R E S E A R C H
Open Access
On the strictly
G-preinvex function
Tai Yong Li
*and Gui Hua Hu
*Correspondence:
[email protected] Tianmu College, Zhejiang A&F University, Hangzhou, 311300, China
Abstract
In this paper, we consider the strictlyG-preinvex functions introduced by Antczak (J. Comput. Appl. Math. 217:212-226, 2008). The relationships between semistrictly
G-preinvex functions and strictlyG-preinvex functions,G-preinvex functions strictly
G-preinvex functions are investigated under mild assumptions. Our results improve and extend the existing ones in the literature.
MSC: 90C26
Keywords: G-preinvex functions; semistrictlyG-preinvex functions; strictly
G-preinvex functions; optimization
1 Introduction
Convexity and some generalizations of convexity play a crucial role in mathematical eco-nomics, engineering, management science, and optimization theory. Therefore, it is im-portant to consider wider classes of generalized convex functions and also to seek practi-cal criteria for convexity or generalized convexity (see [–] and the references therein). A significant generalization of convex functions is the introduction of preinvex functions, which is due to Weir and Mond []. Yang and Li [] presented some properties of prein-vex functions; in [] they introduced two new classes of generalized conprein-vex functions, called semistrictly preinvex functions and strictly preinvex functions. They established relationships between preinvex functions and semistrictly preinvex functions under a cer-tain set of conditions. Recently, Antczak [, ] introduced the concept ofG-preinvex func-tion (strictlyG-preinvex function), which includes the preinvex function (strictly prein-vex function) [] andr-preinvex function [] as special cases. Relations of thisG-preinvex function to preinvex functions and some properties of this class of functions were studied in []. Luo and Wu in [] introduced the semistrictlyG-preinvex functions and established relationships betweenG-preinvex functions and semistrictlyG-preinvex functions under a certain set of conditions.
In this paper, we investigate the relationships between semistrictlyG-preinvex functions and strictlyG-preinvex functions,G-preinvex functions and strictlyG-preinvex functions under mild assumptions. It is worth pointing out that the results obtained here improve and generalize the corresponding ones given in [, ].
The outline of the paper is as follows. In Section , we give some preliminaries. The main results of the paper are presented in Section . Section gives some conclusions.
2 Preliminaries
In this section, we describe some definitions of generalized convexity.
Definition .([]) For a given setK⊆Rnand a given functionη:Rn×Rn→Rn,Kis
said to be an invex set with respect toηiff
∀x,y∈K, ∀λ∈[, ] ⇒ y+λη(x,y)∈K.
Definition .([]) LetK⊆Rnbe an invex set with respect toη:Rn×Rn→Rn. The
functionf :K→Ris said to be preinvex onKiff,∀x,y∈K,∀λ∈[, ],
fy+λη(x,y)≤λf(x) + ( –λ)f(y).
Definition .([]) LetK⊆Rnbe an invex set with respect toη:K×K →Rn. The
functionf :K→Ris said to be semistrictly preinvex onKiff,∀x,y∈K,f(x)=f(y),∀λ∈
(, ),
fy+λη(x,y)<λf(x) + ( –λ)f(y).
Definition .([]) LetK ⊆Rn be an invex set with respect toη:K×K →Rn. The functionf :K→Ris said to be strictly preinvex onKiff,∀x,y∈K,x=y,∀λ∈(, ),
fy+λη(x,y)<λf(x) + ( –λ)f(y).
In [], the relationships between preinvex functions and strictly preinvex functions, semistrictly preinvex functions and strictly preinvex functions were discussed under the following condition.
Condition C([]) Letη:K×K→Rn. We say that the functionηsatisfies Condition C iff,∀x,y∈K,∀λ∈[, ],
ηy,y+λη(x,y)= –λη(x,y),
ηx,y+λη(x,y)= ( –λ)η(x,y).
LetIf(K) be the range off,i.e., the image ofKunderf, and letG–be the inverse ofG.
Definition .([, ]) LetK⊆Rnbe an invex set (with respect toη). The functionf :
K→Ris said to beG-preinvex onKiff there exist a continuous real-valued increasing functionG:If(K)→Rand a vector-valued functionη:K×K→Rnsuch that,∀x,y∈K,
∀λ∈[, ],
fy+λη(x,y)≤G–λGf(x)+ ( –λ)Gf(y).
