R E S E A R C H
Open Access
On interval-valued invex mappings and
optimality conditions for interval-valued
optimization problems
Lifeng Li
1*, Sanyang Liu
2and Jianke Zhang
1*Correspondence:
1School of Science, Xi’an University
of Posts and Telecommunications, Xi’an, China
Full list of author information is available at the end of the article
Abstract
In this paper, we first introduce the concept of interval-valued invex mappings by using gH-differentiability and compare it with interval-valued weakly invex mappings. We can observe that interval-valued invex mappings are more general than
interval-valued weakly invex mappings. In addition, the sufficient optimality condition for interval-valued objective functions is derived under invexity.
Keywords: interval-valued optimization; interval-valued invex mappings; sufficiency
1 Introduction
Convexity plays a vital role in many aspects of mathematical programming including, for example, sufficient optimality conditions and duality theorems. In inequality constrained optimization, the Kuhn-Tucker conditions are sufficient for optimality if the functions in-volved are convex. However, application of the Kuhn-Tucker conditions as sufficient con-ditions for optimality is not restricted to convex problems, and various generalizations of convexity have been made in order to explore the extent of this applicability.
An invex function, introduced by Hanson [], is one of the generalized convex functions. He considered a differentiable functionf :Rn→Rfor which there exists a vector-valued functionη:Rn×Rn→Rnsuch that, for allx,y∈Rn, the inequality
f(x) –f(y)≥ ∇f(y)tη(x,y) ()
holds. Hanson [] proved that if, instead of the usual convexity conditions, the objective function and each of the constraints of a nonlinear constrained optimization problem are all invex for the sameη, then both the sufficiency of Kuhn-Tucker conditions and weak and strong Wolfe duality still hold. Later, Craven [] named functions satisfying () invex (with respect toη).
Ben-Israel and Mond [] considered the preinvex functionf with respect toη(not nec-essarily differentiable) for which there exists a vector-valued functionη:Rn×Rn→Rn such that, for allx,y∈Rn, the inequality
fy+λη(x,y)≤λf(x) + ( –λ)f(y) ()
holds. Moreover, they found that differentiable functions satisfying () satisfy (). Further properties and applications of preinvexity and its some generalizations for some more general problems were studied by Antczak [, ], Bectoret al.[], Mohan and Neogy [], Sunejaet al.[], and others.
However, the majority of real world optimization problems often involve data uncer-tainty or imprecision owing to measurement errors or some unexpected things. Interval-valued optimization [] is an important model to deal with the problems with data uncer-tainty. Many approaches to interval-valued optimization problems have been explored in considerable details (see, for example, [–]). Recently, Wu has extended the concept of convexity for a real-valued function to LU-convexity for an interval-valued function, then he has established the Kuhn-Tucker conditions [, ] for an optimization problem with an interval-valued objective function under the assumption of LU-convexity. In [], Wu studied the Kuhn-Tucker optimality conditions in multiobjective programming problems with an interval-valued objective function. Similar to the concept of non-dominated solu-tion in vector optimizasolu-tion problems, Wu has proposed a solusolu-tion concept in optimizasolu-tion problems with an interval-valued objective function based on a partial ordering on the set of all closed intervals. Then, the interval-valued Wolfe duality theory [] and Lagrangian duality theory [] for interval-valued optimization problems have been proposed. Wu [] studied the duality theory for interval-valued linear programming problems. Chalco-Canoet al.[] gave Kuhn-Tucker type optimality conditions, which are obtained using gH-derivative of interval-valued functions. Also, they discussed the relationship between the approach presented with other well-known approaches given by Wu []. However, these methods given by Chalco-Canoet al.[] cannot solve a kind of optimization prob-lems with interval-valued objective functions, which are not LU-convex but invex. For example, the interval-valued functions such asf(x) = [x– sinx,x–sinx+ ] are not
LU-convex but invex with respect to
η(x,y) =
sinx–siny
cosy
,sinx–siny cosy
t
(the concept of invex can be seen in Definition ). Zhanget al.[] proposed the Kuhn-Tucker optimality conditions for an optimization problem with an interval-valued objec-tive function under the assumptions of preinvexity and weak invexity. The definition of interval-valued invexity in this paper is more general than that of weak invexity given in [] (see Theorem and Example ).
