# Generalized DifferentiableE-Invex Functions and Their Applications in Optimization

Authors: S. Jaiswal; G. Panda
Journal: Advances in Operations Research
 (2012)

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## Abstract

The concept ofE-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.


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## Body

## 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:Rn→Rn, f:M→R, and M are nonempty subset of Rn.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
→
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],
(1.1)f((1-λ)E(x)+λE(y))≤(1-λ)f(E(x))+λf(E(y)).Definition 1.3 (see [3]).
LetM be an E-convex set. f is said to be semi-E-convex on M if for x,y∈M and λ∈[0,1],
(1.2)f(λE(x)+(1-λ)E(y))≤λf(x)+(1-λ)f(y).Definition 1.4 (see [5]).
M is said to be E-invex with respect to η:Rn×Rn→Rn if for x,y∈M and λ∈[0,1],  E(y)+λη(E(x),E(y))∈M.Definition 1.5 (see [6]).
LetM be an E-invex set with respect to η:Rn×Rn→Rn.  Also  f:M→R is said to be E-preinvex with respect to η on M if for x,y∈M and λ∈[0,1],
(1.3)f(E(y)+λη(E(x),E(y)))≤λf(E(x))+(1-λ)f(E(y)).Definition 1.6 (see [7]).
LetM be an E-invex set with respect to η:Rn×Rn→Rn. Also  f:M→R is said to be semi-E-invex with respect to η at y∈M if
(1.4)f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y)
for all x∈M and λ∈[0,1].Definition 1.7 (see [7]).
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn, E:Rn→Rn. Let f:M→R and E(M) be an open set in Rn. Also  f and E are differentiable on M. Then, f is said to be semi-E-quasiinvex at y∈M if
(1.5)f(x)≤f(y)∀x∈M⇒(∇(f∘E)(y))Tη(E(x),E(y))≤0,
or
(1.6)(∇(f∘E)(y))Tη(E(x),E(y))>0∀x∈M⇒f(x)>f(y).Lemma 1.8 (see [1]).
If a setM⊆Rn is E-convex, then E(M)⊆M.Lemma 1.9 (see [5]).
IfM is E-invex, then E(M)⊆M.Lemma 1.10 (see [5]).
If{Mi}i∈I is a collection of E-invex sets and Mi⊆Rn, for all i∈I, then ∩i∈IMi 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 areE-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 onRn may not be convex on a subset of Rn, but E-convex on that set.(3)
AnE-convex function may not be convex on a set M but E-convex on E(M).(4)
It is not necessarily true that ifM is an E-convex set then E(M) is a convex set.In this section we studyE-convex and generalized E-convex functions with differentiability assumption.


### 2.1. Some New Results onE-Convexity with Differentiability

E-convexity at a point may be interpreted as follows.LetM be a nonempty subset of Rn, E:Rn→Rn. A function f:M→R is said to be E-convex at x¯∈M if M is an E-convex set and
(2.1)f(λE(x)+(1-λ)E(x¯))≤λ(f∘E)(x)+(1-λ)(f∘E)(x¯)
for all x∈Nδ(x¯) and λ∈[0,1], where Nδ(x¯) is δ-neighborhood of x¯, for small δ>0.It may be observed that a function may not be convex at a point butE-convex at that point with a suitable mapping E.Example 2.1.
ConsiderM={(x,y)∈R2∣y≥0}. E:R2→R2 is E(x,y)=(0,y) and f(x,y)=x3+y2. Also f is not convex at (-1,1). For all (x,y)∈Nδ(-1,1),   δ>0, and λ∈[0,1],f(λE(x,y)+(1-λ)E(-1,1))-λ(f∘E)(x,y)-(1-λ)(f∘E)(-1,1)=-λ(1-λ)(y-1)2≤0. Hence, f is E-convex at (-1,1).Proposition 2.2.
LetM⊆Rn, E:Rn→Rn be a homeomorphism. If f:M→R attains a local minimum point in the neighborhood of E(x¯), then it is E-convex at x¯.Proof.
Supposef has a local minimum point in a neighborhood Nϵ(E(x¯)) of E(x¯) for some x¯∈M,  ϵ>0. This implies f is convex on Nϵ(E(x¯)). That is,
(2.2)f(λz+(1-λ)E(x¯))≤λf(z)+(1-λ)f(E(x¯))∀z∈Nϵ(E(x¯)).
Since E:Rn→Rn is a homeomorphism, so inverse of the neighborhood Nϵ(E(x¯)) is a neighborhood of x¯ say Nδ(x¯) for some δ>0. Hence, there exists x∈Nδ(x¯) such that E(x)=z,  E(x)∈Nϵ(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 ofE(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.
LetM be an open E-convex subset of Rn,  f and E are differentiable functions, and let E be a homeomorphism. Then, f is E-convex at x¯∈M if and only if
(2.3)(f∘E)(x)≥(f∘E)(x¯)+(∇(f∘E)(x¯))T(E(x)-E(x¯))
for all E(x)∈Nϵ(E(x¯)) where Nϵ(E(x¯)) is ϵ-neighborhood of E(x¯),  ϵ>0.Proof.
SinceM 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 ϵ>0 such that E(x)∈Nϵ(E(x¯)) for all x∈Nδ(x¯), δ>0, very small. So, f is differentiable on E(M). Using expansion of f at E(x¯) in the neighborhood Nϵ(E(x¯)),
(2.4)f(E(x¯)+λ(z-E(x¯)))=f(E(x¯))+λ∇f(E(x¯))T(z-E(x¯))+α[E(x¯),λ(z-E(x¯))]λ∥z-E(x¯)∥,
where z∈Nϵ(E(x¯)) and limλ→0α[E(x¯),λ(z-E(x¯))]=0. Since f is E-convex at x¯∈M, so for all x∈Nδ(x¯), λ∈(0,1], x≠x¯,
(2.5)λ((f∘E)(x)-(f∘E)(x¯))≥f(E(x¯)+λ(E(x)-E(x¯)))-(f∘E)(x¯).
Since E is a homeomorphism, there exists x∈Nδ(x¯) such that E(x)=z. Replacing z by E(x) in (2.4) and using above inequality, we get
(2.6)(f∘E)(x)-(f∘E)(x¯)≥(∇(f∘E)(x¯))T(E(x)-E(x¯)+α[E(x¯),λ(E(x)-E(x¯))]∥E(x)-E(x¯)∥),
where limλ→0α[E(x¯),λ(E(x)-E(x¯))]=0. Hence, (2.3) follows.
The converse part follows directly from (2.4).It is obvious that ifE(x¯) is a local minimum point of f, then ∇(f∘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.
LetM⊆Rn be an open E-convex set, and let f be a differentiable E-convex function at x¯. If E:Rn→Rn is a homeomorphism and ∇(f∘E)(x¯)=0, then E(x¯) is the local minimum of f.


