Generalized Differentiable E -Invex Functions and Their Applications in Optimization

Abstract

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

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Chandal Nahak

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

Received 9 December 2011; Accepted 4 September 2012

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.

1. Introduction

E - convex function was introduced by Youness [ 1] and revised by Yang [ 2]. Chen [ 3] introduced Semi- E -convex function and studied some of its properties. Syau and Lee [ 4] defined E -quasi-convex function, strictly E -quasi-convex function and studied some basic properties. Fulga and Preda [ 5] introduced the class of E -preinvex and E -prequasi-invex functions. All the above E -convex and generalized E -convex functions are defined without differentiability assumptions. Since last few decades, generalized convex functions like quasiconvex, pseudoconvex, invex, B -vex, (p ,r ) -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 E - convexity, we have introduced the concept of differentiable E -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 E -convex function are discussed below, which will be needed in the sequel. Throughout this paper, we consider functions E : ^{R n} [arrow right] ^{R n} , f :M [arrow right]R , and M are nonempty subset of ^{R n} .

Definition 1.1 (see [ 1]).

M is said to be E -convex set if (1 - λ )E (x ) + λE (y ) ∈M for x ,y ∈M , λ ∈ [0,1 ] .

Definition 1.2 (see [ 1]).

f :M [arrow right]R is said to be E -convex on M if M is an E -convex set and for all x ,y ∈M and λ ∈ [0,1 ] , [figure omitted; refer to PDF]

Definition 1.3 (see [ 3]).

Let M be an E -convex set. f is said to be semi- E -convex on M if for x ,y ∈M and λ ∈ [0,1 ] , [figure omitted; refer to PDF]

Definition 1.4 (see [ 5]).

M is said to be E -invex with respect to η : ^{R n} × ^{R n} [arrow right] ^{R n} if for x ,y ∈M and λ ∈ [0,1 ] , E (y ) + λ η (E (x ) ,E (y ) ) ∈M .

Definition 1.5 (see [ 6]).

Let M be an E -invex set with respect to η : ^{R n} × ^{R n} [arrow right] ^{R n} . Also f :M [arrow right]R is said to be E -preinvex with respect to η on M if for x ,y ∈M and λ ∈ [0,1 ] , [figure omitted; refer to PDF]

Definition 1.6 (see [ 7]).

Let M be an E -invex set with respect to η : ^{R n} × ^{R n} [arrow right] ^{R n} . Also f :M [arrow right]R is said to be semi- E -invex with respect to η at y ∈M if [figure omitted; refer to PDF] for all x ∈M and λ ∈ [0,1 ] .

Definition 1.7 (see [ 7]).

Let M be a nonempty E -invex subset of ^{R n} with respect to η : ^{R n} × ^{R n} [arrow right] ^{R n} , E : ^{R n} [arrow right] ^{R n} . Let f :M [arrow right]R and E (M ) be an open set in ^{R n} . Also f and E are differentiable on M . Then, f is said to be semi- E -quasiinvex at y ∈M if [figure omitted; refer to PDF] or [figure omitted; refer to PDF]

Lemma 1.8 (see [ 1]).

If a set M ⊆ ^{R n} is E -convex, then E (M ) ⊆M .

Lemma 1.9 (see [ 5]).

If M is E -invex, then E (M ) ⊆M .

Lemma 1.10 (see [ 5]).

If { _{M i } i ∈I} is a collection of E -invex sets and _{M i} ⊆ ^{R n} , for all i ∈I , then _{∩ i ∈I M i} is E -invex.

2. E -Convexity and Its Generalizations with Differentiability Assumption

E -convexity and convexity are different from each other in several contests. From the previous results on E - convex functions, as discussed by our predecessors, one can observe the following relations between E - convexity and convexity.

(1) All convex functions are E - convex but all E - convex functions are not necessarily convex. (In particular, E - convex function reduces to convex function in case E (x ) =x for all x in the domain of E .)

(2) A real-valued function on ^{R n} may not be convex on a subset of ^{R n} , but E - convex on that set.

(3) An E - convex function may not be convex on a set M but E - convex on E (M ) .

(4) It is not necessarily true that if M is an E - convex set then E (M ) is a convex set.

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

2.1. Some New Results on E -Convexity with Differentiability

E -convexity at a point may be interpreted as follows.