This paper aims at extending the Kuhn-Tucker optimality conditions to nonconvex opti-mization problem. First, we extend the concept of invexity using gH-derivative of interval-valued functions. The concept of invexity by using gH-differentiability of interval-interval-valued functions is more general than the concept of invexity by using weak differentiability (see Theorem and Example ). Second, we present several properties of invex interval-valued functions. Finally, the Kuhn-Tucker optimality conditions are proposed for an interval-valued objective function under the assumptions of invexity.
2 Preliminaries
Definition LetA= [aL,aU] andB= [bL,bU] be inI. We define (i) A+B={a+b:a∈Aandb∈B}= [aL+bL,aU+bU]; (ii) –A={–a:a∈A}= [–aU, –aL];
(iii) A×B={ab:a∈Aandb∈B}= [minab,maxab], where
minab=min{aLbL,aLbU,aUbL,aUbU}andmaxab=max{aLbL,aLbU,aUbL,aUbU}.
Then it is easy to conclude that
A–B=A+ (–B) =aL–bU,aU–bL,
kA={ka:a∈A}=
[kaL,kaU] ifk≥,
|k|[–aU, –aL] ifk< ,
()
wherekis a real number.
Hausdorff metricbetween two closed intervalsAandBdefined as
dH(A,B) =max aL–bL,aU–bU.
Definition LetA= [aL,aU] andB= [bL,bU] inI. We writeABifaL≤bLandaU≤bU, A≺BifABandA=B,i.e., the following (a) or (a), or (a) is satisfied.
(a) aL<bLandaU≤bU; (a) aL≤bLandaU<bU; (a) aL<bLandaU<bU.
LetA,B∈I, if there existsC∈I such thatA=B+C, thenCis called the Hukuhara difference ofAandBand written asC=AB; when we say that the H-differenceC exists, it means thataL–bL≤aU–bUandC= [aL–bL,aU–bU].
Proposition Let A= [aL,aU]and B= [bL,bU]be two closed intervals inI.If aL–bL≤ aU–bU,then the H-difference C exists and C= [aL–bL,aU–bU].
It follows from Proposition that the H-difference is unique, but it does not always exist. To address this issue, a generalization of the Hukuhara difference is proposed in [].
Definition ([]) LetA= [aL,aU] andB= [bL,bU] be two closed intervals, the gH-differ-ence ofAandBis defined by
aL,aUg
bL,bU=minaL–bL,aU–bU,maxaL–bL,aU–bU.
For example, [, ]g[, ] = [, ], [, ]g[, ] = [–, ]. Anda–b= [a,a]g[b,b] = [a–b,a–b] =a–b.
Proposition ([])
(i) For every pairA,B∈I,AgBalways exists andAgB∈I. (ii) AgBif and only ifAB.
written asf(x) = [fL(x),fU(x)], wherefLandfUare two real-valued functions defined on Rnand satisfyfL(x)≤fU(x) for everyx∈Rn. Based on the above concept, Wu [] has introduced the concepts of limit, continuity and two kinds of differentiation of interval-valued functions.
Letf be an interval-valued function defined onRnandA= [aL,aU] be an interval inR,
c∈Rn. If for every> there existsδ> such that, for <x–c<δ, we havedH(f(x),A) < , then
lim
x→cf(x) =A.
Proposition ([]) Let f be an interval-valued function defined on Rnand A= [aL,aU]be an interval in R.Thenlimx→cf(x) =A if and only iflimx→cfL(x) =aLandlimx→cfU(x) =aU.
Proposition ([]) Let f be an interval-valued function defined on Rn.Then f is contin-uous at c∈Rnif and only if both fLand fUare continuous at c.
Proposition ([]) Let X be an open set in R.An interval-valued function f :X→Iwith f(x) = [fL(x),fU(x)]is calledweakly differentiableat x
if the real-valued functions fLand
fUare differentiable at x(in the usual sense).
Definition ([]) LetXbe an open set inR. An interval-valued functionf :X→Iis called H-differentiable atxif there exists a closed intervalA(x)∈Isuch that the limits
lim h→+
f(x+h)f(x)
h
and
lim h→+
f(x)f(x–h)
h
both exist and equalA(x). In this case,A(x) is called the H-derivative off atx.
The following concept is particularization of the fuzzy concepts presented in [] to the interval case. These are defined by using the usual Hukuhara difference.