### 2.2. Some New Results on GeneralizedE-Convexity with Differentiability

Here, we introduce some generalizations ofE-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
→
R is said to be semi-E-convex at x¯∈M if M is an E-convex set and
(2.7)f(λE(x)+(1-λ)E(x¯))≤λf(x)+(1-λ)f(x¯)
for all x∈Nδ(x¯) and λ∈[0,1], where Nδ(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.
Supposef:M→R and E:Rn→Rn 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
(2.8)f(x)≥f(x¯)+(∇(f∘E)(x¯))T(E(x)-E(x¯))
for all E(x)∈Nϵ(E(x¯)), very small ϵ>0.Proof.
Proceeding as in Theorem2.3, we get the following relation from the expansion of f at E(x¯) in the neighborhood Nϵ(E(x¯)), where x- is the fixed point of E. (Since E is a homeomorphism, there exists ϵ>0 such that E(x)∈Nϵ(E(x¯)) for all x∈Nδ(x¯), very small δ>0):
(2.9)f(E(x¯)+λ(E(x)-E(x¯)))=(f∘E)(x¯)+λ(∇(f∘E)(x¯))T(E(x)-E(x¯))+α[E(x¯),λ(E(x)-E(x¯))]λ∥E(x)-E(x¯)∥,
where E(x)∈Nϵ(E(x¯)),  limλ→0α[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∈Nδ(x¯), λ∈(0,1], x≠x¯,
(2.10)λ(f(x)-f(x¯))≥f(E(x¯)+λ(E(x)-E(x¯)))-(f∘E)(x¯).
Using (2.9), the above inequality reduces to
(2.11)f(x)-f(x¯)≥(∇(f∘E)(x¯))T(E(x)-E(x¯)+α[E(x¯),λ(E(x)-E(x¯))]∥E(x)-E(x¯)∥),
where limλ→0α[E(x¯),λ(E(x)-E(x¯))]=0. Hence Inequality (2.8) follows for all E(x)∈Nϵ(E(x¯)).
Conversely, suppose Inequality (2.8) holds at the fixed point x¯ of E for all E(x)∈Nϵ(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. GeneralizedE-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.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn.   Also f:M→R are differentiable on M. E(M) is an open set in Rn. If f is E-preinvex on M then (f∘E)(x)≥(f∘E)(y)+(∇(f∘E)(y))Tη(E(x),E(y))for allx,y∈M.Proof.
IfE(M) is an open set, f and E are differentiable on M, then f∘E is differentiable on M. From Taylor's expansion of f at E(y) for some y∈M and λ>0,
(2.12)f(E(y)+λη(E(x),E(y)))=(f∘E)(y)+λ(∇(f∘E)(y))T(η(E(x),E(y)))+λ∥η(E(x),E(y))∥α(E(y),λη(E(x),E(y))),
where E(x)≠E(y),  limλ→0α(E(y),λη(E(x),E(y)))=0.
Iff is E-preinvex on M with respect to η (Definition 1.5), then as λ→0+, the above inequality reduces to (f∘E)(x)≥(f∘E)(y)+(∇(f∘E)(y))Tη(E(x),E(y)) for all x,y∈M.As a consequence of the above lemma, we may defineE-invexity with differentiability assumption as follows.Definition 2.7.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn. Also f:M→R are differentiable on M. E(M) is an open set in Rn. Then, f is E-invex on M if (f∘E)(x)≥(f∘E)(y)+(∇(f∘E)(y))Tη(E(x),E(y)) for all x,y∈M.From the above discussions onE-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:R2→R2 is E(x,y)=(0,y) and f:M→R is defined by f(x,y)=-x2-y2, and
(2.13)η((x1,y1),(x2,y2))={(x122x2,y122y2),x2≠0,y2≠0,(0,y122y2),x2=0,y2≠0,(x122x2,0),x2≠0,y2=0,(0,0),otherwise.Definition 2.9.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn, let E(M) be an open set in Rn. Suppose f and E are differentiable on M. Then, f is said to be E-quasiinvex on M if
(2.14)(f∘E)(x)≤(f∘E)(y)∀x,y∈M⇒(∇(f∘E)(y))Tη(E(x),E(y))≤0
or
(2.15)(∇(f∘E)(y))Tη(E(x),E(y))>0⇒(f∘E)(x)>(f∘E)(y).A function may not beE-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.
ConsiderM={(x,y)∈R2∣x,y<0}, E:R2→R2 is E(x,y)=(0,y), and f:M→R is f(x,y)=x3+y3,  η:R2×R2→R2 is η((x1,y1),(x2,y2))=(x1-x2,y1-y2). Now for all (x1,y1),(x2,y2)∈M, (f∘E)(x1,y1) − (f∘E)(x2,y2) − ∇(f∘E)(x2,y2)Tη(E(x1,y1),E(x2,y2)) = y13+y23-3y22(y1-y2), which is not always positive. Hence, f is not E-invex with respect to η on M.If we assume that(f∘E)(x1,y1)≤(f∘E)(x2,y2) for all (x1,y1),(x2,y2)∈M, then (∇(f∘E)(x2,y2))Tη(E(x1,y1),E(x2,y2))=3y22(y1-y2)≤0. Hence, f is E-quasiinvex with respect to same η on M.Definition 2.11.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn, let E(M) be an open set in Rn. Suppose f and E are differentiable on M. Then, f is said to be E-pseudoinvex on M if
(2.16)(∇(f∘E)(y))Tη(E(x),E(y))≥0∀x,y∈M⇒(f∘E)(x)≥(f∘E)(y)
or
(2.17)(f∘E)(x)<(f∘E)(y)∀x,y∈M⇒(∇(f∘E)(y))Tη(E(x),E(y))<0.A function may not beE-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.
ConsiderM={(x,y)∈R2∣x,y>0}. E:R2→R2 is E(x,y)=(0,y) and f:M→R is f(x,y)=-x2-y2. For η:R2×R2→R2, defined by η((x1,y1),(x2,y2))=(x1-x2,y1-y2), and for all (x1,y1),(x2,y2)∈M, y1≠y2, (f∘E)(x1,y1)-(f∘E)(x2,y2)-∇(f∘E)(x2,y2)Tη(E(x1,y1),E(x2,y2))=-(y2-y1)2<0. Hence, f is not E-invex with respect to η on M. If ∇(f∘E)(x2,y2))Tη(E(x1,y1),E(x2,y2))≥0, then (f∘E)(x1,y1)-(f∘E)(x2,y2)-f(0,y1)-f(0,y2)=(y2+y1)(y2-y1)≥0. Hence, f is E-pseudoinvex with respect to η on M.If a functionf:M→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.
Iff:M→R is semi-E-invex on an E-invex set M, then f(E(y))≤f(y) for each y∈M.Proof.
Sincef is semi-E-invex on M⊆Rn and M is an E-invex set so for x,y∈M and λ∈[0,1], we have E(y)+λη(E(x),E(y))∈M and f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y). In particular, for λ=0,  f(E(y))≤f(y)for  eachy∈M.AnE-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 whereM={(x,y)∈R2∣x,y<0}, E:R2→R2 is E(x,y)=(0,y) and f:M→R is f(x,y)=x3+y3,  η:R2×R2→R2 is η((x1,y1),(x2,y2))=(x1-x2,y1-y2). It is verified that f is E-invex with respect to η on M. But f(E(2,0))>f(2,0). From Proposition 2.13 it can be concluded that f is not semi-E-invex with respect to same η. Also, using Definition 1.6, for all (x1,y1),(x2,y2)∈M,  λ∈[0,1],f(E(x2,y2)+λη(E(x1,y1),E(x2,y2)))-λf(x1,y1)-(1-λ)f(x2,y2)=-(y2+λy12/2y2)2+λ(x12+y12)+(1-λ)(x22+y22), which is not always negative. Hence, f is not semi-E-invex with respect to η on M.


