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Advances in Operations Research
Volume 2012 (2012), Article ID 175176, 11 pages
http://dx.doi.org/10.1155/2012/175176
Research Article

Generalized Differentiable -Invex Functions and Their Applications in Optimization

Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

Received 9 December 2011; Accepted 4 September 2012

Academic Editor: Chandal Nahak

Copyright © 2012 S. Jaiswal and G. Panda. 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

The concept of -convex function and its generalizations is studied with differentiability assumption. Generalized differentiable -convexity and generalized differentiable -invexity are used to derive the existence of optimal solution of a general optimization problem.

1. Introduction

convex function was introduced by Youness [1] and revised by Yang [2]. Chen [3] introduced Semi--convex function and studied some of its properties. Syau and Lee [4] defined -quasi-convex function, strictly -quasi-convex function and studied some basic properties. Fulga and Preda [5] introduced the class of -preinvex and -prequasi-invex functions. All the above -convex and generalized -convex functions are defined without differentiability assumptions. Since last few decades, generalized convex functions like quasiconvex, pseudoconvex, invex, -vex, -invex, and so forth, have been used in nonlinear programming to derive the sufficient optimality condition for the existence of local optimal point. Motivated by earlier works on convexity and convexity, we have introduced the concept of differentiable -convex function and its generalizations to derive sufficient optimality condition for the existence of local optimal solution of a nonlinear programming problem. Some preliminary definitions and results regarding -convex function are discussed below, which will be needed in the sequel. Throughout this paper, we consider functions , , and are nonempty subset of .

Definition 1.1 (see [1]). is said to be -convex set if for , .

Definition 1.2 (see [1]). is said to be -convex on if is an -convex set and for all and ,

Definition 1.3 (see [3]). Let be an -convex set. is said to be semi--convex on if for and ,

Definition 1.4 (see [5]). is said to be -invex with respect to if for and ,  .

Definition 1.5 (see [6]). Let be an -invex set with respect to .  Also   is said to be -preinvex with respect to on if for and ,

Definition 1.6 (see [7]). Let be an -invex set with respect to . Also   is said to be semi--invex with respect to at if for all and .

Definition 1.7 (see [7]). Let be a nonempty -invex subset of with respect to , . Let and be an open set in  Also   and are differentiable on . Then, is said to be semi--quasiinvex at if or

Lemma 1.8 (see [1]). If a set is -convex, then .

Lemma 1.9 (see [5]). If is -invex, then .

Lemma 1.10 (see [5]). If is a collection of -invex sets and , for all , then is -invex.

2. -Convexity and Its Generalizations with Differentiability Assumption

-convexity and convexity are different from each other in several contests. From the previous results on convex functions, as discussed by our predecessors, one can observe the following relations between convexity and convexity. (1)All convex functions are convex but all convex functions are not necessarily convex. (In particular, convex function reduces to convex function in case for all x in the domain of .) (2)A real-valued function on may not be convex on a subset of , but convex on that set. (3)An convex function may not be convex on a set but convex on . (4)It is not necessarily true that if is an convex set then is a convex set.

In this section we study -convex and generalized -convex functions with differentiability assumption.

2.1. Some New Results on -Convexity with Differentiability

-convexity at a point may be interpreted as follows.

Let be a nonempty subset of , . A function is said to be -convex at if is an -convex set and for all and , where is -neighborhood of , for small .

It may be observed that a function may not be convex at a point but -convex at that point with a suitable mapping .

Example 2.1. Consider . is and . Also is not convex at . For all ,   , and . Hence, is -convex at .

Proposition 2.2. Let , be a homeomorphism. If attains a local minimum point in the neighborhood of , then it is -convex at .

Proof. Suppose has a local minimum point in a neighborhood of for some ,  . This implies is convex on . That is, Since is a homeomorphism, so inverse of the neighborhood is a neighborhood of say for some . Hence, there exists such that ,  . Replacing by in the above inequality, we conclude that is -convex at .

In the above discussion, it is clear that if a local minimum exists in a neighborhood of , then is -convex at . But it is not necessarily true that if is -convex at then is local minimum point. Consider the above example where is -convex at but is not local minimum point of .

Theorem 2.3. Let be an open -convex subset of ,   and are differentiable functions, and let be a homeomorphism. Then, is -convex at if and only if for all where is -neighborhood of ,  .

