# Semi-G-Preinvexity and Optimality in Mathematical Programming

Authors: Zai Yun Peng; Yong Zhao; Kai Zhi Yu; Zhi Lin
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We mainly discuss the properties and applications of semi-G-preinvexity, which was introduced in Peng and Chang 2013. Examples are given to show the existence of semi-G-preinvex functions. Three important properties and characterizations of semi-G-preinvex functions are established. Furthermore, under semi-G-preinvexity, some optimality results are obtained in nonlinear programming problems with inequality constraint. Our results improve and generalize some known results.


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## Body

## 1. Introduction

It is well known that the research on convexity and generalized convexity is one of the most important aspects in mathematical programming and optimization theory; see, for example, [1–5]. In fact, there are a number of nonlinear programming problems whose objective and constraints functions are nonconvex. Therefore, in the recent years, attempts are made by several authors to define various nonconvex classes of functions and to study their optimality criteria in solving such types of problems; see, for example, [6–14]. One of such a generalization of convex function is invexity notion introduced in [13] and called in [15]. Over the years, many generalizations of this concept have been given in the literature. In 1972, Avriel [16] introduced the definition of r-convex functions, which is a generalization of convex functions, and discussed some characterizations of r-convex functions. Later, Ben-Israel and Mond [17] considered a class of nondifferentiable functions that were called preinvex in [18] as a generalization of convexity and studied how and where preinvexity can replace convexity in multiobjective programming problems. Then, the concept of preinvexity was generalized to B-preinvexity [19] and semipreinvexity [20]. Using the definition of a weighted r-mean (where r is a real number) for a sequence of positive numbers, Antczak [21] introduced new classes of functions and called them (p,r)-preinvex with respect to η. Moreover, Antczak [22] considered a class of r-preinvex functions, which is a generalization of r-convex functions, and obtained some optimality results under appropriate r-preinvexity conditions for constrained optimization problems. In 2011, Zhao et al. [23] introduced the definition of r-semipreinvex functions and obtained some basic characterizations and optimality results in nonlinear programming.On the other hand, Avriel et al. [24] introduced the definition of G-convex functions, which is another generalization of convex functions, and obtained some characterizations and the relations between G-convexity and convexity, where G is a continuous real-valued increasing function. As a generalization of G-convexity and invexity, Antczak [25] introduced the concept of G-invexity and obtained some optimality conditions for constrained optimization problems under assumptions of G-invexity. Recently, Antczak [26] introduced the definition of G-preinvex functions, which is a generalization of r-preinvex functions with r>0 [22] with respect to the same η. Then, by using the G-preinvexity, some optimality results for constrained optimization problems were derived. Very recently, Peng and Chang [27] introduced a class of nonconvex functions, semi-G-preinvex functions, which is a generalization of G-preinvex functions [26], semipreinvex functions [20, 28], and r-semipreinvex functions with r>0 [23] with respect to the same η, and then they obtained some interesting properties of semi-G-preinvex functions.But, to the best of our knowledge, there are few optimality results in nonlinear programming (especially, in mathematical programming with constant) that have been concerned under semi-G-preinvex functions in the literature. So, in this paper, we mainly discuss the characterizations and applications in nonlinear programming of semi-G-preinvexity.The rest of the paper is organized as follows. In Section2, we recall some concepts and give some examples to show the existence of semi-G-preinvex functions. In Section 3, we establish three important properties and characterizations of semi-G-preinvex functions, which include that the ratio of semi-G-preinvex functions is semi-preinvex function. In Section 4, under semi-G-preinvexity, we obtain some optimality results in nonlinear programming problems with inequality constraint. Some examples are given for illustration of our results. Our results extend and generalize the corresponding ones in the literature [20, 26–28].


