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

Authors: Hehua Jiao; Sanyang Liu; Xinying Pai
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

A kind of generalized convex set, called as local star-shapedE-invex set with respect to η, is presented, and  some of its important characterizations are derived. Based on this concept, a new class of functions, named as semilocal E-preinvex functions, which is a generalization of semi-E-preinvex functions and semilocal E-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.


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## Body

## 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 E-convex sets, E-convex functions, and E-convex programming, discussed some of their basic properties, and obtained some optimality results on E-convex programming. In 2002, Chen [12] brought forward a class of semi-E-convex functions and also discussed its basic properties. In 2007, by combining the concept of semi-E-convexity and that of semilocal convexity, Hu et al. [13] put forward the concept of generalized convexity called as semilocal E-convexity, studied some of its characterizations, and obtained some optimality conditions and duality results for semilocal E-convex programming. In [14], optimality and duality were further studied for a fractional multiple-objective programming involving semilocal E-convexity. In 2009, Fulga and Preda [15] extended the E-convexity to E-preinvexity and local E-preinvexity and discussed some of their properties and an application. In 2011, Luo and Jian [16] introduced semi-E-preinvex maps in Banach spaces and studied some of their properties.Motivated by research work of [13–16] and references therein, in this paper, we present the concept of semilocal E-preinvexity and discuss its some important properties. Furthermore, as applications of semilocal E-preinvexity, we establish the optimality conditions and duality results for a nonlinear programming. The concept of semilocal E-preinvexity unifies the concepts of semilocal E-convexity and semi-E-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, letRn denote the n-dimensional Euclidean space, and let E:Rn→Rn and η:Rn×Rn→Rn 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 setK⊂Rn is said to be E-convex if there is a map E such that
(2.1)λE(x)+(1-λ)E(y)∈K,∀x,y∈K,0≤λ≤1.Definition 2.2 (see [11]).
A functionf:Rn→R is said to be E-convex on a set K⊂Rn if there is a map E such that K is an E-convex set and
(2.2)f(λE(x)+(1-λ)E(y))≤λf(E(x))+(1-λ)f(E(y)),∀x,y∈K,0≤λ≤1.Definition 2.3 (see [12]).
A functionf:Rn→R is said to be semi-E-convex on a set K⊂Rn if there is a map E such that K is an E-convex set and
(2.3)f(λE(x)+(1-λ)E(y))≤λf(x)+(1-λ)f(y),∀x,y∈K,0≤λ≤1.Definition 2.4 (see [15]).
A setK⊂Rn is said to be E-invex with respect to η if
(2.4)E(y)+λη(E(x),E(y))∈K,∀x,y∈K,0≤λ≤1.Definition 2.5 (see [15]).
LetK⊂Rn be an E-invex set with respect to η. A function f:Rn→R is said to be E-preinvex on K with respect to η if
(2.5)f(E(y)+λη(E(x),E(y)))≤λf(E(x))+(1-λ)f(E(y)),∀x,y∈K,0≤λ≤1.Definition 2.6 (see [16]).
LetK⊂Rn be an E-invex set with respect to η. A function f:Rn→R is said to be semi-E-preinvex on K with respect to η if
(2.6)f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y),∀x,y∈K,0≤λ≤1.Definition 2.7 (see [17]).
A setK⊂Rn is said to be local star-shaped invex with respect to η if for any x,y∈K, there is a maximal positive number a(x,y)≤1 satisfying
(2.7)y+λη(x,y)∈K,∀λ∈[0,a(x,y)].Definition 2.8 (see [13]).
A setK⊂Rn is said to be local star-shaped E-convex if there is a map E such that corresponding to each pair of points x,y∈K, and there is a maximal positive number a(x,y)≤1 satisfying
(2.8)λE(x)+(1-λ)E(y)∈K,∀λ∈[0,a(x,y)].Definition 2.9 (see [13]).
A functionf:Rn→R is said to be semilocal E-convex on a local star-shaped E-convex set K⊂Rn if for each pair of x,y∈K(with a maximal positive number a(x,y)≤1 satisfying (2.8)), there exists a positive number b(x,y)≤a(x,y) satisfying
(2.9)f(λ(E(x))+(1-λ)E(y))≤λf(x)+(1-λ)f(y),∀λ∈[0,b(x,y)].Definition 2.10 (see [18]).
A vector functionf:X0→Rk is said to be a convex-like function if for any x,y∈X0⊂Rn and 0≤λ≤1, there is z∈X0 such that
(2.10)f(z)≤λf(x)+(1-λ)f(y),
where the inequalities are taken component wise.Lemma 2.11 (see [19]).
LetS be a nonempty set in Rn, and let ψ:S→Rk be a convexlike function then either ψ(x)<0 has a solution x∈S or λTψ(x)≥0 for all x∈S and some λ∈Rk, λ≥0, and λ≠0, but both alternatives are never true.


## 3. Local Star-ShapedE-Invex Set

In this section, we introduce the local star-shapedE-invex set with respect to a given mapping η and discuss some of its basic characterizations.Definition 3.1.
A setK⊂Rn is said to be local star-shaped E-invex with respect to a given mapping η if there is a map E such that corresponding to each pair of points x,y∈K, and there is a maximal positive number a(x,y)≤1 satisfying
(3.1)E(y)+λη(E(x),E(y))∈K,0≤λ≤a(x,y).Remark 3.2.