Proof. Since is an -convex set, by Lemma 1.8, . Also, is an open set as is a homeomorphism. Hence, there exists such that for all , , very small. So, is differentiable on . Using expansion of at in the neighborhood , where and . Since is -convex at , so for all , , , Since is a homeomorphism, there exists such that . Replacing by in (2.4) and using above inequality, we get where . Hence, (2.3) follows.
The converse part follows directly from (2.4).

It is obvious that if is a local minimum point of , then . The following result proves the sufficient part for the existence of local optimal solution, proof of which is easy and straightforward. We leave this to the reader.

Corollary 2.4. Let be an open -convex set, and let be a differentiable -convex function at . If is a homeomorphism and , then is the local minimum of .

2.2. Some New Results on Generalized -Convexity with Differentiability

Here, we introduce some generalizations of -convex function like semi--convex, -invex, semi--invex, -pseudoinvex, -quasi-invex and so forth, with differentiability assumption and discuss their properties.

2.2.1. Semi--Convex Function

Chen [3] introduced a new class of semi--convex functions without differentiability assumption. Semi--convexity at a point may be understood as follows:

is said to be semi--convex at if is an -convex set and for all and , where is -neighborhood of .

The following result proves the necessary and sufficient condition for the existence of a semi--convex function at a point.

Theorem 2.5. Suppose and are differentiable functions. Let be a homeomorphism and let be a fixed point of . Then, is semi--convex at if and only if for all , very small .

Proof. Proceeding as in Theorem 2.3, we get the following relation from the expansion of at in the neighborhood , where is the fixed point of . (Since is a homeomorphism, there exists such that for all , very small ): where ,  . Since is semi--convex at , and is a fixed point of , so, for all , , , Using (2.9), the above inequality reduces to where . Hence Inequality (2.8) follows for all .
Conversely, suppose Inequality (2.8) holds at the fixed point of for all . Using (2.9) and in (2.8), we can conclude that is semi--convex at .

2.2.2. Generalized -Invex Function

The class of preinvex functions defined by Ben-Israel and Mond is not necessarily differentiable. Preinvexity, for the differential case, is a sufficient condition for invexity. Indeed, the converse is not generally true. Fulga and Preda [5] defined -invex set, -preinvex function, and -prequasiinvex function where differentiability is not required (Section 1). Chen [3] introduced semi--convex, semi--quasiconvex, and semi--pseudoconvex functions without differentiability assumption. Jaiswal and Panda [7] studied some generalized -invex functions and applied these concepts to study primal dual relations. Here, we define some more generalized -invex functions with and without differentiability assumption, which will be needed in next section. First, we see the following lemma.

Lemma 2.6. Let be a nonempty -invex subset of with respect to .   Also are differentiable on . is an open set in . If is -preinvex on then .

Proof. If is an open set, and are differentiable on , then is differentiable on . From Taylor's expansion of at for some and , where ,  .
If is -preinvex on with respect to (Definition 1.5), then as , the above inequality reduces to for all .

As a consequence of the above lemma, we may define -invexity with differentiability assumption as follows.

Definition 2.7. Let be a nonempty -invex subset of with respect to . Also are differentiable on . is an open set in . Then, is -invex on if for all .

From the above discussions on -invexity and -preinvexity, it is true that preinvexity with differentiability is a sufficient condition for invexity. Also a function which is not -convex may be -invex with respect to some . This may be verified in the following example.

Example 2.8. , is and is defined by , and

Definition 2.9. Let be a nonempty -invex subset of with respect to , let be an open set in . Suppose and are differentiable on . Then, is said to be -quasiinvex on if or

A function may not be -invex with respect to some but -quasiinvex with respect to same . This may be justified in the following example.

Example 2.10. Consider , is , and is ,   is . Now for all , = , which is not always positive. Hence, is not -invex with respect to on .

If we assume that for all , then . Hence, is -quasiinvex with respect to same on .

Definition 2.11. Let be a nonempty -invex subset of with respect to , let be an open set in . Suppose and are differentiable on . Then, is said to be -pseudoinvex on if or

A function may not be -invex with respect to some but -pseudoinvex with respect to same . This can be verified in the following example.

Example 2.12. Consider . is and is . For , defined by , and for all , , . Hence, is not -invex with respect to on . If , then . Hence, is -pseudoinvex with respect to on .