Let M be a nonempty subset of ^{R n} , E : ^{R n} [arrow right] ^{R n} . A function f :M [arrow right]R is said to be E -convex at x ¯ ∈M if M is an E -convex set and [figure omitted; refer to PDF] for all x ∈ _{N δ} ( x ¯ ) and λ ∈ [0,1 ] , where _{N δ} ( x ¯ ) is δ -neighborhood of x ¯ , for small δ >0 .

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

Example 2.1.

Consider M = { (x ,y ) ∈ ^{R 2} |" y ...5;0 } . E : ^{R 2} [arrow right] ^{R 2} is E (x ,y ) = (0 ,y ) and f (x ,y ) = ^{x 3} + ^{y 2} . Also f is not convex at ( -1,1 ) . For all (x ,y ) ∈ _{N δ} ( -1,1 ) , δ >0 , and λ ∈ [0,1 ] , f ( λE (x ,y ) + (1 - λ )E ( -1,1 ) ) - λ (f [composite function]E ) (x ,y ) - (1 - λ ) (f [composite function]E ) ( -1,1 ) = - λ (1 - λ ) (y -1 ^{) 2} ...4;0 . Hence, f is E -convex at ( -1,1 ) .

Proposition 2.2.

Let M ⊆ ^{R n} , E : ^{R n} [arrow right] ^{R n} be a homeomorphism. If f :M [arrow right]R attains a local minimum point in the neighborhood of E ( x ¯ ) , then it is E -convex at x ¯ .

Proof.

Suppose f has a local minimum point in a neighborhood _{N ...} (E ( x ¯ ) ) of E ( x ¯ ) for some x ¯ ∈M , ... >0 . This implies f is convex on _{N ...} (E ( x ¯ ) ) . That is, [figure omitted; refer to PDF] Since E : ^{R n} [arrow right] ^{R n} is a homeomorphism, so inverse of the neighborhood _{N ...} (E ( x ¯ ) ) is a neighborhood of x ¯ say _{N δ} ( x ¯ ) for some δ >0 . Hence, there exists x ∈ _{N δ} ( x ¯ ) such that E (x ) =z , E (x ) ∈ _{N ...} (E ( x ¯ ) ) . Replacing z by E (x ) in the above inequality, we conclude that f is E -convex at x ¯ .

In the above discussion, it is clear that if a local minimum exists in a neighborhood of E ( x - ) , then f is E -convex at x - . But it is not necessarily true that if f is E -convex at x - then E ( x - ) is local minimum point. Consider the above example where f is E -convex at ( -1,1 ) but E ( -1,1 ) is not local minimum point of f .

Theorem 2.3.

Let M be an open E -convex subset of ^{R n} , f and E are differentiable functions, and let E be a homeomorphism. Then, f is E -convex at x ¯ ∈M if and only if [figure omitted; refer to PDF] for all E (x ) ∈ _{N ...} (E ( x ¯ ) ) where _{N ...} (E ( x ¯ ) ) is ... -neighborhood of E ( x ¯ ) , ... >0 .

Proof.

Since M is an E -convex set, by Lemma 1.8, E (M ) ⊆M . Also, E (M ) is an open set as E is a homeomorphism. Hence, there exists ... >0 such that E (x ) ∈ _{N ...} (E ( x ¯ ) ) for all x ∈ _{N δ} ( x ¯ ) , δ >0 , very small. So, f is differentiable on E (M ) . Using expansion of f at E ( x ¯ ) in the neighborhood _{N ...} (E ( x ¯ ) ) , [figure omitted; refer to PDF] where z ∈ _{N ...} (E ( x ¯ ) ) and _{lim λ [arrow right]0} α [E ( x ¯ ) , λ (z -E ( x ¯ ) ) ] =0 . Since f is E -convex at x ¯ ∈M , so for all x ∈ _{N δ} ( x ¯ ) , λ ∈ (0,1 ] , x ...0; x ¯ , [figure omitted; refer to PDF] Since E is a homeomorphism, there exists x ∈ _{N δ} ( x ¯ ) such that E (x ) =z . Replacing z by E (x ) in ( 2.4) and using above inequality, we get [figure omitted; refer to PDF] where _{lim λ [arrow right]0} α [ E ( x ¯ ) , λ ( E ( x ) -E ( x ¯ ) ) ] =0 . Hence, ( 2.3) follows.

The converse part follows directly from ( 2.4).