EveryE-convex set is a local star-shaped E-invex set with respect to η, where η(x,y)=x-y,a(x,y)=1, for all x,y∈Rn. Every local star-shaped E-convex set is a local star-shaped E-invex set with respect to η, where η(x,y)=x-y, for all x,y∈Rn. Every E-invex set with respect to η is a local star-shaped E-invex set with respect to η, where a(x,y)=1, for all x,y∈Rn. But their converses are not necessarily true.The following example shows that local star-shapedE-invex set is more general than E-convex set, E-invex set, and local star-shaped E-convex set.Example 3.3.
LetK=[-4,-1)⋃[1,4],
(3.2)E(x)={x2if|x|≤2,-1if|x|>2,η(x,y)={x-yifx≥0,y≥0,orx≤0,y≤0,-1-yifx>0,y≤0,orx≥0,y<0,1-yifx<0,y≥0,orx≤0,y>0.
We can testify thatK is a local star-shaped E-invex set with respect to η.
However, whenx0=1, y0=3, there exists a λ1∈[0,1] such that  λ1E(x0)+(1-λ1)E(y0)=-1+2λ1∉K, namely, K is not an E-convex set.
Also, there is aλ2∈[0,1] such that E(y0)+λ2η(E(x0),E(y0))=-1∉K, that is, K is not an E-invex set with respect to η.
Similarly, for any positive numbera≤1, there exists a sufficiently small positive number λ3≤a satisfying λ3E(x0)+(1-λ3)E(y0)=-1+2λ3∉K, that is, K is not a local star-shaped E-convex set.Proposition 3.4.
If a setK⊂Rn is local star-shaped E-invex with respect to η, then E(K)⊂K.Proof.
SinceK is local star-shaped E-invex, then for any x,y∈K, there exists a maximal positive number a(x,y)≤1 satisfying E(y)+λη(E(x),E(y))∈K, for all λ∈[0,a(x,y)].
Thus, forλ=0, E(y)∈K.
Hence,E(K)⊂K.Proposition 3.5.
LetE(K) be local star-shaped invex with respect to η, E(K)⊂K, then K is local star-shaped E-invex with respect to the same η.Proof.
Assume thatx,y∈K, then E(x),E(y)∈E(K). Since E(K) is local star-shaped invex with respect to η, thus for E(x),E(y)∈E(K), there exists a positive number a(E(x),E(y))≤1 satisfying
(3.3)E(y)+λη(E(x),E(y))∈E(K)⊂K,∀λ∈[0,a(E(x),E(y))].
Hence, K is local star-shaped E-invex with respect to η.Remark 3.6.
Every local star-shaped invex set with respect toη is local star-shaped E-invex set, where E is an identity map, but its converse is not necessarily true. See the following example.Example 3.7.
LetK1={(x,y)∈R2:(x,y)=λ1(0,0)+λ2(2,3)+λ3(0,2)},K2={(x,y)∈R2:(x,y)=λ1(0,0)+λ2(-2,-3)+λ3(0,-4)}, and K=K1∪K2, where λ1,λ2,λ3≥0 and ∑i=13λi=1. Let η:R2×R2→R2 be defined as η(x,y)=x-y, and let E:R2→R2 be defined as
(3.4)E(x,y)={(0,3)if(x,y)=(2,3),(0,y-1),if(x,y)∈K1∖{(0,0),(2,3)},(0,y)if(x,y)∈K2.
It is not difficult to prove thatK is a local star-shaped E-invex set with respect to η. However, by taking x=(2,3), y=(0,-4), we know that there exists no maximal positive number a(x,y)≤1 such that y+λη(x,y)∈K, for all λ∈[0,a(x,y)].
That is,K is not a local star-shaped invex set with respect to η.Proposition 3.8.
LetKi⊂Rn(i=1,2,…,m) be a collection of local star-shaped E-invex sets with the same map η, then ⋂i=1mKi is a local star-shaped E-invex set with respect to η.Proof.
For allx,y∈⋂i=1mKi, we have x,y∈Ki(i=1,2,…,m).
SinceKi(i=1,2,…,m) are all local star-shaped E-invex sets, then there exist positive numbers ai(x,y)≤1(i=1,2,…,m) such that
(3.5)E(y)+λη(E(x),E(y))∈Ki,∀λ∈[0,ai(x,y)],i=1,2,…,m.
Taking a(x,y)=minai(x,y), i=1,2,…,m, we can get
(3.6)E(y)+λη(E(x),E(y))∈⋂i=1mKi,∀λ∈[0,a(x,y)].
Therefore, the proposition is proved.Remark 3.9.
Even ifK1, K2 are all local star-shaped E-invex set with respect to η, K1∪K2 is not necessarily a local star-shaped E-invex set. See the following example.Example 3.10.
Let the mapη:R2×R2→R2 be defined as η(x,y)=x-y,and the map E:R2→R2 be defined as E(x,y)=(x/2,y/2). Consider the two sets
(3.7)K1={(x,y)∈R2:(x,y)=λ1(0,0)+λ2(2,0)+λ3(0,2)},K2={(x,y)∈R2:(x,y)=λ1(0,0)+λ2(-2,0)+λ3(0,-2)},
where λi≥0, i=1,2,3, and ∑i=13λi=1.