We note that theG-preinvex function in Definition . reduces to the preinvex function in Definition . and ther-preinvex function in [] when settingG(t) =tandG(t) =ert,
r> , respectively.
functionG:If(K)→Rand a vector-valued functionη:K×K→Rnsuch that,∀x,y∈K,
f(x)=f(y),∀λ∈(, ),
fy+λη(x,y)<G–λGf(x)+ ( –λ)Gf(y).
Definition .([]) LetK⊆Rnbe an invex set (with respect toη). The functionf:K→R
is said to be strictlyG-preinvex onK iff there exist a continuous real-valued increasing functionG:If(K)→Rand a vector-valued functionη:K×K→Rnsuch that,∀x,y∈K,
x=y,∀λ∈(, ),
fy+λη(x,y)<G–λGf(x)+ ( –λ)Gf(y).
We also observe that the semistrictlyG-preinvex function in Definition . is a general-ization of the semistrictly preinvex function in Definition ., and the strictlyG-preinvex function in Definition . is a generalization of the strictly preinvex function in Defini-tion . when takingG(t) =t.
Example . This example illustrates that a G-preinvex function is not necessarily a strictlyG-preinvex function. Let
f(x) =ln|x|+ ,
η(x,y) =
x–y ifxy< , –x–y ifxy≥.
Thenf is aG-preinvex function onRwith respect toη, whereG(t) =et. But if we let x= ,y= ,λ=, we have
f(y) =f() =ln, f(x) =f() =
and
fy+λη(x,y)=f
=ln
=G–
=G–λGf(x)+ ( –λ)Gf(y).
So,f is not a strictlyG-preinvex function with respect toηonR.
Example . This example illustrates that a semistrictlyG-preinvex function is not nec-essarily a strictlyG-preinvex function. Let
f(x) =
ifx= ,
ifx= ,x∈[–, –]∪[–, ],
η(x,y) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x–y if –≤x≤, –≤y≤, x–y if –≤x≤–, –≤y≤–, – –y if –≤x≤, –≤y≤–, –y if –≤x≤–, –≤y≤,y= ,
x
Then it is not difficult to show thatfis a semistrictlyG-preinvex function onx∈[–, –]∪ [–, ] with respect toη, whereG(t) =t. However, letx= –=y= ,λ=, and we have
fy+λη(x,y)=f() = > =G–λGf(x)+ ( –λ)Gf(y).
Thus,f is not a strictlyG-preinvex function with respect to the sameηonx∈[–, –]∪ [–, ].
Example . This example illustrates that a strictlyG-preinvex function is not necessarily a strictly preinvex function. Let
f(x) = |x|+ ,
η(x,y) =
x–y ifxy< , –x–y ifxy≥.
Then it is not difficult to show thatf is a strictlyG-preinvex function onRwith respect toη, whereG(t) =lnt. But if we letx= –,y= ,λ=, we have
f(y) =f() =f(x) =f(–) =
and
fy+λη(x,y)=f() = >
=λf(x) + ( –λ)f(y).
So,f is not a strictly preinvex function with respect toηonR.
Remark . From Example . and Example ., we know that strictlyG-preinvex func-tions are different from semistrictlyG-preinvex functions andG-preinvex functions. By Example ., we know that the definition of strictlyG-preinvex functions is a true gener-alization of that of strictly preinvex functions.
3 Properties of strictlyG-preinvex functions
In [], the authors discussed the properties of G-preinvex functions and semistrictly G-preinvex functions. In this section, we derive some properties of strictly G-preinvex functions. We assume always that:
(i) K⊆Rnis an invex set with respect toη:K×K→Rn;
(ii) ηsatisfies Condition C;f is a real-valued function onK.
Theorem . Let f be a semistrictly G-preinvex function with respect toηon K.Ifx¯∈K is a local minimum to the problem of minimizing f(x)subject to x∈K;thenx is a global one¯ .