## 2.1. Some New Results onE-Convexity with Differentiability

E-convexity at a point may be interpreted as follows.LetM be a nonempty subset of Rn, E:Rn→Rn. A function f:M→R is said to be E-convex at x¯∈M if M is an E-convex set and
(2.1)f(λE(x)+(1-λ)E(x¯))≤λ(f∘E)(x)+(1-λ)(f∘E)(x¯)
for all x∈Nδ(x¯) and λ∈[0,1], where Nδ(x¯) is δ-neighborhood of x¯, for small δ>0.It may be observed that a function may not be convex at a point butE-convex at that point with a suitable mapping E.Example 2.1.
ConsiderM={(x,y)∈R2∣y≥0}. E:R2→R2 is E(x,y)=(0,y) and f(x,y)=x3+y2. Also f is not convex at (-1,1). For all (x,y)∈Nδ(-1,1),   δ>0, and λ∈[0,1],f(λE(x,y)+(1-λ)E(-1,1))-λ(f∘E)(x,y)-(1-λ)(f∘E)(-1,1)=-λ(1-λ)(y-1)2≤0. Hence, f is E-convex at (-1,1).Proposition 2.2.
LetM⊆Rn, E:Rn→Rn be a homeomorphism. If f:M→R attains a local minimum point in the neighborhood of E(x¯), then it is E-convex at x¯.Proof.
Supposef has a local minimum point in a neighborhood Nϵ(E(x¯)) of E(x¯) for some x¯∈M,  ϵ>0. This implies f is convex on Nϵ(E(x¯)). That is,
(2.2)f(λz+(1-λ)E(x¯))≤λf(z)+(1-λ)f(E(x¯))∀z∈Nϵ(E(x¯)).
Since E:Rn→Rn is a homeomorphism, so inverse of the neighborhood Nϵ(E(x¯)) is a neighborhood of x¯ say Nδ(x¯) for some δ>0. Hence, there exists x∈Nδ(x¯) such that E(x)=z,  E(x)∈Nϵ(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 ofE(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.
LetM be an open E-convex subset of Rn,  f and E are differentiable functions, and let E be a homeomorphism. Then, f is E-convex at x¯∈M if and only if
(2.3)(f∘E)(x)≥(f∘E)(x¯)+(∇(f∘E)(x¯))T(E(x)-E(x¯))
for all E(x)∈Nϵ(E(x¯)) where Nϵ(E(x¯)) is ϵ-neighborhood of E(x¯),  ϵ>0.Proof.
SinceM 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 ϵ>0 such that E(x)∈Nϵ(E(x¯)) for all x∈Nδ(x¯), δ>0, very small. So, f is differentiable on E(M). Using expansion of f at E(x¯) in the neighborhood Nϵ(E(x¯)),
(2.4)f(E(x¯)+λ(z-E(x¯)))=f(E(x¯))+λ∇f(E(x¯))T(z-E(x¯))+α[E(x¯),λ(z-E(x¯))]λ∥z-E(x¯)∥,
where z∈Nϵ(E(x¯)) and limλ→0α[E(x¯),λ(z-E(x¯))]=0. Since f is E-convex at x¯∈M, so for all x∈Nδ(x¯), λ∈(0,1], x≠x¯,
(2.5)λ((f∘E)(x)-(f∘E)(x¯))≥f(E(x¯)+λ(E(x)-E(x¯)))-(f∘E)(x¯).
Since E is a homeomorphism, there exists x∈Nδ(x¯) such that E(x)=z. Replacing z by E(x) in (2.4) and using above inequality, we get
(2.6)(f∘E)(x)-(f∘E)(x¯)≥(∇(f∘E)(x¯))T(E(x)-E(x¯)+α[E(x¯),λ(E(x)-E(x¯))]∥E(x)-E(x¯)∥),
where limλ→0α[E(x¯),λ(E(x)-E(x¯))]=0. Hence, (2.3) follows.
The converse part follows directly from (2.4).It is obvious that ifE(x¯) is a local minimum point of f, then ∇(f∘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.
LetM⊆Rn be an open E-convex set, and let f be a differentiable E-convex function at x¯. If E:Rn→Rn is a homeomorphism and ∇(f∘E)(x¯)=0, then E(x¯) is the local minimum of f.


