, no. 2, pp. 165–176, 2004.

T. Antczak, “-preinvexity and -invexity in mathematical programming,” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 551–566, 2005.

T. Antczak, “-invex sets and functions,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 355–379, 2001.

T. Antczak, “A class of --invex functions and mathematical programming,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 187–206, 2003.

X. Q. Yang and G. Y. Chen, “A class of nonconvex functions and pre-variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 169, no. 2, pp. 359–373, 1992.

X. M. Yang, X. Q. Yang, and K. L. Teo, “On properties of semipreinvex functions,” Bulletin of the Australian Mathematical Society, vol. 68, no. 3, pp. 449–459, 2003.

T. Antczak, “-pre-invex functions in mathematical programming,” Journal of Computational and Applied Mathematics, vol. 10, pp. 1–15, 2007.

X. J. Long and J. W. Peng, “Semi-B-preinvex functions,” Journal of Optimization Theory and Applications, vol. 131, no. 2, pp. 301–305, 2006.

K. Q. Zhao, X. W. Liu, and Z. Chen, “A class of -semipreinvex functions and optimality in nonlinear programming,” Journal of Global Optimization, vol. 49, no. 1, pp. 37–47, 2011.