We can easily prove that the two setsK1, K2 are all local star-shaped E-invex sets with respect to η. However, when x=(2,0), y=(0,-2), there is not a positive number a(x,y)≤1 such that
(3.8)E(y)+λ(E(x),E(y))=(λ,-1+λ)∈K1∪K2,∀λ∈(0,a(x,y)].
Thus, K1∪K2 is not a local star-shaped E-invex set with respect to η.Proposition 3.11.
LetK⊂Rn be a local star-shaped E1 and E2-invex set with respect to the same η, then K is a local star-shaped (E1∘E2) and (E2∘E1)-invex set with respect to the same η.Proof.
By contradiction, assume that for a pair ofx,y∈K, for all a(x,y)∈(0,1], there exists a λ0∈(0,a(x,y)] such that E1∘E2(y)+λ0η((E1∘E2(x),E1∘E2(y))∉K, that is, E1(E2y)+λ0η((E1(E2x),E1(E2y))∉K.
Since, from Proposition3.4, E2(x),E2(y)∈K, then E1(E2y)+λ0η(E1(E2x),E1(E2y))∉K contradicts the local star-shaped E1-invexity of K.
Hence,K is a local star-shaped (E1∘E2)-invex set.
Similarly,K is a local star-shaped (E2∘E1)-invex set.


## 4. SemilocalE-Preinvex Functions

In the section, we present the concept of semilocalE-preinvex function and study some of its properties. We first recall a relevant definition.Definition 4.1 (see [15]).
A functionf:Rn→R is said to be local E-preinvex on k⊂Rn with respect to η if for any x,y∈K (with a maximal positive number a(x,y)≤1 satisfying (3.1)), there exists 0<b(x,y)≤a(x,y) such that K is a local star-shaped E-invex set and
(4.1)f(E(y)+λη(E(x),E(y)))≤λf(E(x))+(1-λ)f(E(y)),∀λ∈[0,b(x,y)].Definition 4.2.
A functionf:Rn→R is said to be semilocal E-preinvex on k⊂Rn with respect to η if for any x,y∈K (with a maximal positive number a(x,y)≤1 satisfying (3.1)), there exists 0<b(x,y)≤a(x,y) such that K is a local star-shaped E-invex set and
(4.2)f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y),∀λ∈[0,b(x,y)].
If the inequality sign above is strict for anyx,y∈K and x≠y, then f is called a strict semilocal E-preinvex function.
A vector functionf:Rn→Rk is said to be semilocal E-preinvex on a local star-shaped E-invex set K⊂Rn with respect to η if for each pair of points x,y∈K(with a maximal positive number a(x,y)≤1 satisfying (3.1)), there exists a positive number b(x,y)≤a(x,y) satisfying
(4.3)f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y),∀λ∈[0,b(x,y)],
where the inequalities are taken component wise.
The definition of strict semilocalE-preinvex of a vector function f:Rn→Rk is similar to the one for a vector semilocal E-preinvex function.Remark 4.3.
Every semilocalE-convex function on a local star-shaped set K is a semilocal E-preinvex function, where η(x,y)=x-y, for all x,y∈Rn. Every semi-E-preinvex function with respect to η is a semilocal E-preinvex function, where a(x,y)=b(x,y)=1, for all x,y∈Rn. But their converses are not necessarily true.We give below an example of semilocalE-preinvex function, which is neither a semilocal E-convex function nor a semi-E-preinvex function.Example 4.4.
Let the mapE:R→R be defined as
(4.4)E(x)={0ifx<0,1if1<x≤2,xif0≤x≤1orx>2,
and the map η:R×R→R be defined as
(4.5)η(x,y)={0ifx=y,1-xifx≠y.
Obviously, R is a local star-shaped E-convex set and a local star-shaped E-invex set with respect to η. Let f:R→R be defined as
(4.6)f(x)={0if1<x≤2,1ifx>2,-x+1if0≤x≤1,-x+2ifx<0.
We can prove thatf is semilocal E-preinvex on R with respect to η. However, when x0=2, y0=3, and for any b∈(0,1], there exists a sufficiently small λ0∈(0,b] satisfying
(4.7)f(λ0E(x0)+(1-λ0)E(y0))=f(3-2λ0)=1>1-λ0=λ0f(x0)+(1-λ0)f(y0).
That is, f(x) is not a semilocal E-convex function on R.
Similarly, takingx1=1, y1=4, we have
(4.8)f(E(y1)+λ1η(E(x1),E(y1)))=f(4)=1>1-λ1=λ1f(x1)+(1-λ1)f(y1),
for some λ1∈[0,1].
Thus,f(x) is not a semi-E-preinvex function on R with respect to η.Theorem 4.5.
Letf:K⊂Rn→R be a local E-preinvex function on a local star-shaped E-invex set K with respect to η, then f is a semilocal E-preinvex function if and only if f(E(x))≤f(x), for all x∈K.Proof.
Suppose thatf is a semilocal E-preinvex function on set K with respect to η, then for each pair of points x,y∈K (with a maximal positive number a(x,y)≤1 satisfying (3.1)), there exists a positive number b(x,y)≤a(x,y) satisfying
(4.9)f(E(x)+λη(E(y)),E(x))≤λf(y)+(1-λ)f(x),λ∈[0,b(x,y)].
By letting λ=0, we have f(E(x))≤f(x), for all x∈K.