## 2.2. Some New Results on GeneralizedE-Convexity with Differentiability

Here, we introduce some generalizations ofE-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
→
R is said to be semi-E-convex at x¯∈M if M is an E-convex set and
(2.7)f(λE(x)+(1-λ)E(x¯))≤λf(x)+(1-λ)f(x¯)
for all x∈Nδ(x¯) and λ∈[0,1], where Nδ(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.
Supposef:M→R and E:Rn→Rn 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
(2.8)f(x)≥f(x¯)+(∇(f∘E)(x¯))T(E(x)-E(x¯))
for all E(x)∈Nϵ(E(x¯)), very small ϵ>0.Proof.
Proceeding as in Theorem2.3, we get the following relation from the expansion of f at E(x¯) in the neighborhood Nϵ(E(x¯)), where x- is the fixed point of E. (Since E is a homeomorphism, there exists ϵ>0 such that E(x)∈Nϵ(E(x¯)) for all x∈Nδ(x¯), very small δ>0):
(2.9)f(E(x¯)+λ(E(x)-E(x¯)))=(f∘E)(x¯)+λ(∇(f∘E)(x¯))T(E(x)-E(x¯))+α[E(x¯),λ(E(x)-E(x¯))]λ∥E(x)-E(x¯)∥,
where E(x)∈Nϵ(E(x¯)),  limλ→0α[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∈Nδ(x¯), λ∈(0,1], x≠x¯,
(2.10)λ(f(x)-f(x¯))≥f(E(x¯)+λ(E(x)-E(x¯)))-(f∘E)(x¯).
Using (2.9), the above inequality reduces to
(2.11)f(x)-f(x¯)≥(∇(f∘E)(x¯))T(E(x)-E(x¯)+α[E(x¯),λ(E(x)-E(x¯))]∥E(x)-E(x¯)∥),
where limλ→0α[E(x¯),λ(E(x)-E(x¯))]=0. Hence Inequality (2.8) follows for all E(x)∈Nϵ(E(x¯)).
Conversely, suppose Inequality (2.8) holds at the fixed point x¯ of E for all E(x)∈Nϵ(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. GeneralizedE-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.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn.   Also f:M→R are differentiable on M. E(M) is an open set in Rn. If f is E-preinvex on M then (f∘E)(x)≥(f∘E)(y)+(∇(f∘E)(y))Tη(E(x),E(y))for allx,y∈M.Proof.
IfE(M) is an open set, f and E are differentiable on M, then f∘E is differentiable on M. From Taylor's expansion of f at E(y) for some y∈M and λ>0,
(2.12)f(E(y)+λη(E(x),E(y)))=(f∘E)(y)+λ(∇(f∘E)(y))T(η(E(x),E(y)))+λ∥η(E(x),E(y))∥α(E(y),λη(E(x),E(y))),
where E(x)≠E(y),  limλ→0α(E(y),λη(E(x),E(y)))=0.
Iff is E-preinvex on M with respect to η (Definition 1.5), then as λ→0+, the above inequality reduces to (f∘E)(x)≥(f∘E)(y)+(∇(f∘E)(y))Tη(E(x),E(y)) for all x,y∈M.As a consequence of the above lemma, we may defineE-invexity with differentiability assumption as follows.Definition 2.7.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn. Also f:M→R are differentiable on M. E(M) is an open set in Rn. Then, f is E-invex on M if (f∘E)(x)≥(f∘E)(y)+(∇(f∘E)(y))Tη(E(x),E(y)) for all x,y∈M.From the above discussions onE-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:R2→R2 is E(x,y)=(0,y) and f:M→R is defined by f(x,y)=-x2-y2, and