It is obvious that if E ( x ¯ ) is a local minimum point of f , then ∇ (f [composite function]E ) ( x ¯ ) =0 . 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 M ⊆ ^{R n} be an open E -convex set, and let f be a differentiable E -convex function at x ¯ . If E : ^{R n} [arrow right] ^{R n} is a homeomorphism and ∇ (f [composite function]E ) ( x ¯ ) =0 , then E ( x ¯ ) is the local minimum of f .

2.2. Some New Results on Generalized E -Convexity with Differentiability

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

2.2.1. Semi- E -Convex Function

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

f :M [arrow right]R is said to be semi- E -convex at x ¯ ∈M if M is an E -convex set and [figure omitted; refer to PDF] for all x ∈ _{N δ} ( x ¯ ) and λ ∈ [0,1 ] , where _{N δ} ( x ¯ ) is δ -neighborhood of x ¯ .

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

Theorem 2.5.

Suppose f :M [arrow right]R and E : ^{R n} [arrow right] ^{R n} are differentiable functions. Let E be a homeomorphism and let x ¯ be a fixed point of E . Then, f is semi- E -convex at x ¯ ∈M if and only if [figure omitted; refer to PDF] for all E (x ) ∈ _{N ...} (E ( x ¯ ) ) , very small ... >0 .

Proof.

Proceeding as in Theorem 2.3, we get the following relation from the expansion of f at E ( x ¯ ) in the neighborhood _{N ...} (E ( x ¯ ) ) , where x - is the fixed point of E . (Since E is a homeomorphism, there exists ... >0 such that E (x ) ∈ _{N ...} (E ( x ¯ ) ) for all x ∈ _{N δ} ( x ¯ ) , very small δ >0 ): [figure omitted; refer to PDF] where E (x ) ∈ _{N ...} (E ( x ¯ ) ) , _{lim λ [arrow right]0} α [E ( x ¯ ) , λ (E (x ) -E ( x ¯ ) ) ] =0 . Since f is semi- E -convex at x ¯ ∈M , and x - is a fixed point of E , so, for all x ∈ _{N δ} ( x ¯ ) , λ ∈ (0,1 ] , x ...0; x ¯ , [figure omitted; refer to PDF] Using ( 2.9), the above inequality reduces to [figure omitted; refer to PDF] where _{lim λ [arrow right]0} α [E ( x ¯ ) , λ (E (x ) -E ( x ¯ ) ) ] =0 . Hence Inequality ( 2.8) follows for all E (x ) ∈ _{N ...} (E ( x ¯ ) ) .

Conversely, suppose Inequality ( 2.8) holds at the fixed point x ¯ of E for all E (x ) ∈ _{N ...} (E ( x ¯ ) ) . Using ( 2.9) and E ( x ¯ ) = x ¯ in ( 2.8), we can conclude that f is semi- E -convex at x ¯ ∈M .

2.2.2. Generalized E -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 E -invex set, E -preinvex function, and E -prequasiinvex function where differentiability is not required (Section 1). Chen [ 3] introduced semi- E -convex, semi- E -quasiconvex, and semi- E -pseudoconvex functions without differentiability assumption. Jaiswal and Panda [ 7] studied some generalized E -invex functions and applied these concepts to study primal dual relations. Here, we define some more generalized E -invex functions with and without differentiability assumption, which will be needed in next section. First, we see the following lemma.

Lemma 2.6.

Let M be a nonempty E -invex subset of ^{R n} with respect to η : ^{R n} × ^{R n} [arrow right] ^{R n} . Also f :M [arrow right]R are differentiable on M . E (M ) is an open set in ^{R n} . If f is E -preinvex on M then (f [composite function]E ) (x ) ...5; (f [composite function]E ) (y ) + ( ∇ ( f [composite function]E ) ( y ) ^{) T} η ( E ( x ) ,E ( y ) ) for all x ,y ∈M .

Proof.

If E (M ) is an open set, f and E are differentiable on M , then f [composite function]E is differentiable on M . From Taylor's expansion of f at E (y ) for some y ∈M and λ >0 , [figure omitted; refer to PDF] where E (x ) ...0;E (y ) , _{lim λ [arrow right]0} α ( E ( y ) , λ η ( E ( x ) ,E ( y ) ) ) =0 .