Definition ([]) LetT= (a,b) and lett∈T. Givenf :T→I, we say thatfis strongly
generalized differentiable (G-differentiable) attif there exists an elementf(t)∈Isuch
that for allh> sufficiently small,
(i) ∃f(x+h)f(x),f(x)f(x–h)and
lim h→
f(x+h)f(x)
h =hlim→
f(x)f(x–h)
h =f
(x
),
or
(ii) ∃f(x)f(x+h),f(x–h)f(x)and
lim h→
f(x)f(x+h)
–h =hlim→
f(x–h)f(x)
–h =f
(x
or
(iii) ∃f(x+h)f(x),f(x–h)f(x)and
lim h→
f(x+h)f(x)
h =hlim→
f(x–h)f(x)
–h =f
(x
),
or
(iv) ∃f(x)f(x+h),f(x)f(x–h)and
lim h→
f(x)f(x+h)
–h =hlim→
f(x)f(x–h)
h =f
(x
).
Based on the gH-difference, Stefanini [] proposed the following differentiation.
Definition ([]) Letx∈(a,b) andhbe such thatx+h∈(a,b), then the gH-derivative
of a functionf: (a,b)→Iatxis defined as
f(x) =lim
h→
f(x+h)gf(x)
h . ()
If f(x)∈I satisfying () exists, we say that f is generalized Hukuhara differentiable
(gH-differentiable for short) atx.
The next two results express the gH-derivative in terms of the endpoints of the interval-valued function.
Theorem ([]) Let f : (a,b)→Ibe such that f(x) = [fL(x),fU(x)].If fL(x)and fU(x)are differentiable functions at t∈(a,b),then f(x)is gH-differentiable at t,and
f(x) =min fL(x),fU(x),max fL(x),fU(x).
Theorem ([]) Let f: (a,b)→Ibe such that f(x) = [fL(x),fU(x)].Then f(x)is gH-dif-ferentiable at t∈(a,b)if and only if one of the following cases holds:
(a) fL(x)andfU(x)are differentiable att;
(b) the lateral derivatives(fL)–(t),(fL)+(t)and(fU)–(t),(fU)+(t)exist and satisfy (fL)–(t) = (fU)+(t)and(fL)+(t) = (fU)–(t).
Letf be an interval-valued function defined onX⊆Rn, comparing above definitions, the following statements hold.
Proposition
(i) Iff is H-differentiable atx∈X,then it is G-differentiable atx,the converse is not
true.
(ii) Iff is G-differentiable atx∈X,then it is gH-differentiable atx,the converse is not
true.
(iii) Iff is weakly differentiable atx,then it is gH-differentiable atx,the converse is not
true.
Definition ([]) Letf(x) be an interval-valued function defined on, whereis an open subset ofRn. LetDx
to theith variablexi. Assume thatfL(x) andfU(x) have continuous partial derivatives so thatDxifL(x) andDxifU(x) are continuous. Fori= , , . . . ,n, define
Dxif(x) =
minDxif
L(x),Dx
if
U(x),maxDx
if
L(x),Dx
if
U(x),
we will say thatf(x) is differentiable atx, and we write
∇f(x) =Dxf(x),Dxf(x), . . . ,Dxnf(x)
t .
We call∇f(x) the gradient of the interval-valued function atx.
Example Letf :R→I be defined byf(x) = [x
+x, x+ x+ ]. SofL(x) =x+x
andfU(x) = x+ x+ .Dxf
L(x) = x
,Dxf
L(x) = x
,Dxf
U(x) = x
,Dxf
U(x) = x
.
Thus,
Dxf(x) =
[x, x] ifx≥,
[x, x] ifx< ,
()
Dxf(x) =
[x, x] ifx≥,
[x, x] ifx< .
()
Thus,
∇f(x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
([x, x], [x, x])t ifx≥,x≥,
([x, x], [x, x])t ifx≥,x< ,
([x, x], [x, x])t ifx< ,x≥,
([x, x], [x, x])t ifx< ,x< .
()
Further,
∇Lf(x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
(x, x)t ifx≥,x≥,
(x, x)t ifx≥,x< ,
(x, x)t ifx< ,x≥,
(x, x)t ifx< ,x< ,
()
∇Uf(x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
(x, x)t ifx≥,x≥,
(x, x)t ifx≥,x< ,
(x, x)t ifx< ,x≥,
(x, x)t ifx< ,x< .
()
3 Preinvexity and invexity of interval-valued functions
In what follows, we show the connection between preinvex and invex interval-valued mappings. Here, we recall the definition of preinvex interval-valued mappings.