## 2. Preliminaries

Throughout this paper, letX be a nonempty subset of Rn. Let f:X→R be a real-valued function and η:X×X→Rn a vector-valued function. And let If(X) be the range of f, that is, the image of X under f, and let f-1 be the inverse of f.Now we recall some useful definitions.Definition 1 (see [24]).
LetG be a continuous real-valued strictly monotonic function defined on D⊂R, and f:X→R, where X is a nonempty convex set in Rn. f is said to be G-convex on X if and only if, for any x,y∈X, λ∈[0,1],
(1)f(y+λ(x-y))≤G-1(λG(f(x))+(1-λ)G(f(y))),
where G-1 is the inverse of G, f(X)⊂D.Remark 2.
All convex functions areG-convex functions with G(x)=x, but the converse is not necessarily true.Weir et al. [5, 18] presented the concepts of invex sets and preinvex functions which are generalization of convex sets and convex functions, respectively, as follows.Definition 3.
A setX is said to be invex if there exists a vector-valued function η:X×X→Rn such that
(2)x,y∈X,λ∈[0,1]⟹y+λη(x,y)∈X.Definition 4.
LetK⊆Rn be an invex set with respect to η:K×K→Rn and let f:K→R be a mapping. We say that f is preinvex if and only if
(3)f(y+λη(x,y))≤λf(x)+(1-λ)f(y),∀x,y∈K,λ∈[0,1].
Antczak presented the concepts ofG-invex functions [25] and G-preinvex functions [26] which are generalization of G-convex functions and preinvex functions, respectively, as follows.Definition 5.
LetX be a nonempty invex (with respect to η) subset of Rn and let f:X→R be a differentiable function defined on X. Further, we assume that there exists a differentiable real-valued increasing function G:If(X)→R. Then f is said to be G-invex at y on X with respect to η if there exists a vector-valued function η:X×X→Rn such that, for all x∈X,
(4)G(f(x))-G(f(y))≥G′(f(y))∇f(y)η(x,y).Definition 6.
LetX be a nonempty invex (with respect to η) subset of Rn; f:X→R is said to be (strictly) G-preinvex at y with respect to η if and only if there exist a continuous real-valued increasing function G:If(X)→R and a vector-valued function η:X×X→Rn such that, for all x∈X(x≠y) and any λ∈[0,1](λ∈(0,1)),
(5)f(y+λη(x,y))≤G-1(λG(f(x))+(1-λ)G(f(y)))(<).
If the inequality above is satisfied for any y∈X, then f is (strictly) G-preinvex on X with respect to η.Remark 7.
Everyr-preinvex function with respect to η with r>0 introduced in [22] is G-preinvex with respect to the same function η, where G:If(X)→R is defined by G(t)=ert. Every preinvex function with respect to η is G-preinvex function with respect to the same η, where G(x)=x.Definition 8 (see [20]).
X
⊆
R
n is said to be semiconnected set if there exists η:X×X×[0,1]→Rn such that, for any x,y∈X, λ∈[0,1], y+λη(x,y,λ)∈X.Definition 9 (see [20, 28]).
LetX⊆Rn be semiconnected set with respect to η:X×X×[0,1]→Rn; f:X→R is said to be semipreinvex with respect to η if and only if, for any x,y∈X, λ∈[0,1], lim⁡λ→0λη(x,y,λ)=0 and f(y+λη(x,y,η))≤λf(x)+(1-λ)f(y).In [27], Peng and Chang presented the definition of semi-G-preinvex functions as follows.Definition 10.