Conversely, assume thatf is a local E-preinvex function on a local star-shaped E-invex set K, then for any x,y∈K, there exist a(x,y)∈(0,1] satisfying (3.1) and b(x,y)∈(0,a(x,y)] such that
(4.10)f(E(y)+λη(E(x),E(y)))≤λf(E(x))+(1-λ)f(E(y)),∀λ∈[0,b(x,y)].
Since f(E(x))≤f(x), for all x∈K, then
(4.11)f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y),∀λ∈[0,b(x,y)].
The proof is completed.Remark 4.6.
A localE-preinvex function on a local star-shaped E-invex set with respect to η is not necessarily a semilocal E-preinvex function.Example 4.7.
LetK,E, and η be the same as the ones of Example 3.3 and f:R→R be defined by f(x)=x2,then f is local E-preinvex on K with respect to η.
Sincef(E(2))=16>f(2)=4, from Theorem 4.5, it follows that f is not a semilocal E-preinvex function.Theorem 4.8.
Letf:Rn→R be a semilocal E-preinvex function on a local star-shaped E-invex set K⊂Rn with respect to η, and let φ:R→R be a nondecreasing and convex function, then φ(f(x)) is semilocal E-preinvex on K with respect to η.The proof is easy and is omitted.Theorem 4.9.
If the functionsfi:Rn→R(i=1,2,…,m) are all semilocal E-preinvex on a local star-shaped E-invex set K⊂Rn with respect to the same η, then the function f(x)=∑i=1maifi(x) is semilocal E-preinvex on K with respect to η for all ai≥0, i=1,2,…,m.Proof.
SinceK is a local star-shaped E-invex set with respect to η, then for all x,y∈K, there exists a positive number a(x,y)≤1 such that
(4.12)E(y)+λη(E(x),E(y))∈K,∀λ∈[0,a(x,y)].
On the other hand, fi(i=1,2,…,m) are all semilocal E-preinvex on K with respect to the same η; thus, there exist positive numbers bi(x,y)≤a(x,y) such that
(4.13)fi(E(y)+λη(E(x),E(y)))≤λfi(x)+(1-λ)fi(y),∀λ∈[0,bi(x,y)],i=1,2,…,m.
Now, letting b(x,y)=minbi(x,y),i=1,2,…,m, we have
(4.14)∑i=1maifi(E(y)+λη(E(x),E(y)))≤λ∑i=1maifi(x)+(1-λ)∑i=1maifi(y),∀λ∈[0,b(x,y)].
That is, f(x) is semilocal E-preinvex on Kwith respect to η.Definition 4.10.
The setG={(x,α):x∈K⊂Rn,α∈R} is said to be a local star-shaped E-invex set with respect to η corresponding to Rn if there are two maps η, E and a maximal positive number a((x,α1),(y,α2))≤1, for each (x,α1),(y,α2)∈G such that
(4.15)(E(y)+λη(E(x),E(y)),λα1+(1-λ)α2)∈G,∀λ∈[0,a((x,α1),(y,α2))].Theorem 4.11.
LetK⊂Rn be a local star-shaped E-invex set with respect to η, then f is a semilocal E-preinvex function on K with respect to η if and only if its epigraph Gf={(x,α):x∈K,f(x)≤α,α∈R} is a local star-shaped E-invex set with respect to η corresponding to Rn.Proof.
Assume thatf is semilocal E-preinvex on K with respect to η and (x,α1),(y,α2)∈Gf, then x,y∈K, and f(x)≤α1, f(y)≤α2. Since K is a local star-shaped E-invex set, there is a maximal positive number a(x,y)≤1 such that
(4.16)E(y)+λη(E(x),E(y))∈K,∀λ∈[0,a(x,y)].
In addition, in view of f being a semilocal E-preinvex function on K with respect to η, there is a positive number b(x,y)≤a(x,y) such that
(4.17)f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y)≤λα1+(1-λ)α2,∀λ∈[0,b(x,y)].
That is, (E(y)+λη(E(x),E(y)),λα1+(1-λ)α2)∈Gf, for all λ∈[0,b(x,y)].
Therefore,Gf={(x,α):x∈K,f(x)≤α,α∈R} is a local star-shaped E-invex set with respect to η corresponding to Rn.
Conversely, ifGf is a local star-shaped E-invex set with respect to η corresponding to Rn, then for any points (x,f(x)),(y,f(y))∈Gf, there is a maximal positive number a((x,f(x)),(y,f(y)))≤1 such that
(4.18)(E(y)+λη(E(x),E(y)),λf(x)+(1-λ)f(y))∈Gf,∀λ∈[0,a((x,f(x)),(y,f(y)))].
That is, E(y)+λη(E(x),E(y))∈K,
(4.19)f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y),∀λ∈[0,a((x,f(x)),(y,f(y)))].
Thus, K is a local star-shaped E-invex set, and f is a semilocal E-preinvex function on K with respect to η.Theorem 4.12.
Iff is a semilocal E-preinvex function on a local star-shaped E-invex set K⊂Rn with respect to η, then the level set Sα={x∈K:f(x)≤α} is a local star-shaped E-invex set for any α∈R.Proof.
For anyα∈R and x,y∈Sα, then x,y∈K and f(x)≤α, f(y)≤α. Since K is a local star-shaped E-invex set, there is a maximal positive number a(x,y)≤1 such that
(4.20)E(y)+λη(E(x),E(y))∈K,∀λ∈[0,a(x,y)].