(2.13)η((x1,y1),(x2,y2))={(x122x2,y122y2),x2≠0,y2≠0,(0,y122y2),x2=0,y2≠0,(x122x2,0),x2≠0,y2=0,(0,0),otherwise.Definition 2.9.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn, let E(M) be an open set in Rn. Suppose f and E are differentiable on M. Then, f is said to be E-quasiinvex on M if
(2.14)(f∘E)(x)≤(f∘E)(y)∀x,y∈M⇒(∇(f∘E)(y))Tη(E(x),E(y))≤0
or
(2.15)(∇(f∘E)(y))Tη(E(x),E(y))>0⇒(f∘E)(x)>(f∘E)(y).A function may not beE-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.
ConsiderM={(x,y)∈R2∣x,y<0}, E:R2→R2 is E(x,y)=(0,y), and f:M→R is f(x,y)=x3+y3,  η:R2×R2→R2 is η((x1,y1),(x2,y2))=(x1-x2,y1-y2). Now for all (x1,y1),(x2,y2)∈M, (f∘E)(x1,y1) − (f∘E)(x2,y2) − ∇(f∘E)(x2,y2)Tη(E(x1,y1),E(x2,y2)) = y13+y23-3y22(y1-y2), which is not always positive. Hence, f is not E-invex with respect to η on M.If we assume that(f∘E)(x1,y1)≤(f∘E)(x2,y2) for all (x1,y1),(x2,y2)∈M, then (∇(f∘E)(x2,y2))Tη(E(x1,y1),E(x2,y2))=3y22(y1-y2)≤0. Hence, f is E-quasiinvex with respect to same η on M.Definition 2.11.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn, let E(M) be an open set in Rn. Suppose f and E are differentiable on M. Then, f is said to be E-pseudoinvex on M if
(2.16)(∇(f∘E)(y))Tη(E(x),E(y))≥0∀x,y∈M⇒(f∘E)(x)≥(f∘E)(y)
or
(2.17)(f∘E)(x)<(f∘E)(y)∀x,y∈M⇒(∇(f∘E)(y))Tη(E(x),E(y))<0.A function may not beE-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.
ConsiderM={(x,y)∈R2∣x,y>0}. E:R2→R2 is E(x,y)=(0,y) and f:M→R is f(x,y)=-x2-y2. For η:R2×R2→R2, defined by η((x1,y1),(x2,y2))=(x1-x2,y1-y2), and for all (x1,y1),(x2,y2)∈M, y1≠y2, (f∘E)(x1,y1)-(f∘E)(x2,y2)-∇(f∘E)(x2,y2)Tη(E(x1,y1),E(x2,y2))=-(y2-y1)2<0. Hence, f is not E-invex with respect to η on M. If ∇(f∘E)(x2,y2))Tη(E(x1,y1),E(x2,y2))≥0, then (f∘E)(x1,y1)-(f∘E)(x2,y2)-f(0,y1)-f(0,y2)=(y2+y1)(y2-y1)≥0. Hence, f is E-pseudoinvex with respect to η on M.If a functionf:M→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.
Iff:M→R is semi-E-invex on an E-invex set M, then f(E(y))≤f(y) for each y∈M.Proof.
Sincef is semi-E-invex on M⊆Rn and M is an E-invex set so for x,y∈M and λ∈[0,1], we have E(y)+λη(E(x),E(y))∈M and f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y). In particular, for λ=0,  f(E(y))≤f(y)for  eachy∈M.AnE-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 whereM={(x,y)∈R2∣x,y<0}, E:R2→R2 is E(x,y)=(0,y) and f:M→R is f(x,y)=x3+y3,  η:R2×R2→R2 is η((x1,y1),(x2,y2))=(x1-x2,y1-y2). It is verified that f is E-invex with respect to η on M. But f(E(2,0))>f(2,0). From Proposition 2.13 it can be concluded that f is not semi-E-invex with respect to same η. Also, using Definition 1.6, for all (x1,y1),(x2,y2)∈M,  λ∈[0,1],f(E(x2,y2)+λη(E(x1,y1),E(x2,y2)))-λf(x1,y1)-(1-λ)f(x2,y2)=-(y2+λy12/2y2)2+λ(x12+y12)+(1-λ)(x22+y22), which is not always negative. Hence, f is not semi-E-invex with respect to η on M.


