`Journal of Applied MathematicsVolume 2012, Article ID 629194, 19 pageshttp://dx.doi.org/10.1155/2012/629194`
Research Article

## A Class of Semilocal ??-Preinvex Functions and Its Applications in Nonlinear Programming

1College of Science, Xidian University, Xi'an, Shanxi 710071, China
2College of Mathematics and Computer Science, Yangtze Normal University, Fuling, Chongqing 408100, China
3Department of Mathematics, China University of Petroleum, Qingdao, Shandong 266555, China

Received 20 April 2011; Accepted 5 December 2011

Copyright © 2012 Hehua Jiao 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 kind of generalized convex set, called as local star-shaped -invex set with respect to is presented, and some of its important characterizations are derived. Based on this concept, a new class of functions, named as semilocal -preinvex functions, which is a generalization of semi--preinvex functions and semilocal -convex functions, is introduced. Simultaneously, some of its basic properties are discussed. Furthermore, as its applications, some optimality conditions and duality results are established for a nonlinear programming.

#### 1. Introduction

It is well known that convexity and generalized convexity have been playing a key role in many aspects of optimization, such as duality theorems, optimality conditions, and convergence of optimization algorithms. Therefore, the research on characterizations and generalizations of convexity is one of the most important aspects in mathematical programming and optimization theory in [1, 2]. During the past several decades, many significant generalizations of convexity have been proposed.

In 1977, Ewing [3] presented a generalized convexity known as semilocal convexity, where the concept is applied to provide sufficient optimality conditions in variational and control problems. Generalizations of semilocal convex functions and their properties have been studied by Kaul and Kaur [4, 5] and Kaur [6]. In [7], optimality conditions and duality results were established for nonlinear programming involving semilocal preinvex and related functions. These results are extended in [8] for a multiple-objective programming problems. In [9, 10], Lyall et al. investigated the optimality conditions and duality results for fractional single- (multiple-) objective programming involving semilocal preinvex and related functions, respectively.

On the other hand, in 1999, Youness [11] introduced the concepts of -convex sets, -convex functions, and -convex programming, discussed some of their basic properties, and obtained some optimality results on -convex programming. In 2002, Chen [12] brought forward a class of semi--convex functions and also discussed its basic properties. In 2007, by combining the concept of semi--convexity and that of semilocal convexity, Hu et al. [13] put forward the concept of generalized convexity called as semilocal -convexity, studied some of its characterizations, and obtained some optimality conditions and duality results for semilocal -convex programming. In [14], optimality and duality were further studied for a fractional multiple-objective programming involving semilocal -convexity. In 2009, Fulga and Preda [15] extended the -convexity to -preinvexity and local -preinvexity and discussed some of their properties and an application. In 2011, Luo and Jian [16] introduced semi--preinvex maps in Banach spaces and studied some of their properties.

Motivated by research work of [1316] and references therein, in this paper, we present the concept of semilocal -preinvexity and discuss its some important properties. Furthermore, as applications of semilocal -preinvexity, we establish the optimality conditions and duality results for a nonlinear programming. The concept of semilocal -preinvexity unifies the concepts of semilocal -convexity and semi--preinvexity. Thus, we extend the work of [10, 12, 13] and generalize the results obtained in the literatures on this topic.

#### 2. Preliminaries

Throughout the paper, let denote the -dimensional Euclidean space, and let and be two fixed mappings. In this section, we review some related definitions and some results which will be used in this paper.

Definition 2.1 (see [11]). A set is said to be -convex if there is a map such that

Definition 2.2 (see [11]). A function is said to be -convex on a set if there is a map such that is an -convex set and

Definition 2.3 (see [12]). A function is said to be semi--convex on a set if there is a map such that is an -convex set and

Definition 2.4 (see [15]). A set is said to be -invex with respect to if

Definition 2.5 (see [15]). Let be an -invex set with respect to . A function is said to be -preinvex on with respect to if

Definition 2.6 (see [16]). Let be an -invex set with respect to . A function is said to be semi--preinvex on with respect to if

Definition 2.7 (see [17]). A set is said to be local star-shaped invex with respect to if for any , there is a maximal positive number satisfying

Definition 2.8 (see [13]). A set is said to be local star-shaped -convex if there is a map such that corresponding to each pair of points , and there is a maximal positive number satisfying

Definition 2.9 (see [13]). A function is said to be semilocal -convex on a local star-shaped -convex set if for each pair of (with a maximal positive number satisfying (2.8)), there exists a positive number satisfying

Definition 2.10 (see [18]). A vector function is said to be a convex-like function if for any and , there is such that where the inequalities are taken component wise.