If f is E -preinvex on M with respect to η (Definition 1.5), then as λ [arrow right] ^{0 +} , the above inequality reduces to (f [composite function]E ) (x ) ...5; (f [composite function]E ) (y ) + ( ∇ (f [composite function]E ) (y ) ^{) T} η (E (x ) ,E (y ) ) for all x ,y ∈M .

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

Definition 2.7.

Let M be a nonempty E -invex subset of ^{R n} with respect to η : ^{R n} × ^{R n} [arrow right] ^{R n} . Also f :M [arrow right]R are differentiable on M . E (M ) is an open set in ^{R n} . Then, f is E -invex on M if (f [composite function]E ) (x ) ...5; (f [composite function]E ) (y ) + ( ∇ (f [composite function]E ) (y ) ^{) T} η (E (x ) ,E (y ) ) for all x ,y ∈M .

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

Example 2.8.

M = { (x ,y ) ∈ ^{R 2} |" x ,y >0 } , E : ^{R 2} [arrow right] ^{R 2} is E (x ,y ) = (0 ,y ) and f :M [arrow right]R is defined by f (x ,y ) = - ^{x 2} - ^{y 2} , and [figure omitted; refer to PDF]

Definition 2.9.

Let M be a nonempty E -invex subset of ^{R n} with respect to η : ^{R n} × ^{R n} [arrow right] ^{R n} , let E (M ) be an open set in ^{R n} . Suppose f and E are differentiable on M . Then, f is said to be E -quasiinvex on M if [figure omitted; refer to PDF] or [figure omitted; refer to PDF]

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

Example 2.10.

Consider M = { (x ,y ) ∈ ^{R 2} |" x ,y <0 } , E : ^{R 2} [arrow right] ^{R 2} is E (x ,y ) = (0 ,y ) , and f :M [arrow right]R is f (x ,y ) = ^{x 3} + ^{y 3} , η : ^{R 2} × ^{R 2} [arrow right] ^{R 2} is η ( ( _{x 1} , _{y 1} ) , ( _{x 2} , _{y 2} ) ) = ( _{x 1} - _{x 2} , _{y 1} - _{y 2} ) . Now for all ( _{x 1} , _{y 1} ) , ( _{x 2} , _{y 2} ) ∈M , (f [composite function]E ) ( _{x 1} , _{y 1} ) - (f [composite function]E ) ( _{x 2} , _{y 2} ) - ∇ (f [composite function]E ) ( _{x 2} , _{y 2} ^{) T} η (E ( _{x 1} , _{y 1} ) ,E ( _{x 2} , _{y 2} ) ) = ^{y 1 3} + ^{y 2 3} -3 ^{y 2 2} ( _{y 1} - _{y 2} ) , which is not always positive. Hence, f is not E -invex with respect to η on M .

If we assume that (f [composite function]E ) ( _{x 1} , _{y 1} ) ...4; (f [composite function]E ) ( _{x 2} , _{y 2} ) for all ( _{x 1} , _{y 1} ) , ( _{x 2} , _{y 2} ) ∈M , then ( ∇ (f [composite function]E ) ( _{x 2} , _{y 2} ) ^{) T} η (E ( _{x 1} , _{y 1} ) ,E ( _{x 2} , _{y 2} ) ) =3 ^{y 2 2} ( _{y 1} - _{y 2} ) ...4;0 . Hence, f is E -quasiinvex with respect to same η on M .

Definition 2.11.

Let M be a nonempty E -invex subset of ^{R n} with respect to η : ^{R n} × ^{R n} [arrow right] ^{R n} , let E (M ) be an open set in ^{R n} . Suppose f and E are differentiable on M . Then, f is said to be E -pseudoinvex on M if [figure omitted; refer to PDF] or [figure omitted; refer to PDF]

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

Example 2.12.