## 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
→
R is said to be semi-E-convex at x¯∈M if M is an E-convex set and
(2.7)f(λE(x)+(1-λ)E(x¯))≤λf(x)+(1-λ)f(x¯)
for all x∈Nδ(x¯) and λ∈[0,1], where Nδ(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.
Supposef:M→R and E:Rn→Rn 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
(2.8)f(x)≥f(x¯)+(∇(f∘E)(x¯))T(E(x)-E(x¯))
for all E(x)∈Nϵ(E(x¯)), very small ϵ>0.Proof.
Proceeding as in Theorem2.3, we get the following relation from the expansion of f at E(x¯) in the neighborhood Nϵ(E(x¯)), where x- is the fixed point of E. (Since E is a homeomorphism, there exists ϵ>0 such that E(x)∈Nϵ(E(x¯)) for all x∈Nδ(x¯), very small δ>0):
(2.9)f(E(x¯)+λ(E(x)-E(x¯)))=(f∘E)(x¯)+λ(∇(f∘E)(x¯))T(E(x)-E(x¯))+α[E(x¯),λ(E(x)-E(x¯))]λ∥E(x)-E(x¯)∥,
where E(x)∈Nϵ(E(x¯)),  limλ→0α[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∈Nδ(x¯), λ∈(0,1], x≠x¯,
(2.10)λ(f(x)-f(x¯))≥f(E(x¯)+λ(E(x)-E(x¯)))-(f∘E)(x¯).
Using (2.9), the above inequality reduces to
(2.11)f(x)-f(x¯)≥(∇(f∘E)(x¯))T(E(x)-E(x¯)+α[E(x¯),λ(E(x)-E(x¯))]∥E(x)-E(x¯)∥),
where limλ→0α[E(x¯),λ(E(x)-E(x¯))]=0. Hence Inequality (2.8) follows for all E(x)∈Nϵ(E(x¯)).
Conversely, suppose Inequality (2.8) holds at the fixed point x¯ of E for all E(x)∈Nϵ(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. GeneralizedE-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.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn.   Also f:M→R are differentiable on M. E(M) is an open set in Rn. If f is E-preinvex on M then (f∘E)(x)≥(f∘E)(y)+(∇(f∘E)(y))Tη(E(x),E(y))for allx,y∈M.Proof.
IfE(M) is an open set, f and E are differentiable on M, then f∘E is differentiable on M. From Taylor's expansion of f at E(y) for some y∈M and λ>0,
(2.12)f(E(y)+λη(E(x),E(y)))=(f∘E)(y)+λ(∇(f∘E)(y))T(η(E(x),E(y)))+λ∥η(E(x),E(y))∥α(E(y),λη(E(x),E(y))),
where E(x)≠E(y),  limλ→0α(E(y),λη(E(x),E(y)))=0.
Iff is E-preinvex on M with respect to η (Definition 1.5), then as λ→0+, the above inequality reduces to (f∘E)(x)≥(f∘E)(y)+(∇(f∘E)(y))Tη(E(x),E(y)) for all x,y∈M.As a consequence of the above lemma, we may defineE-invexity with differentiability assumption as follows.Definition 2.7.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn. Also f:M→R are differentiable on M. E(M) is an open set in Rn. Then, f is E-invex on M if (f∘E)(x)≥(f∘E)(y)+(∇(f∘E)(y))Tη(E(x),E(y)) for all x,y∈M.From the above discussions onE-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:R2→R2 is E(x,y)=(0,y) and f:M→R is defined by f(x,y)=-x2-y2, and
(2.13)η((x1,y1),(x2,y2))={(x122x2,y122y2),x2≠0,y2≠0,(0,y122y2),x2=0,y2≠0,(x122x2,0),x2≠0,y2=0,(0,0),otherwise.Definition 2.9.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn, let E(M) be an open set in Rn. Suppose f and E are differentiable on M. Then, f is said to be E-quasiinvex on M if
(2.14)(f∘E)(x)≤(f∘E)(y)∀x,y∈M⇒(∇(f∘E)(y))Tη(E(x),E(y))≤0
or
(2.15)(∇(f∘E)(y))Tη(E(x),E(y))>0⇒(f∘E)(x)>(f∘E)(y).A function may not beE-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.