Proof Suppose thatx¯∈Kis a local minimum to the problem of minimizingf(x) subject tox∈K. Then there is an-neighborhoodN(x¯) aroundx¯such that
Ifx¯is not a global minimum off overK, then there existsx∗∈Ksuch that
fx∗<f(x¯).
By assumption,f :K→Ris a semistrictlyG-preinvex function with respect toηonK. Thus,
fx¯+ληx∗,x¯<G–λGfx∗+ ( –λ)Gf(x¯)<f(x¯)
for allλ∈(, ). For a sufficiently smallλ> , it follows that
¯
x+ληx∗,x¯∈K∩N(x¯),
which is a contradiction to (.). This completes the proof.
Remark . It is easy to see that strictG-preinvexity implies semistrictG-preinvexity, so by Theorem ., we have the following.
Corollary . Let f :K→R be a strictly G-preinvex function with respect toηon K.If ¯
x∈K is a local minimum to the problem of minimizing f(x)subject to x∈K,thenx is a¯ global one.
By Corollary ., we can conclude that strictlyG-preinvex functions constitute an im-portant class of generalized convex functions in mathematical programming.
Theorem . Let f be a G-preinvex function on K.For each pair x,y∈K,x=y,if there existsα∈(, )such that
fy+αη(x,y)<G–αGf(x)+ ( –α)Gf(y), (.)
then f is a strictly G-preinvex function on K.
Proof By contradiction: suppose that there existx,y∈K,x=y,λ∈(, ) such that
fy+λη(x,y)≥G–λGf(x)+ ( –λ)Gf(y). (.)
Denote
z=y+λη(x,y).
Sincef isG-preinvex, we have
f(z)≤G–λGf(x)+ ( –λ)Gf(y). (.)
The above two inequalities imply that
We note that the pairx,zand the pairz,yare both distinct under (.), there existsβ,β∈
(, ) such that
fz+βη(x,z)
<G–βG
f(x)+ ( –β)G
f(z), (.)
fy+βη(z,y)
<G–βG
f(z)+ ( –β)G
f(y). (.)
Denote
¯
x=z+βη(x,z), y¯=y+βη(z,y).
From Condition C
¯
x=z+βη(x,z) =y+λη(x,y) +βη
x,y+λη(x,y)
=y+λη(x,y) + ( –λ)βη(x,y)
=y+λ+ ( –λ)β
η(x,y),
¯
y=y+βη(z,y) =y+βη
y+λη(x,y),y
=y+βη
y+λη(x,y),y+λη(x,y) –λη(x,y)
=y+βη
y+λη(x,y),y+λη(x,y) +ηy,y+λη(x,y)
=y–βη
y,y+λη(x,y)
=y+λβη(x,y).
Letμ=λ+ ( –λ)β,μ=λβ,μ=μλ–μ–μ. It is easy to verify thatμ,μ,μ∈(, ). Again from Condition C,
¯
y+μη(x¯,y¯) =y+μη(x,y) +μη
y+μη(x,y),y+μη(x,y)
=y+μη(x,y) +μη
y+μη(x,y),y+μη(x,y) + (μ–μ)η(x,y)
=y+μη(x,y) +μη
y+μη(x,y),y+μη(x,y)
+
μ–μ
–μ
ηx,y+μη(x,y)
=y+μη(x,y) –
μ(μ–μ)
–μ
ηx,y+μη(x,y)
=y+μη(x,y) –μ(μ–μ)η(x,y)
=y+μ–μ(μ–μ)
η(x,y)
=y+λη(x,y)
=z.
Sincef isG-preinvex, we have
By the assumption,Gis an increasing function and hence, by Lemma of [],G–is also
increasing. Thus, from (.)-(.), we have
f(z)≤G–μGf(x¯)+ ( –μ)Gf(y¯)
<G–μβG
f(x)+ ( –β)G
f(z)
+ ( –μ)βG
f(z)+ ( –β)G
f(y)
=G–μβG
f(x)+μ( –β) + ( –μ)β
Gf(z)
+ ( –μ)( –β)G
f(y)
=G–μβ+λμ( –β) +λβ( –μ)
Gf(x)
+( –μ)( –β) + ( –λ)μ( –β) + ( –λ)β( –μ)
Gf(y)
=G–λGf(x)+ ( –λ)Gf(y),
where
μβ+λμ( –β) +λβ( –μ)
=μλ+β( –λ) –λβ
+λβ
=μλ+β( –λ) –μ
+μ
=μ(μ–μ) +μ
=λ,
( –μ)( –β) + ( –λ)μ( –β) + ( –λ)β( –μ)
= ( –μ)( –μ) +μ( –μ)
= –μ+μ(μ–μ)
= –λ,
which contradicts (.). This completes the proof.