Consider M = { (x ,y ) ∈ ^{R 2} |" x ,y >0 } . E : ^{R 2} [arrow right] ^{R 2} is E (x ,y ) = (0 ,y ) and f :M [arrow right]R is f (x ,y ) = - ^{x 2} - ^{y 2} . For η : ^{R 2} × ^{R 2} [arrow right] ^{R 2} , defined by η ( ( _{x 1} , _{y 1} ) , ( _{x 2} , _{y 2} ) ) = ( _{x 1} - _{x 2} , _{y 1} - _{y 2} ) , and for all ( _{x 1} , _{y 1} ) , ( _{x 2} , _{y 2} ) ∈M , _{y 1} ...0; _{y 2} , (f [composite function]E ) ( _{x 1} , _{y 1} ) - (f [composite function]E ) ( _{x 2} , _{y 2} ) - ∇ (f [composite function]E ) ( _{x 2} , _{y 2} ^{) T} η (E ( _{x 1} , _{y 1} ) ,E ( _{x 2} , _{y 2} ) ) = - ( _{y 2} - _{y 1} ^{) 2} <0 . Hence, f is not E -invex with respect to η on M . If ∇ (f [composite function]E ) ( _{x 2} , _{y 2} ) ^{) T} η (E ( _{x 1} , _{y 1} ) ,E ( _{x 2} , _{y 2} ) ) ...5;0 , then (f [composite function]E ) ( _{x 1} , _{y 1} ) - (f [composite function]E ) ( _{x 2} , _{y 2} ) -f (0 , _{y 1} ) -f (0 , _{y 2} ) = ( _{y 2} + _{y 1} ) ( _{y 2} - _{y 1} ) ...5;0 . Hence, f is E -pseudoinvex with respect to η on M .

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

Proposition 2.13.

If f :M [arrow right]R is semi- E -invex on an E -invex set M , then f (E (y ) ) ...4;f (y ) for each y ∈M .

Proof.

Since f is semi- E -invex on M ⊆ ^{R n} and M is an E -invex set so for x ,y ∈M and λ ∈ [0,1 ] , we have E (y ) + λ η (E (x ) ,E (y ) ) ∈M and f ( E ( y ) + λ η ( E ( x ) ,E ( y ) ) ) ...4; λf ( x ) + ( 1 - λ ) f ( y ) . In particular, for λ =0 , f (E (y ) ) ...4;f (y ) for each y ∈M .

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

Example 2.14.

Consider the previous example where M = { (x ,y ) ∈ ^{R 2} |" x ,y <0 } , E : ^{R 2} [arrow right] ^{R 2} is E (x ,y ) = (0 ,y ) and f :M [arrow right]R is f (x ,y ) = ^{x 3} + ^{y 3} , η : ^{R 2} × ^{R 2} [arrow right] ^{R 2} is η ( ( _{x 1} , _{y 1} ) , ( _{x 2} , _{y 2} ) ) = ( _{x 1} - _{x 2} , _{y 1} - _{y 2} ) . It is verified that f is E -invex with respect to η on M . But f (E (2,0 ) ) >f (2,0 ) . From Proposition 2.13it can be concluded that f is not semi- E -invex with respect to same η . Also, using Definition 1.6, for all ( _{x 1} , _{y 1} ) , ( _{x 2} , _{y 2} ) ∈M , λ ∈ [0,1 ] , f ( E ( _{x 2} , _{y 2} ) + λ η ( E ( _{x 1} , _{y 1} ) ,E ( _{x 2} , _{y 2} ) ) ) - λf ( _{x 1} , _{y 1} ) - (1 - λ )f ( _{x 2} , _{y 2} ) = - ( _{y 2} + λ ^{y 1 2} /2 _{y 2} ^{) 2} + λ ( ^{x 1 2} + ^{y 1 2} ) + (1 - λ ) ( ^{x 2 2} + ^{y 2 2} ) , which is not always negative. Hence, f is not semi- E -invex with respect to η on M .

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 [figure omitted; refer to PDF] where f :M [arrow right]R , _{g i} :M [arrow right] ^{R m} , M ⊆ ^{R n} ,g = ( _{g 1} , _{g 2} , ... , _{g m} ^{) T} . ^{M [variant prime]} = {x ∈M : _{g i} (x ) ...4;0 , i =1 , ... ,m } is the set of feasible solutions.

Theorem 3.1 (sufficient optimality condition).

Let M be a nonempty open E -convex subset of ^{R n} , f :M [arrow right]R , g :M [arrow right] ^{R m} , and E : ^{R n} [arrow right] ^{R n} are differentiable functions. Let E be a homeomorphism and let x ¯ be a fixed point of E . If f and g are semi- E -convex at x ¯ ∈ ^{M [variant prime]} and ( x ¯ , y ¯ ) ∈ ^{M [variant prime]} × ^{R m} satisfies [figure omitted; refer to PDF] then x ¯ is local optimal solution of (P ) .

Proof.