In addition, due to the semilocal E-preinvexity of f, there is a positive number b(x,y)≤a(x,y) such that
(4.21)f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y)≤λα+(1-λ)α=α,∀λ∈[0,b(x,y)].
That is, E(y)+λη(E(x),E(y))∈Sα, for all λ∈[0,b(x,y)].
Therefore,Sα is a local star-shaped E-invex set with respect to η for any α∈R.Theorem 4.13.
Letf be a real-valued function defined on a local star-shaped E-invex set K⊂Rn, then f is a semilocal E-preinvex function with respect to η if and only if for each pair of points x,y∈K (with a maximal positive number a(x,y)≤1 satisfying (3.1)), there exists a positive number b(x,y)≤a(x,y) such that
(4.22)f(E(y)+λη(E(x),E(y)))<λα+(1-λ)β,∀λ∈[0,b(x,y)],
whenever f(x)<α, f(y)<β.Proof.
Letx,y∈K and α,β∈R such that f(x)<α, f(y)<β. Due to the local star-shaped E-invexity of K, there is a maximal positive number a(x,y)≤1 such that
(4.23)E(y)+λη(E(x),E(y))∈K,∀λ∈[0,a(x,y)].
In addition, owing to the semilocal E-preinvexity of f, there is a positive number b(x,y)≤a(x,y) such that
(4.24)f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y)<λα+(1-λ)β,∀λ∈[0,b(x,y)].
Conversely, let(x,α)∈Gf, (y,β)∈Gf (see epigraph Gf in Theorem 4.11), then x,y∈K, f(x)≤α, and f(y)≤β. Hence, f(x)<α+ϵ and f(y)<β+ϵ hold for any ϵ>0. According to the hypothesis, for x,y∈K (with a positive number a(x,y)≤1 satisfying (3.1)), there exists a positive number b(x,y)≤a(x,y) such that
(4.25)f(E(y)+λη(E(x),E(y)))<λα+(1-λ)β+ϵ,∀λ∈[0,b(x,y)].
Let ϵ→0+, then
(4.26)f(E(y)+λη(E(x),E(y)))≤λα+(1-λ)β,∀λ∈[0,b(x,y)].
That is, (E(y)+λη(E(x),E(y)),λα+(1-λ)β)∈Gf, for all λ∈[0,b(x,y)].
Therefore,Gf is a local star-shaped E-invex set corresponding to Rn.
From Theorem4.11, it follows that f is semilocal E-preinvex on K with respect to η.


## 5. Nonlinear Programming

In this section, we discuss the optimality conditions and Mond-Weir type duality for nonlinear programming involving semilocalE-preinvex and related functions.We firstly consider the nonlinear programming problem without constraint as follows:(5.1)(P)min⁡f(x),x∈K,
where K is a local star-shaped E-invex set and the objective function f is a semilocal E-preinvex function on K with respect to η.Theorem 5.1.
The following statements hold for programming(P).(i)
The optimal solution setω for (P) is a local star-shaped E-invex set with respect to η.(ii)
Ifx0 is a local minimum for (P) and E(x0)=x0, then x0 is a global minimum for (P).(iii)
If the real-valued functionf is a strict semilocal E-preinvex function on K with respect to η, then the global optimal solution for (P) is unique.Proof.
(i) Assume thatx,y∈ω, then x,y∈K and f(x)=f(y). On account of K being a local star-shaped E-invex set with respect to η, there is a maximal positive number a(x,y)≤1 such that
(5.2)E(y)+λη(E(x),E(y))∈K,∀λ∈[0,a(x,y)].
Besides, due to semilocal E-preinvexity of f, there is a positive number b(x,y)≤a(x,y) such that
(5.3)f(E(y)+λη(E(x),E(y)))≤λf(x)+(1-λ)f(y)=f(x),∀λ∈[0,b(x,y)].
From the optimality of x, we have f(E(y)+λη(E(x),E(y)))≥f(x).
Hence,f(E(y)+λη(E(x),E(y)))=f(x), that is, E(y)+λη(E(x),E(y))∈ω, for all λ∈[0,b(x,y)].
This shows thatω is a local star-shaped E-invex set with respect to η.
(ii) Assume thatNϵ(x0) is a neighbourhood of x0 with radius ϵ>0, and f attains its local minimum at x0∈Nϵ(x0)∩K. For x,x0∈K, there is a maximal positive number a(x,x0)≤1 such that E(x0)+λη(E(x),E(x0))∈K, for all λ∈[0,a(x,x0)].
Moreover, there is a positive numberb(x,x0)≤a(x,x0) such that
(5.4)f(E(x0)+λη(E(x),E(x0)))≤λf(x)+(1-λ)f(x0),∀λ∈[0,b(x,x0)].
Owing to E(x0)=x0, so for sufficiently small λ>0,
(5.5)E(x0)+λη(E(x),E(x0))∈Nϵ(x0)∩K.
Thus, f(x0)≤f(E(x0)+λη(E(x),E(x0)))≤λf(x)+(1-λ)f(x0), namely, f(x0)≤f(x).
This means thatx0 is a global minimum for (P).
(iii) By contradiction, assume thatx1,x2∈K are two global optimal solutions for (P) and x1≠x2. For x1,x2∈K, there is a maximal positive number a(x1,x2)≤1 such that
(5.6)E(x2)+λη(E(x1),E(x2))∈K,∀λ∈[0,a(x1,x2)].