Corollary . Let f be a G-preinvex function on K.If there existsα∈(, )such that,for each pair x,y∈K,x=y,
fy+αη(x,y)<G–αGf(x)+ ( –α)Gf(y),
then f is a strictly G-preinvex function on K.
Corollary . Let f be a preinvex function on K.For each pair x,y∈K,x=y,if there exists
α∈(, )such that
fy+αη(x,y)<αf(x) + ( –α)f(y),
Remark . Even Corollary . is a new result, it is an improvement of [, Theorem ].
Corollary .([]) Let f be a preinvex function on K.If there existsα∈(, )such that, for each pair x,y∈K,x=y,
fy+αη(x,y)<αf(x) + ( –α)f(y),
then f is a strictly preinvex function on K.
Theorem . f is a strictly G-preinvex function on K if and only if f is a semistrictly G-preinvex function on K and satisfies the following condition:there existsα∈(, )such that,for each pair x,y∈K,x=y,
fy+αη(x,y)<G–αGf(x)+ ( –α)Gf(y). (.)
Proof The necessity is obvious from Definition .. So we only prove the sufficiency. Since f is a semistrictlyG-preinvex function onK, we only show thatf(x) =f(y),x=yimplies that
fy+λη(x,y)<G–λGf(x)+ ( –λ)Gf(y)=f(x), ∀λ∈(, ).
Letx¯=y+αη(x,y). From (.) and for eachx,y∈K,f(x) =f(y),x=y, we have
f(x¯) =fy+αη(x,y)<G–αGf(x)+ ( –α)Gf(y)=f(x). (.)
For eachλ∈(, ), ifλ<α, takingμ=α–λα , thenμ∈(, ), and from Condition C, we have
¯
x+μη(y,x¯) =y+αη(x,y) +α–λ
α η
y,y+η(x,y)
=y+αη(x,y) + (λ–α)η(x,y)
=y+λη(x,y).
From the semistrictG-preinvexity off and (.), it follows that
fy+λη(x,y)=fx¯+μη(y,x¯)
<G–μGf(y)+ ( –μ)Gf(x¯)
≤G–μGf(y)+ ( –μ)Gf(x)
=f(x).
Ifλ>α, takingν=λ–α–α, thenν∈(, ), and from Condition C, we have
¯
x+νη(x,x¯) =y+αη(x,y) +λ–α –αη
x,y+αη(x,y)
=y+αη(x,y) + (λ–α)η(x,y)
From the semistrictG-preinvexity off and (.), it follows that
fy+λη(x,y)=fx¯+νη(x,x¯)
<G–νGf(x)+ ( –ν)Gf(x¯)
≤G–μGf(x)+ ( –μ)Gf(x)
=f(x).
This completes the proof.
Theorems .-. improve and generalize the corresponding ones given in [, ] from the strictly preinvex case to the strictlyG-preinvex case.
4 Conclusions
In this paper, we firstly obtain one property of strictlyG-preinvex functions, we consider the strictlyG-preinvex functions introduced by Antczak []. The relationships between semistrictlyG-preinvex functions and strictlyG-preinvex functions,G-preinvex functions and strictlyG-preinvex functions are investigated under weaker conditions. Our results improve and extend the existing ones in the literature.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main theorems are proved by TYL. Both authors drafted the manuscript, read and approved the final manuscript.
Acknowledgements
The research was supported by the National Natural Science Foundation of China Grant 10461007, the Foundation of Department of Education of Zhejiang Province under Grant Y201121204 and the Foundation of Tianmu College Grant TMYY1210. The authors would like to express their thanks to the referees for helpful suggestions.
Received: 11 February 2014 Accepted: 13 May 2014 Published:23 May 2014 References
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