Since f and g are semi- E -convex at x ¯ ∈M by Theorem 2.5, [figure omitted; refer to PDF] Adding the above two inequalities, we have [figure omitted; refer to PDF]

If ( 3.2) hold, then f (x ) -f ( x ¯ ) + ^{y ¯ T} g (x ) ...5;0 for all E (x ) ∈ _{N ...} (E ( x ¯ ) ) . Since g (x ) ...4;0 for all x ∈ ^{M [variant prime]} and y ¯ ...5;0 , so ^{y ¯ T} g (x ) ...4;0 . Hence, f (x ) -f ( x ¯ ) ...5;0 for all E (x ) ∈ _{N ...} (E ( x ¯ ) ) . Since E is a homeomorphism, there exists δ >0 such that x ∈ _{N δ} ( x ¯ ) for all E (x ) ∈ _{N ...} (E ( x ¯ ) ) , which means f (x ) ...5;f ( x ¯ ) for all x ∈ _{N δ} ( x ¯ ) . Hence, x ¯ is a local optimal solution of (P ) .

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

Lemma 3.2.

Let M be a nonempty E -invex subset of ^{R n} with respect to some η : ^{R n} × ^{R n} [arrow right] ^{R n} . Let _{g i} :M [arrow right]R , i =1 , ... ,m be semi- E -quasiinvex functions with respect to η on M . Then, ^{M [variant prime]} is an E -invex set.

Proof.

Let _{M i} = {x ∈M : _{g i} (x ) ...4;0 } , i =1 , ... ,m . ^{M [variant prime]} = ^{∩ i =1 m} _{M i} and ^{M [variant prime]} ⊆M . Since _{g i} , i =1 , ... ,m are semi- E -quasiinvex function on M , so for all x ,y ∈ _{M i} and λ ∈ [0,1 ] , _{g i} (E (y ) + λ η (E (x ) ,E (y ) ) ) ...4;max { _{g i} (x ) , _{g i} (y ) } ...4;0 . Hence, E (y ) + λ η (E (x ) ,E (y ) ) ∈ _{M i} for all x ,y ∈ _{M i} . So _{M i} is E -invex with respect to same η . From Lemma 1.10, ^{M [variant prime]} = ^{∩ i =1 m} _{M i} is E -invex with respect to same η .

Corollary 3.3.

Let M be a nonempty E -invex subset of ^{R n} with respect to some η : ^{R n} × ^{R n} [arrow right] ^{R n} . Let _{g i} :M [arrow right]R , i =1 , ... ,m , be semi- E -quasiinvex functions with respect to η on M . If x is a feasible solution of (P ) , then E (x ) is also a feasible solution of (P ) .

Proof.

Since x is a feasible solution of (P ) , so x ∈ ^{M [variant prime]} [implies]E (x ) ∈E ( ^{M [variant prime]} ) . Since each _{g i} , i =1 , ... ,m is semi- E -quasiinvex function on M , from Lemma 3.2, ^{M [variant prime]} is an E -invex set. Also E ( ^{M [variant prime]} ) ⊆ ^{M [variant prime]} . Hence, E (x ) ∈ ^{M [variant prime]} . That is, E (x ) is a feasible solution of (P ) .

Theorem 3.4 (sufficient optimality condition).

Let M be a nonempty E -invex subset of ^{R n} with respect to η : ^{R n} × ^{R n} [arrow right] ^{R n} . Let E (M ) be an open set in ^{R n} . Suppose f : ^{R n} [arrow right]R , g : ^{R n} [arrow right] ^{R m} and E are differentiable functions on M . If f is E -pseudoinvex function with respect to η and for u ...5;0 , ^{u T} g is semi- E -quasiinvex function with respect to the same η at x ∈ ^{M [variant prime]} , where x is a fixed point of the map E and (x ,u ) ∈ ^{M [variant prime]} × ^{R m} , u ...5;0 satisfies the following system: [figure omitted; refer to PDF] then x is a local optimal solution of (P ) .

Proof.