Furthermore, there is a positive number b(x1,x2)≤a(x1,x2) such that
(5.7)f(E(x2)+λη(E(x1),E(x2)))<λf(x1)+(1-λ)f(x2)=f(x1),∀λ∈[0,b(x1,x2)].
This contradicts the fact that x1 is a global optimal solution for (P).
Therefore, the global optimal solution for(P) is unique.Theorem 5.2.
Letu∈K and E(u)=u. If f is differentiable on K, then u is a minimum for programming (P) if and only if u satisfies the inequality ∇f(u)Tη(E(v),u)≥0, for all v∈K.Proof.
Assume thatu is a minimum for (P). Since K is a local star-shaped E-invex set with respect to η, for any v∈K, there is a maximal positive number a(u,v)≤1 such that
(5.8)E(u)+λη(E(v),E(u))∈K,∀λ∈(0,a(u,v)].
From the differentiability of f and E(u)=u, we get
(5.9)f(u)≤f(E(u)+λη(E(v),E(u)))=f(u)+λ∇f(u)Tη(E(v),u)+o(λ).
Dividing the inequality above by λ and letting λ→0+, we have
(5.10)∇f(u)Tη(E(v),u)≥0.
Conversely, if∇f(u)Tη(E(v),u)≥0, owing to semilocal E-preinvexity of f on K, there is a positive number b(u,v)≤a(u,v) such that
(5.11)f(E(u)+λη(E(v),E(u)))≤λf(v)+(1-λ)f(u),∀λ∈(0,b(u,v)],
which together with E(u)=u implies
(5.12)f(v)-f(u)≥f(u+λη(E(v),u))-f(u)λ.
Letting λ→0+ in the inequality above, we obtain
(5.13)f(v)-f(u)≥∇f(u)Tη(E(v),u)≥0.
This follows that u is a minimum for (P).Next, we consider the optimization problem with inequality constraint(5.14)(NP){minf(x)s.t.gi(x)≤0,i∈I,x∈K.Denote the feasible set of(NP) by K0={x∈K,gi(x)≤0,i∈I}, where I={1,2,…,m}, and K⊂Rn is an open local star-shaped E-invex set with respect to η.If the constraint functionsgi(x) (i∈I) are all semilocal E-preinvex on K with respect to the same map η, then, from Theorem 4.12 and Proposition 3.8, we can conclude that the feasible set K0 is a local star-shaped E-invex set with respect to η. Moreover, from Theorem 5.1, we can obtain the following theorem easily.Theorem 5.3.
Assume thatf,gi(i∈I) are all semilocal E-preinvex on K with respect to η, then(i)
K0 is a local star-shaped E-invex set with respect to η;(ii)
the optimal solution setω for (NP) is a local star-shaped E-invex set with respect to η;(iii)
ifx0 is a local minimum for (NP) and E(x0)=x0, then x0 is a global minimum for (NP);(iv)
if the real-valued functionf is a strict semilocal E-preinvex function on K with respect to η, then the global optimal solution for (NP) is unique.For convenience of discussion, we give the following notation:(5.15)g(x)=(gi(x),i∈I),gJ(x)=(gj(x),j∈J),whereJ⊂I.
For x*∈K0, denote I(x*)={i∈I:gi(x*)=0}, I¯(x*)=I∖I(x*).To discuss the necessary optimality conditions for the corresponding programming, we first give a lemma as follows.Lemma 5.4.
Letx* be a local optimal solution for (NP). Assume that gj is continuous at x* for any j∈I¯(x*), and f, gI(x*) possess the directional derivatives at x* along the direction η(E(x),x*) for each x∈K, then the system:
(5.16)f′(x*;η(E(x),x*))<0,gI(x*)′(x*;η(E(x),x*))<0
has no solution in K, where f′(x*;d) denotes the directional derivative of f at x* along the direction d and gI(x*)′(x*;d)=(gi′(x*;d),i∈I(x*)).The proof of this lemma is similar to the one of [10, Lemma 13].Theorem 5.5.
Letx* be a local optimal solution for (NP). Assume that functions gj are continuous at x* for any j∈I¯(x*), and f, g possess the directional derivatives with respect to η(E(x),x*) at x* for  each x∈K. If (f'(x*;η(E(x),x*)),gI(x*)′(x*;η(E(x),x*))) is a convex-like function and E(x*)=x*, then there are λ0*∈R, λ¯*∈Rm such that
(5.17)λ0*f′(x*;η(E(x),x*))+λ¯*Tg′(x*;η(E(x),x*))≥0,∀x∈K,(5.18)λ¯*Tg(x*)=0,g(x*)≤0,0≠(λ0*,λ¯*)≥0.
Additionally, if g is a semilocal E-preinvex function on K with respect to η and there is x̂∈K such that g(x̂)<0, then there exists λ*∈Rm such that
(5.19)f′(x*;η(E(x),x*))+λ*Tg′(x*;η(E(x),x*))≥0,∀x∈K,(5.20)λ*Tg(x*)=0,g(x*)≤0,λ*≥0.Proof.