Definition Lety∈X⊆Rn. Then we say thatXis invex atywith respect toη:X×X→ Rnif for eachx∈X,λ∈[, ],y+λη(x,y)∈X.Xis said to be an invex set with respect to ηifXis invex at eachy∈X.
Definition ([]) LetK⊆Rnbe an invex set with respect toη:K×K→Rn,f(x) = [fL(x),fU(x)] be an interval-valued function defined onK. We say thatf ispreinvexatx∗ with respect toηif
fx+ληx∗,xλfx∗+ ( –λ)f(x)
for eachλ∈[, ] and eachx∈K.
Theorem Let K be an invex subset of Rnwith respect toη:K×K→Rnand f be an interval-valued function defined on K.Then f is preinvex at x∗if and only if fLand fUare preinvex at x∗with respect to the sameη:K×K→Rn,i.e.,
fLx+ληx∗,x≤λfLx∗+ ( –λ)fL(x), ()
fUx+ληx∗,x≤λfUx∗+ ( –λ)fU(x) ()
for eachλ∈[, ]and each x∈K.
Definition ([]) LetK⊆Rnbe an invex set with respect toη:K×K→Rn,f(x) = [fL(x),fU(x)] be an interval-valued function defined onK. We say thatf isinvexatx∗ if
the real-valued functionsfLandfUare invex atx∗,i.e.,
fL(x) –fLx∗≥ηx,x∗t∇fLx∗, ()
fU(x) –fUx∗≥ηx,x∗t∇fUx∗ ()
for eachx∈K.
Remark Since the definition of interval-valued invex functions defined in [] consid-ered the end-point functions, we call them weakly invex functions in this paper.
Based on the gH-differentiability, we give the definition of interval-valued invex func-tions as follows.
Definition A gH-differentiable interval-valued mappingf :X→Iis said to be invex on the invex setX⊆Rnwith respect toηif for anyx,y∈X,λ∈[, ],
f(x)gf
x∗ηx,x∗t∇fx∗ ()
Example Consider the interval-valued mapping f(x) = [, ]x, x∈R. Then f(x) is
gH-differentiable onRby Theorem , and
∇f(x) =
[x, x], x≥, [x, x], x< .
Letη(x,y) =x–y, thus
η(x,y)t∇f(y) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
[y(x–y), y(x–y)], y≥,x–y≥, [y(x–y), y(x–y)], y< ,x–y< , [y(x–y), y(x–y)], y≥,x–y< , [y(x–y), y(x–y)], y< ,x–y≥,
()
f(x)gf(y) =
[x–y, x– y], x≥y,
[x– y,x–y], x<y. ()
We can observe thatf(x) is invex with respect toη(x,y) =x–yby Definition .
The following theorem shows the relationship between interval-valued invex functions and weakly invex functions.
Theorem Let K be an invex subset of Rn with respect toη:K×K→Rnand f(x) = [fL(x),fU(x)]be an interval-valued function defined on K.If f is weakly invex,then it is invex,but the converse is not true in general.
Proof Sincef is weakly invex atx∗, we have that real-valued functionsfLandfUare invex atx∗,i.e.,
fL(x) –fLx∗≥ηx,x∗t∇fLx∗, ()
fU(x) –fUx∗≥ηx,x∗t∇fUx∗ ()
for eachλ∈[, ] and eachx∈K.
() On the condition ofη(x,x∗)t∇fL(x∗)≤η(x,x∗)t∇fU(x∗), we have
ηx,x∗t∇fx∗=ηx,x∗t∇fLx∗,ηx,x∗t∇fUx∗.
Iff(x)gf(x∗) = [fL(x) –fL(x∗),fU(x) –fU(x∗)], then from () and () we have
f(x)gf
x∗ηx,x∗t∇fx∗.
Iff(x)gf(x∗) = [fU(x) –fU(x∗),fL(x) –fL(x∗)], then
fL(x) –fLx∗≥fU(x) –fUx∗≥ηx,x∗t∇fUx∗≥ηx,x∗t∇fLx∗.
We have
f(x)gf
() On the condition ofη(x,x∗)t∇fL(x∗) >η(x,x∗)t∇fU(x∗), we have
ηx,x∗t∇fx∗=ηx,x∗t∇fUx∗,ηx,x∗t∇fLx∗.
Iff(x)gf(x∗) = [fU(x) –fU(x∗),fL(x) –fL(x∗)], then from () and () we have
f(x)gf
x∗ηx,x∗t∇fx∗.