Suppose (x ,u ) ∈ ^{M [variant prime]} × ^{R m} satisfies ( 3.5) and ( 3.6). For all y ∈ ^{M [variant prime]} , g (y ) ...4;0 . Also, u ...5;0 . Hence, ^{u T} g (y ) ...4;0 for all y ∈ ^{M [variant prime]} . From ( 3.6), Y9;u , (g [composite function]E ) (x ) YA; =0 , that is, ^{u T} g (E (x ) ) =0 . x is a fixed point of E that is E (x ) =x . So ^{u T} g (x ) =0 . Hence, [figure omitted; refer to PDF] Since u ...5;0 and ^{u T} g is semi- E -quasiinvex function with respect to η at x , so the above inequality implies [figure omitted; refer to PDF] From ( 3.5), ∇f (E (x ) ) = - ∇ ^{u T} g (E (x ) ) . Putting this value in the above inequality, we have ∇f ( E ( x ) ) η ( E ( x ) ,E ( y ) ) ...5;0 .

f is E -pseudoinvex at x with respect to η . Hence, ∇f (E (x ) ) η (E (x ) ,E (y ) ) ...5;0 implies [figure omitted; refer to PDF] Hence, x is the optimal solution (P ) on E ( ^{M [variant prime]} ) .

The following example justifies the above theorem.

Example 3.5.

Consider the optimization problem, [figure omitted; refer to PDF] where M = { (x ,y ) ∈ ^{R 2} |" x ,y >0 } . E : ^{R 2} [arrow right] ^{R 2} is E (x ,y ) = (0 ,y ) . This is not a convex programming problem. Consider η : ^{R 2} × ^{R 2} [arrow right] ^{R 2} defined by η ( ( _{x 1} , _{y 1} ) , ( _{x 2} , _{y 2} ) ) = ( _{x 1} - _{x 2} , _{y 1} - _{y 2} ) . Here, ^{M [variant prime]} = { (x ,y ) ∈M : ^{x 2} + ^{y 2} -4 ...4;0 } , and E ( ^{M [variant prime]} ) = { (0 ,y ) :y ...5;0 , ^{y 2} -4 ...4;0 } . The sufficient conditions (7-8) reduce to [figure omitted; refer to PDF] whose solution is y =2 ,u =1 and E (0,2 ) = (0,2 ) .

In Example 2.12, we have already proved that f (x ,y ) = - ^{x 2} - ^{y 2} is E -pseudoinvex function with respect to η . Using Definition 1.7, one can verify that ug (x ,y ) is semi- E -quasi-invex with respect to same η at ( 0,2 ) ∈ ^{M [variant prime]} , where ug (x ,y ) = ^{x 2} + ^{y 2} -4 . So (0,2 ) is the optimal solution of (P ) on E ( ^{M [variant prime]} ) .

4. Conclusion

E -convexity and its generalizations are studied by many authors earlier without differentiability assumption. Here, we have studied the the properties of E -convexity, E -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 E -convex at that point with a suitable mapping E , and if a local minimum of f exists in a neighborhood of E (x ) , then f is E -convex at x . But it is not necessarily true that if f is E -convex at x then E (x ) is local minimum point.

(2) From the relation between E -invexity and its generalizations, one may observe that a function which is not E -convex may be E -invex with respect to some η and E - preinvexity with differentiability is a sufficient condition for E - invexity. Moreover, a function may not be E -invex with respect to some η but E -quasi-invex with respect to same η , a function may not E -invex with respect to some η but E -pseudoinvex with respect to the same η and an E -invex function with respect to some η may not be semi- E -invex with respect to same η .

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

References

[1] E. A. Youness, " E -convex sets, E -convex functions, and E -convex programming," Journal of Optimization Theory and Applications , vol. 102, no. 2, pp. 439-450, 1999.

[2] X. M. Yang, "On E -convex sets, E -convex functions, and E -convex programming," Journal of Optimization Theory and Applications , vol. 109, no. 3, pp. 699-704, 2001.

[3] X. Chen, "Some properties of semi- E -convex functions," Journal of Mathematical Analysis and Applications , vol. 275, no. 1, pp. 251-262, 2002.

[4] Y. R. Syau, E. S. Lee, "Some properties of E -convex functions," Applied Mathematics Letters , vol. 18, no. 9, pp. 1074-1080, 2005.

[5] C. Fulga, V. Preda, "Nonlinear programming with E -preinvex and local E -preinvex functions," European Journal of Operational Research , vol. 192, no. 3, pp. 737-743, 2009.

[6] D. I. Duca, L. Lupsa, "On the E -epigraph of an E -convex function," Journal of Optimization Theory and Applications , vol. 129, no. 2, pp. 341-348, 2006.

[7] S. Jaiswal, G. Panda, "Duality results using higher order generalised E -invex functions," International Journal of Computing Science and Mathematics , vol. 3, no. 3, pp. 288-297, 2010.

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