Define vector functionψ(x)=(f′(x*;η(E(x),x*)),gI(x*)′(x*;η(E(x),x*))). Then ψ(x) is a convexlike function. By Lemma 5.4, the system φ(x)<0 has no solution in K. Thus, from Lemma 2.11, there are λ0*∈R,λ¯I(x*)*∈R|I(x*)| such that
(5.21)λ0*f′(x*;η(E(x),x*))+λ¯I(x*)*TgI(x*)′(x*;η(E(x),x*))≥0,∀x∈K,0≠(λ0*,λ¯I(x*)*)≥0.
Hence, by letting λ¯*=(λ¯I(x*)*,0I¯(x*)), we further have
(5.22)λ0*f′(x*;η(E(x),x*))+λ¯*Tg′(x*;η(E(x),x*))≥0,∀x∈K,λ¯*Tg(x*)=0,g(x*)≤0,0≠(λ0*,λ¯*)≥0.
Subsequently, we testify that λ0*≠0, and if this is not true, we get from above
(5.23)λ¯*Tg′(x*;η(E(x),x*))≥0,∀x∈K,λ¯*Tg(x*)=0,g(x*)≤0,0≠λ¯*≥0.
On account of g being semilocal E-preinvex on K with respect to η and E(x*)=x*, from Proposition 5.9(i) below, we have g(x̂)-g(x*)≥g′(x*;η(E(x̂),x*)).
Soλ¯*Tg(x̂)≥0. But this contradicts the fact that g(x̂)<0 and λ¯*>0.
Thus,λ0*>0. Dividing (5.17) and the first equality of (5.18) by λ0* and letting λ*=λ¯*∖λ0*, we know that (5.19) and (5.20) hold.
Consequently, the whole proof is finished.To discuss the sufficient optimality conditions for(NP), we further generalize the concept of semilocal E-preinvex function as follows.Definition 5.6.
A real-valued functionf defined on a local star-shaped E-invex set k⊂Rn is said to be quasisemilocal E-preinvex (with respect to η) if for all x,y∈K (with a maximal positive number a(x,y)≤1 satisfying (3.1)) satisfying f(x)≤f(y), there is a positive number b(x,y)≤a(x,y) such that
(5.24)f(E(y)+λη(E(x),E(y)))≤f(y),∀λ∈[0,b(x,y)].
The definition of quasi-semilocalE-preinvex of a vector function f:Rn→Rk is similar to the one for a vector semilocal E-preinvex function.Definition 5.7.
A real-valued functionf defined on a local star-shaped E-invex set K⊂Rn is said to be pseudosemilocal E-preinvex (with respect to η) if for all x,y∈K (with a maximal positive number a(x,y)≤1 satisfying (3.1)) satisfying f(x)<f(y), there are a positive number b(x,y)≤a(x,y) and a positive number c(x,y) such that
(5.25)f(E(y)+λη(E(x),E(y)))≤f(y)-λc(x,y),∀λ∈[0,b(x,y)].
The definition of pseudo-semilocalE-preinvex of a vector function f:Rn→Rk is similar to the one for a vector semilocal E-preinvex function.Remark 5.8.
Every semilocalE-preinvex function on a local star-shaped E-invex set K with respect to η is both a quasi-semilocal E-preinvex function and a pseudo-semilocal E-preinvex function.We now present one of their elementary properties.Proposition 5.9.
Letf be a real-valued function on a local star-shaped E-invex set K⊂Rn, and f possesses directional derivative with respect to the direction η(E(x),y) at y for all x,y∈K. If E(y)=y, then the following statements hold true:(i)
iff is semilocal E-preinvex on K with respect to η, then f′(y;η(E(x),y))≤f(x)-f(y),(ii)
iff is quasi-semilocal E-preinvex on K with respect to η, then f(x)≤f(y) implies that f′(y;η(E(x),y))≤0,(iii)
iff is pseudo-semilocal E-preinvex on K with respect to η, then f(x)<f(y) implies that f′(y;η(E(x),y))<0.The proof is obvious by using the related definitions and is omitted.Theorem 5.10.
Letx*∈K0 and E(x*)=x*. Suppose that f,g possess directional derivatives with respect to the direction η(E(x),x*) at x* for any x∈K, and assume that there is λ*∈Rm such that (5.19) and (5.20) hold. If f is pseudo-semilocal E-preinvex on K and gI(x*) is quasi-semilocal E-preinvex on K with respect to η, then x* is an optimal solution for (NP).Proof.
Due togI(x*)(x)≤gI(x*)(x*)=0, for all x∈K0, it follows from Proposition 5.9(ii) that gI(x*)′(x*,η(E(x),x*))≤0, for all x∈K0, which together with λI(x*)*≥0 implies
(5.26)λI(x*)*TgI(x*)′(x*,η(E(x),x*))≤0,∀x∈K0.
Moreover, using λ*Tg(x*)=0, g(x*)≤0, and λ*≥0, we get
(5.27)λ*Tg′(x*;η(E(x),x*))≤0,∀x∈K0.
Therefore, from (5.19), we have f′(x*;η(E(x),x*))≥0, for all x∈K0.
Thus, from Proposition5.9(iii), this implies f(x)≥f(x*), for all x∈K0.
That is,x* is an optimal solution for (NP).Theorem 5.11.
Letx*∈K0 and E(x*)=x*. Suppose that f,g possess directional derivatives with respect to the direction η(E(x),x*) at x* for any x∈K, and there is a λ*∈Rm such that (5.19) and (5.20) hold. If (f+λI(x*)*TgI(x*)) is pseudo-semilocal E-preinvex on K with respect to η, then x* is an optimal solution for (NP).Proof.