Lemma 2.11 (see [19]). Let be a nonempty set in , and let be a convexlike function then either has a solution or for all and some , , and , but both alternatives are never true.

#### 3. Local Star-Shaped ??-Invex Set

In this section, we introduce the local star-shaped -invex set with respect to a given mapping and discuss some of its basic characterizations.

Definition 3.1. A set is said to be local star-shaped -invex with respect to a given mapping if there is a map such that corresponding to each pair of points , and there is a maximal positive number satisfying

Remark 3.2. Every -convex set is a local star-shaped -invex set with respect to , where , for all . Every local star-shaped -convex set is a local star-shaped -invex set with respect to , where , for all . Every -invex set with respect to is a local star-shaped -invex set with respect to , where , for all . But their converses are not necessarily true.

The following example shows that local star-shaped -invex set is more general than -convex set, -invex set, and local star-shaped -convex set.

Example 3.3. Let ,
We can testify that is a local star-shaped -invex set with respect to .
However, when , , there exists a such that??, namely, is not an -convex set.
Also, there is a such that , that is, is not an -invex set with respect to .
Similarly, for any positive number , there exists a sufficiently small positive number satisfying , that is, is not a local star-shaped -convex set.

Proposition 3.4. If a set is local star-shaped -invex with respect to , then .

Proof. Since is local star-shaped -invex, then for any , there exists a maximal positive number satisfying , for all .
Thus, for , .
Hence, .

Proposition 3.5. Let be local star-shaped invex with respect to , , then is local star-shaped -invex with respect to the same .

Proof. Assume that , then . Since is local star-shaped invex with respect to , thus for , there exists a positive number satisfying Hence, is local star-shaped -invex with respect to .

Remark 3.6. Every local star-shaped invex set with respect to is local star-shaped -invex set, where is an identity map, but its converse is not necessarily true. See the following example.

Example 3.7. Let , , and , where and . Let be defined as , and let be defined as
It is not difficult to prove that is a local star-shaped -invex set with respect to . However, by taking , , we know that there exists no maximal positive number such that , for all .
That is, is not a local star-shaped invex set with respect to .

Proposition 3.8. Let be a collection of local star-shaped -invex sets with the same map , then is a local star-shaped -invex set with respect to .

Proof. For all, we have .
Since are all local star-shaped -invex sets, then there exist positive numbers such that Taking , , we can get Therefore, the proposition is proved.

Remark 3.9. Even if , are all local star-shaped -invex set with respect to , is not necessarily a local star-shaped -invex set. See the following example.

Example 3.10. Let the map be defined as , and the map be defined as . Consider the two sets where , , and .
We can easily prove that the two sets , are all local star-shaped -invex sets with respect to . However, when , , there is not a positive number such that Thus, is not a local star-shaped -invex set with respect to .

Proposition 3.11. Let be a local star-shaped and -invex set with respect to the same , then is a local star-shaped and -invex set with respect to the same .

Proof. By contradiction, assume that for a pair of , for all , there exists a such that , that is, .
Since, from Proposition 3.4, , then contradicts the local star-shaped -invexity of .
Hence, is a local star-shaped -invex set.
Similarly, is a local star-shaped -invex set.

#### 4. Semilocal ??-Preinvex Functions

In the section, we present the concept of semilocal -preinvex function and study some of its properties. We first recall a relevant definition.

Definition 4.1 (see [15]). A function is said to be local -preinvex on with respect to if for any (with a maximal positive number satisfying (3.1)), there exists such that is a local star-shaped -invex set and

Definition 4.2. A function is said to be semilocal -preinvex on with respect to if for any (with a maximal positive number satisfying (3.1)), there exists such that is a local star-shaped -invex set and
If the inequality sign above is strict for any and , then is called a strict semilocal -preinvex function.
A vector function is said to be semilocal -preinvex on a local star-shaped -invex set with respect to if for each pair of points (with a maximal positive number satisfying (3.1)), there exists a positive number satisfying where the inequalities are taken component wise.
The definition of strict semilocal -preinvex of a vector function is similar to the one for a vector semilocal -preinvex function.