If a function is semi--invex with respect to at each point of an -invex set , then is said to be semi--invex with respect to on . Semi--invex functions and some of its generalizations are studied in [7]. Here, we discuss some more results on generalized semi--invex functions.

Proposition 2.13. If is semi--invex on an -invex set , then for each .

Proof. Since is semi--invex on and is an -invex set so for and , we have and . In particular, for ,  .

An -invex function with respect to some may not be semi--invex with respect to same may be verified in the following example.

Example 2.14. Consider the previous example where , is and is ,   is . It is verified that is -invex with respect to on . But . From Proposition 2.13 it can be concluded that is not semi--invex with respect to same . Also, using Definition 1.6, for all ,  ,, which is not always negative. Hence, is not semi--invex with respect to on .

3. Application in Optimization Problem

In this section, the results of previous section are used to derive the sufficient optimality condition for the existence of solution of a general nonlinear programming problem. Consider a nonlinear programming problem where , , .   is the set of feasible solutions.

Theorem 3.1 (sufficient optimality condition). Let be a nonempty open -convex subset of ,,  , and are differentiable functions. Let be a homeomorphism and let be a fixed point of . If and are semi--convex at and satisfies then is local optimal solution of .

Proof. Since and are semi--convex at by Theorem 2.5, Adding the above two inequalities, we have
If (3.2) hold, then . Since for all and , so . Hence, for all . Since is a homeomorphism, there exists such that for all , which means . Hence, is a local optimal solution of .

Also we see that a fixed point of is a local optimal solution of under generalized -invexity assumptions.

Lemma 3.2. Let be a nonempty -invex subset of with respect to some . Let , be semi--quasiinvex functions with respect to on . Then, is an -invex set.

Proof. Let .   and . Since , are semi--quasiinvex function on , so for all and ,. Hence, for all . So is -invex with respect to same . From Lemma 1.10, is -invex with respect to same .

Corollary 3.3. Let be a nonempty -invex subset of with respect to some . Let , be semi--quasiinvex functions with respect to on . If is a feasible solution of , then is also a feasible solution of .

Proof. Since is a feasible solution of , so . Since each , is semi--quasiinvex function on , from Lemma 3.2, is an -invex set. Also . Hence, . That is, is a feasible solution of .

Theorem 3.4 (sufficient optimality condition). Let be a nonempty -invex subset of with respect to . Let be an open set in . Suppose , and are differentiable functions on . If is -pseudoinvex function with respect to and for is semi--quasiinvex function with respect to the same at , where is a fixed point of the map and satisfies the following system: then is a local optimal solution of .

Proof. Suppose satisfies (3.5) and (3.6). For all ,  . Also, . Hence, for all . From (3.6), , that is, .   is a fixed point of that is . So . Hence, Since and is semi--quasiinvex function with respect to at , so the above inequality implies From (3.5), . Putting this value in the above inequality, we have .
is -pseudoinvex at with respect to . Hence, implies Hence, is the optimal solution on .

The following example justifies the above theorem.

Example 3.5. Consider the optimization problem, where . is . This is not a convex programming problem. Consider defined by . Here, , and . The sufficient conditions (7-8) reduce to whose solution is and .
In Example 2.12, we have already proved that is -pseudoinvex function with respect to . Using Definition 1.7, one can verify that is semi--quasi-invex with respect to same at (, where . So is the optimal solution of on .

4. Conclusion

-convexity and its generalizations are studied by many authors earlier without differentiability assumption. Here, we have studied the the properties of -convexity, -invexity, and their generalizations with differentiable assumption. From the developments of this paper, we conclude the following interesting properties. (1)A function may not be convex at a point but -convex at that point with a suitable mapping , and if a local minimum of exists in a neighborhood of , then is -convex at . But it is not necessarily true that if is -convex at then is local minimum point. (2)From the relation between -invexity and its generalizations, one may observe that a function which is not -convex may be -invex with respect to some and preinvexity with differentiability is a sufficient condition for invexity. Moreover, a function may not be -invex with respect to some but -quasi-invex with respect to same , a function may not -invex with respect to some but -pseudoinvex with respect to the same and an -invex function with respect to some may not be semi--invex with respect to same .

Here, we have studied -convexity for first-order differentiable functions. Higher-order differentiable -convex functions may be studied in a similar manner to derive the necessary and sufficient optimality conditions for a general nonlinear programming problems.

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