ConsiderM={(x,y)∈R2∣x,y<0}, E:R2→R2 is E(x,y)=(0,y), and f:M→R is f(x,y)=x3+y3,  η:R2×R2→R2 is η((x1,y1),(x2,y2))=(x1-x2,y1-y2). Now for all (x1,y1),(x2,y2)∈M, (f∘E)(x1,y1) − (f∘E)(x2,y2) − ∇(f∘E)(x2,y2)Tη(E(x1,y1),E(x2,y2)) = y13+y23-3y22(y1-y2), which is not always positive. Hence, f is not E-invex with respect to η on M.If we assume that(f∘E)(x1,y1)≤(f∘E)(x2,y2) for all (x1,y1),(x2,y2)∈M, then (∇(f∘E)(x2,y2))Tη(E(x1,y1),E(x2,y2))=3y22(y1-y2)≤0. Hence, f is E-quasiinvex with respect to same η on M.Definition 2.11.
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn, let E(M) be an open set in Rn. Suppose f and E are differentiable on M. Then, f is said to be E-pseudoinvex on M if
(2.16)(∇(f∘E)(y))Tη(E(x),E(y))≥0∀x,y∈M⇒(f∘E)(x)≥(f∘E)(y)
or
(2.17)(f∘E)(x)<(f∘E)(y)∀x,y∈M⇒(∇(f∘E)(y))Tη(E(x),E(y))<0.A function may not beE-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.
ConsiderM={(x,y)∈R2∣x,y>0}. E:R2→R2 is E(x,y)=(0,y) and f:M→R is f(x,y)=-x2-y2. For η:R2×R2→R2, defined by η((x1,y1),(x2,y2))=(x1-x2,y1-y2), and for all (x1,y1),(x2,y2)∈M, y1≠y2, (f∘E)(x1,y1)-(f∘E)(x2,y2)-∇(f∘E)(x2,y2)Tη(E(x1,y1),E(x2,y2))=-(y2-y1)2<0. Hence, f is not E-invex with respect to η on M. If ∇(f∘E)(x2,y2))Tη(E(x1,y1),E(x2,y2))≥0, then (f∘E)(x1,y1)-(f∘E)(x2,y2)-f(0,y1)-f(0,y2)=(y2+y1)(y2-y1)≥0. Hence, f is E-pseudoinvex with respect to η on M.If a functionf:M→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.
Iff:M→R is semi-E-invex on an E-invex set M, then f(E(y))≤f(y) for each y∈M.Proof.
Sincef is semi-E-invex on M⊆Rn and M is an E-invex set so for x,y∈M and λ∈[0,1], we have E(y)+λη(E(x),E(y))∈M and f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y). In particular, for λ=0,  f(E(y))≤f(y)for  eachy∈M.AnE-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 whereM={(x,y)∈R2∣x,y<0}, E:R2→R2 is E(x,y)=(0,y) and f:M→R is f(x,y)=x3+y3,  η:R2×R2→R2 is η((x1,y1),(x2,y2))=(x1-x2,y1-y2). It is verified that f is E-invex with respect to η on M. But f(E(2,0))>f(2,0). From Proposition 2.13 it can be concluded that f is not semi-E-invex with respect to same η. Also, using Definition 1.6, for all (x1,y1),(x2,y2)∈M,  λ∈[0,1],f(E(x2,y2)+λη(E(x1,y1),E(x2,y2)))-λf(x1,y1)-(1-λ)f(x2,y2)=-(y2+λy12/2y2)2+λ(x12+y12)+(1-λ)(x22+y22), 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(3.1)(P)minf(x)subject  tog(x)≤0,
where f:M→R, gi:M→Rm, M⊆Rn,g=(g1,g2,…,gm)T.  M′={x∈M:gi(x)≤0,i=1,…,m} is the set of feasible solutions.Theorem 3.1 (sufficient optimality condition).
LetM be a nonempty open E-convex subset of Rn,f:M→R,  g:M→Rm, and E:Rn→Rn 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′ and (x¯,y¯)∈M′×Rm satisfies
(3.2)∇[(f∘E)(x)+〈y,(g∘E)(x)〉]=0,〈y,(g∘E)(x)〉=0,y≥0,
then x¯ is local optimal solution of (P).Proof.
Sincef and g are semi-E-convex at x¯∈M by Theorem 2.5,
(3.3)f(x)-f(x¯)≥(∇(f∘E)(x¯))T(E(x)-E(x¯))∀E(x)∈Nϵ(E(x¯)),g(x)-g(x¯)≥(∇(g∘E)(x¯))T(E(x)-E(x¯))∀E(x)∈Nϵ(E(x¯)).
Adding the above two inequalities, we have
(3.4)[f(x)-f(x¯)]+y¯T[g(x)-g(x¯)]≥∇[(f∘E)(x¯)+y¯T((g∘E)(x¯))]T(E(x)-E(x¯)).
If (3.2) hold, then f(x)-f(x¯)+y¯Tg(x)≥0for allE(x)∈Nϵ(E(x¯)). Since g(x)≤0 for all x∈M′ and y¯≥0, so y¯Tg(x)≤0. Hence, f(x)-f(x¯)≥0 for all E(x)∈Nϵ(E(x¯)). Since E is a homeomorphism, there exists δ>0 such that x∈Nδ(x¯) for all E(x)∈Nϵ(E(x¯)), which means f(x)≥f(x¯)for  allx∈Nδ(x¯). Hence, x¯ is a local optimal solution of (P).Also we see that a fixed point ofE is a local optimal solution of (P) under generalized E-invexity assumptions.Lemma 3.2.