Consideringλ*Tg(x*)=0, g(x*)≤0, λ*≥0, and the given conditions, we have (f+λI(x*)*TgI(x*))′(x*;η(E(x),x*))=f′(x*;η(E(x),x*))+λ*Tg'(x*;η(E(x),x*))≥0, for all x∈K0. Hence, from Proposition 5.9(iii), we get
(5.28)(f+λI(x*)*TgI(x*))(x)≥(f+λI(x*)*TgI(x*))(x*),∀x∈K0.
The inequality above together with λI(x*)*TgI(x*)(x*)=0 follows:
(5.29)f(x)+λI(x*)*TgI(x*)(x)≥f(x*),∀x∈K0.
On account of λI(x*)*≥0 and gI(x*)(x)≤0, we obtain
(5.30)f(x)≥f(x*),∀x∈K0.
Therefore, x* is an optimal solution for (NP).The following conclusion is a direct corollary of Theorem5.10 or Theorem 5.11.Corollary 5.12.
Letx*∈K0 and E(x*)=x*. Suppose that f,g possess directional derivatives with respect to the direction η(E(x),x*) at x* for any x∈K, and assume that there is a λ*∈Rm such that (5.19) and (5.20) hold. If f and gI(x*) are semilocal E-preinvex functions on K with respect to η, then x* is an optimal solution for (NP).Finally, we consider the following Mond-Weir type dual problem of(NP):(5.31)(DP){maxf(u)s.t.f′(u;η(E(x),u))+λTg′(u;η(E(x),u))≥0,∀x∈K,λTg(u)≥0,λ∈Rm,λ≥0,u∈K.Theorem 5.13 (weak duality).
Letx and (u,λ) be arbitrary feasible solutions of (NP) and (DP), respectively. If f and g are all semilocal E-preinvex functions on K with respect to η, and they possess directional derivatives with respect to the direction η(E(x),u) at u, where E(u)=u, x∈K, then f(x)≥f(u).Proof.
Consideringf and g being semilocal E-preinvex on K with respect to η and E(u)=u, we get from Proposition 5.9(i)
(5.32)f(x)≥f(u)+f′(u;η(E(x),u)),g(x)≥g(u)+g′(u;η(E(x),u)).
Combining the first constraint condition of (DP) and the inequalities above, we have
(5.33)f(x)≥f(u)-λTg′(u;η(E(x),u))≥f(u)+λT(g(u)-g(x)).
Hence, on account of λ≥0,g(x)≤0, and λTg(u)≥0, we obtain f(x)≥f(u).Theorem 5.14 (strong duality).
Assume thatx* is an optimal solution for (NP), E(x*)=x*, and E(u)=u for any feasible point (u,λ) of (DP). Suppose that f and g are semilocal E-preinvex on K with respect to η and gj is continuous at x* for any j∈I¯(x*), and they possess directional derivatives with respect to the direction η(E(x),x*) at x* and the direction η(x*,u) at u, respectively, where x∈K. Further, assume that there is x̂∈K such that g(x̂)<0. If f′(x*;η(E(x),x*)),gI(x*)′(x*;η(E(x),x*)) is a convex-like function, then there is a λ*∈Rm such that (x*,λ*) is an optimal solution for (DP).Proof.
From the assumptions and Theorem5.5, we can conclude that there is λ*≥0 such that (x*,λ*) is a feasible point for (DP). Assume that (u,λ) is a feasible solution of (DP). On account of f and g being semilocal E-preinvex on K with respect to η and E(u)=u, we get from Proposition 5.9(i)
(5.34)f(x*)-f(u)≥f′(u;η(x*,u)),g(x*)-g(u)≥g′(u;η(x*,u)).
Combining the first constraint condition of (DP) and the relationships above, we have
(5.35)f(x*)-f(u)≥-λTg′(u;η(x*,u))≥λT(g(u)-g(x*)).
Noticing that λ≥0, g(x*)≤0, and λTg(u)≥0, we know f(x*)≥f(u).
Therefore,(x*,λ*) is an optimal solution for (DP).Theorem 5.15 (converse duality).
Suppose thatx*∈K0 and (u¯,λ¯) is a feasible point for (DP). Further, suppose that f and g are semilocal E-preinvex on K with respect to η, and f, g possess directional derivatives with respect to the direction η(E(x),u¯) at u¯ for any x∈K. If f(x*)=f(u¯) and E(u¯)=u¯, then x* is an optimal solution for (NP).Proof.
Sincef and g are semilocal E-preinvex on K with respect to η and E(u¯)=u¯, we have from Proposition 5.9(i)
(5.36)f(x)-f(u¯)≥f′(u¯;η(E(x),u¯)),g(x)-g(u¯)≥g′(u¯;η(E(x),u¯)),∀x∈K0.
On account of (u¯,λ¯) being a feasible point for (DP), we get from the first constraint inequality of (DP) and the two relationships above
(5.37)f(x)-f(u¯)≥-λ¯Tg′(u¯;η(E(x),u¯))≥λ¯T(g(u¯)-g(x)).
This together with λ¯≥0, g(x)≤0, f(x*)=f(u¯), and λ¯Tg(u¯)≥0 follows that
(5.38)f(x)≥f(x*).
Thus, x* is an optimal solution for (NP).



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