Remark 4.3. Every semilocal -convex function on a local star-shaped set is a semilocal -preinvex function, where , for all . Every semi--preinvex function with respect to is a semilocal -preinvex function, where , for all . But their converses are not necessarily true.

We give below an example of semilocal -preinvex function, which is neither a semilocal -convex function nor a semi--preinvex function.

Example 4.4. Let the map be defined as and the map be defined as Obviously, is a local star-shaped -convex set and a local star-shaped -invex set with respect to . Let be defined as
We can prove that is semilocal -preinvex on with respect to . However, when , , and for any , there exists a sufficiently small satisfying That is, is not a semilocal -convex function on .
Similarly, taking , , we have for some .
Thus, is not a semi--preinvex function on with respect to .

Theorem 4.5. Let be a local -preinvex function on a local star-shaped -invex set with respect to , then is a semilocal -preinvex function if and only if , for all .

Proof. Suppose that is a semilocal -preinvex function on set with respect to , then for each pair of points (with a maximal positive number satisfying (3.1)), there exists a positive number satisfying By letting , we have , for all .
Conversely, assume that is a local -preinvex function on a local star-shaped -invex set , then for any , there exist satisfying (3.1) and such that Since , for all , then The proof is completed.

Remark 4.6. A local -preinvex function on a local star-shaped -invex set with respect to is not necessarily a semilocal -preinvex function.

Example 4.7. Let , , and be the same as the ones of Example 3.3 and be defined by , then is local -preinvex on with respect to .
Since , from Theorem 4.5, it follows that is not a semilocal -preinvex function.

Theorem 4.8. Let be a semilocal -preinvex function on a local star-shaped -invex set with respect to , and let be a nondecreasing and convex function, then is semilocal -preinvex on with respect to .

The proof is easy and is omitted.

Theorem 4.9. If the functions are all semilocal -preinvex on a local star-shaped -invex set with respect to the same , then the function is semilocal -preinvex on with respect to for all , .

Proof. Since is a local star-shaped -invex set with respect to , then for all , there exists a positive number such that On the other hand, are all semilocal -preinvex on with respect to the same ; thus, there exist positive numbers such that Now, letting , we have That is, is semilocal -preinvex on with respect to .

Definition 4.10. The set is said to be a local star-shaped -invex set with respect to corresponding to if there are two maps , and a maximal positive number , for each such that

Theorem 4.11. Let be a local star-shaped -invex set with respect to , then is a semilocal -preinvex function on with respect to if and only if its epigraph is a local star-shaped -invex set with respect to corresponding to .

Proof. Assume that is semilocal -preinvex on with respect to and , then , and , . Since is a local star-shaped -invex set, there is a maximal positive number such that In addition, in view of being a semilocal -preinvex function on with respect to , there is a positive number such that That is, , for all .
Therefore, is a local star-shaped -invex set with respect to corresponding to .
Conversely, if is a local star-shaped -invex set with respect to corresponding to , then for any points , there is a maximal positive number such that That is, , Thus, is a local star-shaped -invex set, and is a semilocal -preinvex function on with respect to .

Theorem 4.12. If is a semilocal -preinvex function on a local star-shaped -invex set with respect to , then the level set is a local star-shaped -invex set for any .

Proof. For any and , then and , . Since is a local star-shaped -invex set, there is a maximal positive number such that In addition, due to the semilocal -preinvexity of , there is a positive number such that That is, , for all .
Therefore, is a local star-shaped -invex set with respect to for any .

Theorem 4.13. Let be a real-valued function defined on a local star-shaped -invex set , then is a semilocal -preinvex function with respect to if and only if for each pair of points (with a maximal positive number satisfying (3.1)), there exists a positive number such that whenever , .