LetX⊆Rn be semiconnected set with respect to η:X×X×[0,1]→Rn; f:X→R is said to be (strictly) semi-G-preinvex at y with respect to η if and only if there exist a continuous real-valued increasing function G:If(X)→R and a vector-valued function η:X×X×[0,1]→Rn such that, for all x∈X(x≠y), λ∈[0,1](λ∈(0,1)), lim⁡λ→0λη(x,y,λ)=0 and
(6)f(y+λη(x,y,λ))≤G-1(λG(f(x))+(1-λ)G(f(y)))(<).
If the inequality above is satisfied for any y∈X, then f is (strictly) semi-G-preinvex on X with respect to η.Remark 11.
In order to define an analogous class of (strictly) semi-G-preincave functions with respect to η, the direction of the inequality in the definition of these functions should be changed to the opposite one.Remark 12.
Everyr-semipreinvex function [23] (r>0) with respect to η is semi-G-preinvex function with respect to the same function η, where G(t)=ert.Remark 13.
EveryG-preinvex function [26] with respect to η is semi-G-preinvex function with respect to the same function G, where η(x,y,λ)=η(x,y). However, the converse is not true.Remark 14.
EveryG-convex function [24] is semi-G-preinvex function with respect to the same function G, where η(x,y,λ)=x-y. However, the converse is not true.Remark 15.
Every semipreinvex function [20] with respect to η is semi-G-preinvex function with respect to the same η, where G(t)=t. However, the converse is not true.Example 16.
This example illustrates that a semi-G-preinvex function is not necessarily a semipreinvex function with respect to the same η.
LetX=R; it is easy to check that X is a semiconnected set with respect to η(x,y,λ) and lim⁡λ→0λη(x,y,λ)=0, where
(7)η(x,y,λ)={x-yλ,x<0,y<0,x>y,0<λ≤1;λ(x-y),x≥0,y≥0,x≥y;λ(x-y),x<0,y<0,x≤y;x-y,x≥0,y≥0,x<y;-x-y,x≥0,y<0,x<-y;-12x-y,x<0,y≥0,x>-y;-y,x≥0,y<0,x≥-y;0,x<0,y≥0,x≤-y.
Let f:X→R be defined by f(x)=ln⁡(|x|+5/2). From example 2.4 of [27], we know that f is a semi-G-preinvex function with respect to η, where G(t)=et, but it is not a semipreinvex function with respect to the same η.Now, we give another example that illustrates that semi-G-preinvex functions exist.Example 17.
LetX=(0,π/2); it is easy to check that X is a semiconnected set with respect to η(x,y,λ) and lim⁡λ→0λη(x,y,λ)=0, where
(8)η(x,y,λ)={λx-y,ifx≥y;x-y,ifx<y.
Let f:X→R be defined by f(x)=sinx.
From the definition of semi-G-preinvexity, it is not difficult to conclude that f is a semi-G-preinvex function with respect to η, where G(t)=arcsint.
In the sequel, we will use the following lemmas.Lemma 18.
LetG:If(X)→R be a continuous real-valued function; then G-1 is increasing if and only if G is increasing.Lemma 19.
LetG:If(X)→R be a continuous real-valued function. If G is increasing and concave, then G-1 is convex. If G is increasing and convex, then G-1 is concave.