LetM be a nonempty E-invex subset of Rn with respect to some η:Rn×Rn→Rn. Let gi:M→R, i=1,…,m be semi-E-quasiinvex functions with respect to η on M. Then, M′ is an E-invex set.Proof.
LetMi={x∈M:gi(x)≤0},i=1,…,m.  M′=∩i=1mMi and M′⊆M. Since gi, i=1,…,m are semi-E-quasiinvex function on M, so for all x,y∈Mi and λ∈[0,1],gi(E(y)+λη(E(x),E(y)))≤max{gi(x),gi(y)}≤0. Hence, E(y)+λη(E(x),E(y))∈Mi for all x,y∈Mi. So Mi is E-invex with respect to same η. From Lemma 1.10, M′=∩i=1mMi is E-invex with respect to same η.Corollary 3.3.
LetM be a nonempty E-invex subset of Rn with respect to some η:Rn×Rn→Rn. Let gi:M→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.
Sincex is a feasible solution of (P), so x∈M′⇒E(x)∈E(M′). Since each gi, i=1,…,m is semi-E-quasiinvex function on M, from Lemma 3.2, M′ is an E-invex set. Also E(M′)⊆M′. Hence, E(x)∈M′. That is, E(x) is a feasible solution of (P).Theorem 3.4 (sufficient optimality condition).
LetM be a nonempty E-invex subset of Rn with respect to η:Rn×Rn→Rn. Let E(M) be an open set in Rn. Suppose f:Rn→R, g:Rn→Rm and E are differentiable functions on M. If f is E-pseudoinvex function with respect to η and for u≥0,uTg is semi-E-quasiinvex function with respect to the same η at x∈M′, where x is a fixed point of the map E and (x,u)∈M′×Rm,u≥0 satisfies the following system:
(3.5)∇[(f∘E)(x)+〈u,(g∘E)(x)〉]=0,(3.6)〈u,(g∘E)(x)〉=0,
then x is a local optimal solution of (P).Proof.
Suppose(x,u)∈M′×Rm satisfies (3.5) and (3.6). For all y∈M′,  g(y)≤0. Also, u≥0. Hence, uTg(y)≤0 for all y∈M′. From (3.6), 〈u,(g∘E)(x)〉=0, that is, uTg(E(x))=0.  x is a fixed point of E that is E(x)=x. So uTg(x)=0. Hence,
(3.7)uTg(y)≤uTg(x)∀y∈M′.
Since u≥0 and uTg is semi-E-quasiinvex function with respect to η at x, so the above inequality implies
(3.8)∇uTg(E(x))η(E(y),E(x))≤0.
From (3.5), ∇f(E(x))=-∇uTg(E(x)). Putting this value in the above inequality, we have ∇f(E(x))η(E(x),E(y))≥0.
f is E-pseudoinvex at x with respect to η. Hence, ∇f(E(x))η(E(x),E(y))≥0 implies
(3.9)f(E(y))≥f(E(x))=f(x)∀y∈M′.
Hence, x is the optimal solution (P) on E(M′).The following example justifies the above theorem.Example 3.5.
Consider the optimization problem,(3.10)(P)min-x2-y2subject  tox2+y2-4≤0,
where M={(x,y)∈R2∣x,y>0}. E:R2→R2 is E(x,y)=(0,y). This is not a convex programming problem. Consider η:R2×R2→R2 defined by η((x1,y1),(x2,y2))=(x1-x2,y1-y2). Here, M′={(x,y)∈M:x2+y2-4≤0}, and E(M′)={(0,y):y≥0,y2-4≤0}. The sufficient conditions (7-8) reduce to
(3.11)-2y+u2y=0,u(y2-4)=0,u≥0,
whose solution is y=2,u=1 and E(0,2)=(0,2).
In Example2.12, we have already proved that f(x,y)=-x2-y2 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′, where ug(x,y)=x2+y2-4. So (0,2) is the optimal solution of (P) on E(M′).


## 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 butE-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 betweenE-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 studiedE-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.



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