Proof. Let and such that , . Due to the local star-shaped -invexity of , there is a maximal positive number such that In addition, owing to the semilocal -preinvexity of , there is a positive number such that
Conversely, let , (see epigraph in Theorem 4.11), then , , and . Hence, and hold for any . According to the hypothesis, for (with a positive number satisfying (3.1)), there exists a positive number such that Let , then That is, , for all .
Therefore, is a local star-shaped -invex set corresponding to .
From Theorem 4.11, it follows that is semilocal -preinvex on with respect to .

#### 5. Nonlinear Programming

In this section, we discuss the optimality conditions and Mond-Weir type duality for nonlinear programming involving semilocal -preinvex and related functions.

We firstly consider the nonlinear programming problem without constraint as follows: where is a local star-shaped -invex set and the objective function is a semilocal -preinvex function on with respect to .

Theorem 5.1. The following statements hold for programming .(i)The optimal solution set for is a local star-shaped -invex set with respect to .(ii)If is a local minimum for and , then is a global minimum for .(iii)If the real-valued function is a strict semilocal -preinvex function on with respect to , then the global optimal solution for is unique.

Proof. (i) Assume that , then and . On account of being a local star-shaped -invex set with respect to , there is a maximal positive number such that Besides, due to semilocal -preinvexity of , there is a positive number such that From the optimality of , we have .
Hence, , that is, , for all .
This shows that is a local star-shaped -invex set with respect to .
(ii) Assume that is a neighbourhood of with radius , and attains its local minimum at . For , there is a maximal positive number such that , for all .
Moreover, there is a positive number such that Owing to , so for sufficiently small , Thus, , namely, .
This means that is a global minimum for .
(iii) By contradiction, assume that are two global optimal solutions for and . For , there is a maximal positive number such that Furthermore, there is a positive number such that This contradicts the fact that is a global optimal solution for .
Therefore, the global optimal solution for is unique.

Theorem 5.2. Let and . If is differentiable on , then is a minimum for programming if and only if satisfies the inequality , for all .

Proof. Assume that is a minimum for . Since is a local star-shaped -invex set with respect to , for any , there is a maximal positive number such that From the differentiability of and , we get Dividing the inequality above by and letting , we have
Conversely, if , owing to semilocal -preinvexity of on , there is a positive number such that which together with implies Letting in the inequality above, we obtain This follows that is a minimum for .

Next, we consider the optimization problem with inequality constraint

Denote the feasible set of by , where , and is an open local star-shaped -invex set with respect to .

If the constraint functions () are all semilocal -preinvex on with respect to the same map , then, from Theorem 4.12 and Proposition 3.8, we can conclude that the feasible set is a local star-shaped -invex set with respect to . Moreover, from Theorem 5.1, we can obtain the following theorem easily.

Theorem 5.3. Assume that are all semilocal -preinvex on with respect to , then(i) is a local star-shaped -invex set with respect to ;(ii) the optimal solution set for is a local star-shaped -invex set with respect to ;(iii) if is a local minimum for and , then is a global minimum for ;(iv) if the real-valued function is a strict semilocal -preinvex function on with respect to , then the global optimal solution for is unique.

For convenience of discussion, we give the following notation: For , denote , .

To discuss the necessary optimality conditions for the corresponding programming, we first give a lemma as follows.

Lemma 5.4. Let be a local optimal solution for . Assume that is continuous at for any , and , possess the directional derivatives at along the direction for each , then the system: has no solution in , where denotes the directional derivative of at along the direction and .

The proof of this lemma is similar to the one of [10, Lemma 13].

Theorem 5.5. Let be a local optimal solution for . Assume that functions are continuous at for any , and , possess the directional derivatives with respect to at for each . If is a convex-like function and , then there are , such that Additionally, if is a semilocal -preinvex function on with respect to and there is such that , then there exists such that

Proof. Define vector function . Then is a convexlike function. By Lemma 5.4, the system has no solution in . Thus, from Lemma 2.11, there are such that Hence, by letting , we further have Subsequently, we testify that , and if this is not true, we get from above On account of being semilocal -preinvex on with respect to and , from Proposition 5.9(i) below, we have .
So . But this contradicts the fact that and .
Thus, . Dividing (5.17) and the first equality of (5.18) by and letting , we know that (5.19) and (5.20) hold.
Consequently, the whole proof is finished.