## 3. Properties and Characterizations of  Semi-G-Preinvex Function

In this section, we establish three interesting properties for semi-G-preinvex function, which extend and generalize the corresponding ones in the literature [27, 28].Firstly, we show that the ratio of semi-G-preinvex functions is semi-G-preinvex, which extends earlier results by Craven and Mond [11] and Yang et al. [28]. Since the proof is obvious, it has been omitted in this paper.Theorem 20.
LetX⊆Rn be semiconnected set with respect to η and let f and g be real-valued functions defined on X. If f(x)≥0, g(x)>0, f(x) is a semi-G1-preinvex function, -g(x) is a semi-G2-preinvex function with respect to η on X, where G1 is concave and G2 is convex. Then,(i)
f(x)/g(x) is a semipreinvex function with respect to η¯(x,y,λ)=(g(y)/(λg(y)+(1-λ)g(x)))η(x,y,λ);(ii)
iflim⁡λ→0η-(x,y,λ)=η*(x,y), we also have f(x)/g(x) be an invex function with respect to η*(x,y)=(g(y)/g(x))η(x,y).Theorem 21.
LetX⊆Rn be semi-connected set with respect to η(x,y,λ); f:X→R is a semi-G-preinvex function with respect to the same η if and only if the set F(f)={(x,u):x∈X,u∈R,f(x)<u} is semiconnected with respect to η′:F(f)×F(f)×[0,1]→Rn+1, where
(9)η′((y,v),(x,u),λ)={η(y,x,0),λ=0;(η(y,x,λ),G-1(λG(u)+(1-λ)G(v))-uλ),0<λ≤1,
for all (x,u),(y,v)∈F(f).Proof.
In a similar way to the proof of Theorem 3.5 of [27], with suitable modifications, we can obtain the conclusion.Theorem 22.
LetX⊆Rn+1 and f(x)=inf⁡{u:u∈R,(x,u)∈X}, for all x∈Rn; G is continuous real-valued increasing. If X is a semiconnected set with respect to η*:X×X×[0,1]→Rn+1,
(10)η*((y,v),(x,u),λ)={η(y,x,0),λ=0;(η(y,x,λ),G-1(λG(u)+(1-λ)G(v))-uλ),0<λ≤1,
for all (x,u),(y,v)∈X, where η:Rn×Rn×[0,1]→Rn. Then f:Rn→R is a semi-G-preinvex function with respect to η.Proof.
(i)  Let0<λ≤1, and let x,y∈Rn. Since X is a semiconnected set with respect to η*((y,v),(x,u),λ), then, for any (x,u),(y,v)∈X,
(11)(x,u)+λη*((y,v),(x,u),λ)∈X,∀λ∈(0,1].
From
(12)η*((y,v),(x,u),λ)=(η(y,x,λ),G-1(λG(u)+(1-λ)G(v))-uλ),
it follows that
(13)(x,u)+λη*((y,v),(x,u),λ)=(x+λη(y,x,λ),G-1(λG(u)+(1-λ)G(v)))∈X,sssssssssssssssssssssssssssssssssssssssssssssss∀λ∈(0,1].
Thus, by the definition of f, we have
(14)f(y+λη(x,y,λ))≤G-1(λG(f(x))+(1-λ)G(f(y))),ssssssssssssssssssssssssssssssssssssssssssssssssss∀λ∈(0,1].
Hence, f is semi-G-preinvex function with respect to η on Rn.
(ii)  Whenλ=0, the conclusion is correct obviously.