Iff(x)gf(x∗) = [fL(x) –fL(x∗),fU(x) –fU(x∗)], then
fU(x) –fUx∗≥fL(x) –fLx∗≥ηx,x∗t∇fLx∗≥ηx,x∗t∇fUx∗.
Thus
f(x)gf
x∗ηx,x∗t∇fx∗.
Example Considering the interval-valued functionf(x) = [x– sinx,x–sinx+ ],
x∈R, we can prove that bothfL(x) andfU(x) are weakly invex with respect to
η(x,y) =
sinx–siny
cosy
,sinx–siny cosy
t .
Thenf(x) = [x– sinx,x–sinx+ ],x∈Ris invex with respect to the sameη(x,y) by
Theorem .
Example Considering the interval-valued functionf(x) = [–|x|,|x|],x∈R,
η(x,y) =
x–y, xy≥, x+y, xy< .
From Theorem , it follows thatf(x) is gH-differentiable onR, and∇f(y) = [–, ], thus
η(x,y)t∇f(y) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
[–(x–y),x–y], x≥y≥ ory≤x≤, [x–y, –(x–y)], y≥x≥ orx≤y≤, [–(x+y),x+y], xy< andx+y> , [x+y, –(x+y)], xy< andx+y< ,
f(x)gf(y) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
[–(x–y),x–y], x≥y≥ ory≤x≤, [x–y, –(x–y)], y≥x≥ orx≤y≤, [–(x+y),x+y], x> >yandx+y> , [x+y, –(x+y)], x> >yandx+y< , [–(x+y),x+y], x< <y.
Then
f(x)gf(y)η(x,y)∇f(y)t.
Letx= ,y= –,η(x,y) = ,
fLy+λη(x,y)= –λ> – +λ=λfL(y) + ( –λ)fL(x).
Thus,f(x) is not preinvex sincefL(x) is not preinvex with respect toη.
The following theorem given in [] illustrates the relations between weakly invex interval-valued and preinvex interval-valued functions.
Theorem Let K ⊆ Rn be an invex set with respect to η: K × K →Rn; if f(x) = [fL(x),fU(x)]is a weakly continuously differentiable and preinvex interval-valued func-tion defined on K,then f is also a weakly invex interval-valued function with respect to the sameηdefined on K.
We can prove the following result.
Theorem Let K ⊆ Rn be an invex set with respect to η: K × K → Rn. If f(x) = [fL(x),fU(x)]is a weakly differentiable and preinvex interval-valued function defined on K,then f is also an interval-valued invex function with respect to the sameηdefined on K.
Proof Sincef is interval-valued preinvex and weakly differentiable, we have
fLy+λη(x,y)–fL(y)≤λfL(x) –fL(y),
fUy+λη(x,y)–fU(y)≤λfU(x) –fU(y),
which forλ∈(, ] implies
fL(y+λη(x,y)) –fL(y)
λ ≤f
L(x) –fL(y),
fU(y+λη(x,y)) –fU(y)
λ ≤f
U(x) –fU(y).
By taking limits forλ→+, sincef is weakly differentiable, we get
η(x,y)t∇fL(y)≤fL(x) –fL(y), ()
η(x,y)t∇fU(y)≤fU(x) –fU(y). ()
On the other hand, from Definition and Theorem , we have
f(x)gf(y) =
[fL(x) –fL(y),fU(x) –fU(y)], fL(x) –fL(y)≤fU(x) –fU(y),
[fU(x) –fU(y),fL(x) –fL(y)], fL(x) –fL(y) >fU(x) –fU(y) ()
and
η(x,y)t∇f(y)
=η(x,y)tmin ∇fL(y),∇fU(y),max ∇fL(y),∇fU(y) ()
=
[η(x,y)t∇fL(y),η(x,y)t∇fU(y)], η(x,y)t∇fL(y)≤η(x,y)t∇fU(y),
From ()-() it follows that
f(x)gf(y)η(x,y)t∇f(y).
Thus,f is an invex interval-valued function.
The following example given in [] shows that a weakly invex interval-valued function may not be a preinvex interval-valued function, from Theorem we can conclude that the converse of Theorem is not true.
Example The interval-valued functionf(x) = [, ]·ex,x∈Ris invex with respect to η= –, but not preinvex with respect to the same functionη.