To discuss the sufficient optimality conditions for , we further generalize the concept of semilocal -preinvex function as follows.

Definition 5.6. A real-valued function defined on a local star-shaped -invex set is said to be quasisemilocal -preinvex (with respect to ) if for all (with a maximal positive number satisfying (3.1)) satisfying , there is a positive number such that
The definition of quasi-semilocal -preinvex of a vector function is similar to the one for a vector semilocal -preinvex function.

Definition 5.7. A real-valued function defined on a local star-shaped -invex set is said to be pseudosemilocal -preinvex (with respect to ) if for all (with a maximal positive number satisfying (3.1)) satisfying , there are a positive number and a positive number such that
The definition of pseudo-semilocal -preinvex of a vector function is similar to the one for a vector semilocal -preinvex function.

Remark 5.8. Every semilocal -preinvex function on a local star-shaped -invex set with respect to is both a quasi-semilocal -preinvex function and a pseudo-semilocal -preinvex function.

We now present one of their elementary properties.

Proposition 5.9. Let be a real-valued function on a local star-shaped -invex set , and possesses directional derivative with respect to the direction at for all . If , then the following statements hold true:(i) if is semilocal -preinvex on with respect to , then ,(ii)if is quasi-semilocal -preinvex on with respect to , then implies that ,(iii)if is pseudo-semilocal -preinvex on with respect to , then implies that .

The proof is obvious by using the related definitions and is omitted.

Theorem 5.10. Let and . Suppose that possess directional derivatives with respect to the direction at for any , and assume that there is such that (5.19) and (5.20) hold. If is pseudo-semilocal -preinvex on and is quasi-semilocal -preinvex on with respect to , then is an optimal solution for .

Proof. Due to , for all , it follows from Proposition 5.9(ii) that , for all , which together with implies Moreover, using , , and , we get Therefore, from (5.19), we have , for all .
Thus, from Proposition 5.9(iii), this implies , for all .
That is, is an optimal solution for .

Theorem 5.11. Let and . Suppose that possess directional derivatives with respect to the direction at for any , and there is a such that (5.19) and (5.20) hold. If is pseudo-semilocal -preinvex on with respect to , then is an optimal solution for .

Proof. Considering , , , and the given conditions, we have , for all . Hence, from Proposition 5.9(iii), we get The inequality above together with follows: On account of and , we obtain Therefore, is an optimal solution for .

The following conclusion is a direct corollary of Theorem 5.10 or Theorem 5.11.

Corollary 5.12. Let and . Suppose that possess directional derivatives with respect to the direction at for any , and assume that there is a such that (5.19) and (5.20) hold. If and are semilocal -preinvex functions on with respect to , then is an optimal solution for .

Finally, we consider the following Mond-Weir type dual problem of :

Theorem 5.13 (weak duality). Let and be arbitrary feasible solutions of and , respectively. If and are all semilocal -preinvex functions on with respect to , and they possess directional derivatives with respect to the direction at , where , , then .

Proof. Considering and being semilocal -preinvex on with respect to and , we get from Proposition 5.9(i) Combining the first constraint condition of and the inequalities above, we have Hence, on account of , and , we obtain .

Theorem 5.14 (strong duality). Assume that is an optimal solution for , , and for any feasible point of . Suppose that and are semilocal -preinvex on with respect to and is continuous at for any , and they possess directional derivatives with respect to the direction at and the direction at , respectively, where . Further, assume that there is such that . If is a convex-like function, then there is a such that is an optimal solution for .

Proof. From the assumptions and Theorem 5.5, we can conclude that there is such that is a feasible point for . Assume that is a feasible solution of . On account of and being semilocal -preinvex on with respect to and , we get from Proposition 5.9(i) Combining the first constraint condition of and the relationships above, we have Noticing that , , and , we know .
Therefore, is an optimal solution for .

Theorem 5.15 (converse duality). Suppose that and is a feasible point for . Further, suppose that and are semilocal -preinvex on with respect to , and , possess directional derivatives with respect to the direction at for any . If and , then is an optimal solution for .

Proof. Since and are semilocal -preinvex on with respect to and , we have from Proposition 5.9(i) On account of being a feasible point for , we get from the first constraint inequality of and the two relationships above