## 4. Semi-G-Preinvexity and Optimality in Nonlinear Programming

In this section, we consider nonlinear programming problems with inequality constraint and obtain some optimality results under semi-G-preinvexity. Meanwhile, two examples are given for illustration of the main results. Because semi-G-preinvexity is a generalization of G-(pre-)invexity and (semi-)preinvexity, some of our results generalize the corresponding results of [20, 26].We consider the mathematical programming problem with inequality constraint as follows:(P)minf(x)s.t.gi(x)≤0,i∈J={1,…,m},x∈X,
where X is a nonempty subset of Rn, f,gi:X→R, i=1,2,…,m. Denote the set of all feasible solutions for (P) by
(15)D∶={x∈X:gi(x)≤0,i∈J}.Theorem 23.
Let the set of all feasible solutionsD in problem (P) be semiconnected set with respect to η. If f is a nonconstant semi-G-preincave function with respect to η on D, then no interior point of D is an optimal solution of (P), or equivalently, any optimal solution x- in problem (P), if it exists, must be a boundary point of D.Proof.
If problem(P) has no solution, the theorem is trivially true. Let x- be an optimal solution in problem (P). By assumption, f is nonconstant on D. Then, there exists a feasible point x~∈D such that
(16)f(x~)>f(x-).
Let z be an interior point of D. By assumption, D is semiconnected set with respect to η. Then there exists y∈D such that, for some λ∈[0,1],
(17)z=x~+λη(y,x~,λ).
Since f is a semi-G-preincave function with respect to η at any optimal solution x- on D, then
(18)f(z)=f(x~+λη(y,x~,λ))≥G-1(λG(f(y))+(1-λ)G(f(x~)))>G-1(λG(f(x-))+(1-λ)G(f(x-)))=f(x-).
From the inequality above, we know that f does not attain its minimum at an interior point z. This completes the proof of the theorem.LetUδ(x-) denote a neighborhood of x- of radius δ.Theorem 24.
Let the set of all feasible solutionsD in problem (P) be semiconnected set with respect to η and let x- be a local minimum in problem (P). Suppose that for every δ>0 and for every x∈D there exists λ~∈(0,1) such that x-+λ~η(x,x-,λ~)∈Uδ(x-). If f is strictly semi-G-preinvex with respect to η at x- on D, then x- is a strictly global minimum in (P).Proof.
By assumption,f is strictly semi-G-preinvex with respect to η at x- on D. Then
(19)f(x-+λη(x,x-,λ))<G-1(λG(f(x))+(1-λ)G(f(x-)))
holds for all x∈D(x≠x-) and any λ∈(0,1). Since x- is a local minimum in problem (P), then there exists Uδ(x-) such that the inequality
(20)f(x-)≤f(x)
holds for all x∈Uδ(x-)∩D.
Now, letx∈D, such that x≠x-. Since D is semiconnected set with respect to η, then
(21)x-+λη(x,x-,λ)∈D
for all x∈D and any λ∈(0,1). By assumption, for every δ>0 and for every x∈D, there exists λ~∈(0,1) such that x-+λ~η(x,x-,λ~)∈Uδ(x-). From (20), there exists λ~∈(0,1) such that
(22)f(x-+λ~η(x,x-,λ~))≥f(x-).
By virtue of (19) and (22), we obtain
(23)f(x-)≤f(x-+λ~η(x,x-,λ~))<G-1(λ~G(f(x))+(1-λ~)G(f(x-)))≤G-1(λ~G(max⁡{f(x),f(x-)})ssssssss+(1-λ~)G(max⁡{f(x),f(x-)}))=G-1(G(max⁡{f(x),f(x-)}))=max⁡{f(x),f(x-)}.
If max⁡{f(x),f(x-)}=f(x-), from (23), we can get f(x-)<f(x-), which is impossible. Thus, max⁡{f(x),f(x-)}=f(x); it follows from (23) that the following inequality
(24)f(x)>f(x-)
holds. By the arbitrariness of x, we get the conclusion of the theorem; that is, x- is a strictly global minimum in (P).Theorem 25.
Let the objective functionf be the semi-G-preinvex with respect to η on D, and let the constraint functions gi(i∈J) be semi-Gi-preinvex with respect to the same function η on D. Then, the set of all optimal solutions A is semiconnected set with respect to the same η.Proof.
Letx1 and x2 be optimal solutions in (P) such that x1≠x2. Then f(x1)=f(x2)=min⁡x∈Df(x). By assumption, f is semi-G-preinvex with respect to η on D. Since x1,x2∈D, from Definition 10, we have
(25)f(x2+λη(x1,x2,λ))≤G-1(λG(f(x1))+(1-λ)G(f(x2))).
From f(x1)=f(x2), one has
(26)f(x2+λη(x1,x2,λ))≤G-1(λG(f(x1))+(1-λ)G(f(x2)))=f(x1)=f(x2).
To prove that x2+λη(x1,x2,λ)∈A for any λ∈[0,1], it is sufficient to show that x2+λη(x1,x2,λ)∈D for any λ∈[0,1]. By assumption, gi,i∈J, are semi-Gi-preinvex with respect to the same function η on D. Therefore, for i∈J,