However, Mohan and Neogy [] have proved that a differentiable invex real-valued func-tion is also preinvex under the following condifunc-tion.
Condition C We say that the functionη:Rn→Rnsatisfies Condition C if for anyx,y∈X,
ηx,y+λη(x,y)= –λη(x,y), ηy,y+λη(x,y)= ( –λ)η(x,y).
We can conclude that a continuously weakly differentiable invex interval-valued func-tionf :K→Iis also a preinvex interval-valued function onKif the functionηsatisfies Condition C.
Theorem Suppose that K is an invex set of Rn with respect toη:K×K→Rnand f :K→I is a continuously weakly differentiable interval-valued function on an open set containing K. If f is invex on K with respect toηandηsatisfies ConditionC,then f is preinvex with respect toηon K.
Proof Suppose thatx,x∈X. Let <λ< be given and look atx=x+λη(x,x).
Note thatx∈X, by invexity of˜f, we have
f(x)gf(x)η(x,x)t∇f(x)
and
f(x)gf(x)η(x,x)t∇f(x),
i.e.,
min fL(x) –fL(x),fU(x) –fU(x)
,max fL(x) –fL(x),fU(x) –fU(x)
η(x,x)t
min ∇fL(x),∇fU(x),max ∇fL(x),∇fU(x)
and
min fL(x) –fL(x),fU(x) –fU(x)
,max fL(x) –fL(x),fU(x) –fU(x)
η(x,x)t
() On the condition offL(x
) –fL(x)≤fU(x) –fU(x),η(x,x)t∇fL(x)≤η(x,x)t∇fU(x),
fL(x
) –fL(x)≤fU(x) –fU(x), andη(x,x)t∇fL(x)≤η(x,x)t∇fU(x), we have
f(x)gf(x) =
fL(x) –fL(x),fU(x) –fU(x)
,
f(x)gf(x) =
fL(x) –fL(x),fU(x) –fU(x)
and
η(x,x)t∇f(x) =
η(x,x)t∇fL(x),η(x,x)t∇fU(x)
,
η(x,x)t∇f(x) =
η(x,x)t∇fL(x),η(x,x)t∇fU(x)
.
Thus,
fL(x) –fL(x)≥η(x,x)t∇fL(x),
fU(x) –fU(x)≥η(x,x)t∇fU(x)
and
fL(x) –fL(x)≥η(x,x)t∇fL(x),
fU(x) –fU(x)≥η(x,x)t∇fU(x).
Therefore,
λfL(x) + ( –λ)fL(x) –fL(x)≥
λη(x,x) + ( –λ)η(x,x)
t
∇fL(x).
However, by Condition C,λη(x,x) + ( –λ)η(x,x) = . Hence,
fLx+λη(x,x)
≤λfL(x) + ( –λ)fL(x).
By a similar way,
fUx+λη(x,x)
≤λfU(x) + ( –λ)fU(x).
Thusf is preinvex with respect toηby Theorem . () On the condition offL(x
) –fL(x)≤fU(x) –fU(x),η(x,x)t∇fL(x)≤η(x,x)t∇fU(x),
fL(x
) –fL(x)≤fU(x) –fU(x), andη(x,x)t∇fL(x) >η(x,x)t∇fU(x), we have
f(x)gf(x) =
fL(x) –fL(x),fU(x) –fU(x)
,
f(x)gf(x) =
fL(x) –fL(x),fU(x) –fU(x)
and
η(x,x)t∇f(x) =
η(x,x)t∇fL(x),η(x,x)t∇fU(x)
,
η(x,x)t∇f(x) =
η(x,x)t∇fU(x),η(xx)t∇fL(x)
Thus,
fL(x) –fL(x)≥η(x,x)t∇fL(x),
fU(x) –fU(x)≥η(x,x)t∇fU(x)
and
fL(x) –fL(x)≥η(x,x)t∇fU(x),
fU(x) –fU(x)≥η(x,x)t∇fL(x).
Supposemin{η(x,x)t∇fL(x),η(x,x)tfU(x)}=η(x,x)t∇fL(x). Therefore,
λfL(x) + ( –λ)fL(x) –fL(x)≥λη(x,x)t∇fL(x) + ( –λ)η(x,x)t∇fU(x)
≥λη(x,x)t∇fL(x) + ( –λ)η(x,x)t∇fL(x)
=λη(x,x) + ( –λ)η(x,x)
t
∇fL(x).