(27)gi(x2+λη(x1,x2,λ))≤Gi-1(λGi(gi(x1))+(1-λ)Gi(gi(x2))).
From the definition of semi-G-preinvexity, G is an increasing function on its domain. Using x1,x2∈D together with Lemma 18, for any λ∈[0,1],
(28)Gi-1(λGi(gi(x1))+(1-λ)Gi(gi(x2)))≤Gi-1(λGi(0)+(1-λ)Gi(0))=Gi-1(Gi(0))=0.
Thus, for any λ∈[0,1],
(29)gi(x2+λη(x1,x2,λ))≤0.
So the set A of all optimal solutions for (P) is semiconnected set with respect to η.Now, we illustrate the theorem above by a nonconvex optimization problem.Example 26.
We consider the following optimization problem:(30)f(x)={ln⁡(3-|x|),|x|≤20,|x|>2⟶min⁡,g(x)=arctan(32-|x|)≤0.
Note that the set of all feasible solutions D={x∈R:-∞<x≤-3/2or3/2≤x<+∞} and the set of all optimal solutions A={x∈R:-∞<x≤-2or2≤x<+∞}.
It can be proved by Definition10 that all functions constituting the considered optimization problem are semi-G-preinvex on R with respect to the same function η, where
(31)η(x,y,λ)={λ(x-y),x≥0,y≥0,x≤y,x-yλ,x≤0,y≤0,x≤y,0<λ≤1,λ(x-y),x≤0,y≤0,x>y,x-y,x≥0,y≥0,x>y,-x-y,xy<0.
Indeed, the objective functionf is semi-G1-preinvex on R with respect to G1(t)=exp⁡(t)-1 and the constraint function g is semi-G2-preinvex on R with respect to G2(t)=tant. Since all hypotheses of Theorem 25 are fulfilled, then the set of all optimal solutions A in the considered optimization problem is semiconnected with respect to the function η.
Indeed, it is not difficult to show by Definition8 that the set of all optimal solutions A in the considered optimization problem is semiconnected with respect to the same η. However, the set of all optimal solutions A is not convex.Theorem 27.
Lety∈D be optimal in (P). Moreover, suppose that f is strictly semi-G-preinvex with respect to η at y on D and the constraint functions gi,i∈J, are semi-Gi-preinvex with respect to the same function η on D. Then, y is a unique optimal solution in problem (P).Proof.
By contradiction, we assume that there existsx-∈D(x-≠y), being another optimal solution in problem (P). By assumption, the constraint functions gi,i∈J, are semi-Gi-preinvex with respect to the same function η on D. It follows from Definition 10 and x-,y∈D, for all i∈J and any λ∈[0,1]; we have
(32)gi(y+λη(x-,y,λ))≤Gi-1(λGi(gi(x-))+(1-λ)Gi(gi(y)))≤Gi-1(λGi(0)+(1-λ)Gi(0))=0.
Then, for all i∈J and any λ∈[0,1], one has
(33)y+λη(x-,y,λ)∈D.
Since f is strictly semi-G-preinvex with respect to η at y on D, from Definition 10, the following inequality
(34)f(y+λη(x,y,λ))<G-1(λG(f(x))+(1-λ)G(f(y)))
holds for any λ∈(0,1) and all x∈D(x≠y). So, we can get
(35)f(y+λη(x-,y,λ))<G-1(λG(f(x-))+(1-λ)G(f(y)))=G-1(G(f(y)))=f(y),
for x=x-.
Sincey+λη(x-,y,λ)∈D, we obtain that the inequality above (35) is a contradiction to the optimality of y in problem (P). The proof is complete.Now, we give an example to illustrate Theorem27 as follows.Example 28.
Consider the following optimization problem:(36)f(x)=exp⁡(arctan|3x|)⟶min,g(x)=arctan(|x|-2)≤0.
Note that the set of all feasible solutions D=(x∈R:-2≤x≤2) and the feasible solution x-=0 are optimal in the considered optimization problem.
From Definition10, we can verify that the objective function f is strictly semi-G-preinvex on R with respect to η and G1(t)=tan(ln⁡t) and the constraint function g is semi-G2-preinvex on R with respect to η and G2=tant, where the function η is defined by
(37)η(x,y,λ)={λx-yλ,x≥0,y≥0,x<y,0<λ≤1;x2-yλ,x<0,y<0,x>y,0<λ≤1;-y,x<0,y<0,x≤y;x2-y,x≥0,y≥0,x≥y;x3-yλ,x≥0,y<0,x<-y,0<λ≤1;x3-yλ,x<0,y≥0,x>-y,0<λ≤1;-y,x≥0,y<0,x≥-y;-y,x<0,y≥0,x≤-y.
All assumptions of Theorem 27 are satisfied; by virtue of Theorem 27, the optimal solution x-=0 is unique in the considered optimization problem. While, in Example 26, we notice that the objective function f is only semi-G1-preinvex on D, but it is not strictly semi-G1-preinvex. Therefore, the considered nonconvex optimization problem has no unique optimal solution; what is more, the set of all optimal solutions is not bounded in Example 26.Remark 29.
From Theorems23–25, Theorem 27, and Examples 26 and 28, we can know that this class of semi-G-preinvexity plays an important role in optimization theory and applications.



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