However, by Condition C,λη(x,x) + ( –λ)η(x,x) = . Hence,
fLx+λη(x,x)
≤λfL(x) + ( –λ)fL(x).
By a similar way,
fUx+λη(x,x)
≤λfU(x) + ( –λ)fU(x).
Thusf is preinvex with respect toηby Theorem .
() On the condition offL(x) –fL(x)≤fU(x) –fU(x),η(x,x)t∇fL(x)≤η(x,x)t∇fU(x),
fL(x) –fL(x) >fU(x) –fU(x), andη(x,x)t∇fL(x)≤η(x,x)t∇fU(x), we have
f(x)gf(x) =
fL(x) –fL(x),fU(x) –fU(x)
,
f(x)gf(x) =
fU(x) –fU(x),fL(x) –fL(x)
and
η(x,x)t∇f(x) =
η(x,x)t∇fL(x),η(x,x)t∇fU(x)
,
η(x,x)t∇f(x) =
η(x,x)t∇fL(x),η(x,x)t∇fU(x)
.
Thus,
fL(x) –fL(x)≥η(x,x)t∇fL(x),
fU(x) –fU(x)≥η(x,x)t∇fU(x)
and
fL(x) –fL(x)≥η(x,x)t∇fU(x),
Supposemin{η(x,x)t∇fL(x),η(x,x)t∇fU(x)}=η(x,x)t∇fL(x). Thus,
λfL(x) + ( –λ)fL(x) –fL(x)≥λη(x,x)t∇fU(x) + ( –λ)η(x,x)t∇fL(x)
≥λη(x,x)t∇fL(x) + ( –λ)η(x,x)t∇fL(x)
=λη(x,x) + ( –λ)η(x,x)
t
∇fL(x).
However, by Condition C,λη(x,x) + ( –λ)η(x,x) = . Hence,
fLx+λη(x,x)
≤λfL(x) + ( –λ)fL(x).
By a similar way,
fUx+λη(x,x)
≤λfU(x) + ( –λ)fU(x).
Thusf is preinvex with respect toηby Theorem .
() On the condition offL(x) –fL(x)≤fU(x) –fU(x),η(x,x)t∇fL(x)≤η(x,x)t∇fU(x),
fL(x
) –fL(x) >fU(x) –fU(x), andη(x,x)t∇fL(x) >η(x,x)t∇fU(x), we have
f(x)gf(x) =
fL(x) –fL(x),fU(x) –fU(x)
,
f(x)gf(x) =
fU(x) –fU(x),fL(x) –fL(x)
and
η(x,x)t∇f(x) =
η(x,x)t∇fL(x),η(x,x)t∇fU(x)
,
η(x,x)t∇f(x) =
η(x,x)t∇fU(x),η(x,x)t∇fL(x)
.
Thus,
fL(x) –fL(x)≥η(x,x)t∇fL(x),
fU(x) –fU(x)≥η(x,x)t∇fU(x)
and
fL(x) –fL(x)≥η(x,x)t∇fL(x),
fU(x) –fU(x)≥η(x,x)t∇fU(x).
Thus,
λfL(x) + ( –λ)fL(x) –fL(x)≥
λη(x,x) + ( –λ)η(x,x)
t
∇fL(x).
However, by Condition C,λη(x,x) + ( –λ)η(x,x) = . Hence,
fLx+λη(x,x)
By a similar way,
fUx+λη(x,x)
≤λfU(x) + ( –λ)fU(x).
As a consequence,f is preinvex with respect toηby Theorem .
We can prove the result on the other four conditions by a similar way.
4 The Kuhn-Tucker optimality conditions with interval-valued objective functions
An interval-valued objective minimization problem is
(IVOP) minf(x) =fL(x),fU(x)
subject togi(x)≤, i= , , . . . ,m.
LetP={x∈Rn:g
i(x)≤,i= , , . . . ,m}be a feasible set of (IVOP).
Definition Letx∗be a feasible solution of the primal problem (IVOP). We say thatx∗ is a non-dominated solution of problem (IVOP) if and only if there exists nox∈Psuch thatf(x)≺f(x∗). In this case,f(x∗) is called the non-dominated objective value off.
Theorem Let f(x)be gH-differentiable on X⊆Rnand interval-valued invex with respect toη:X×X→Rn,and gi(x) (i= , , . . . ,m)be invex with respect to the sameη.If there exist x∗∈P and vi(i= , , . . . ,m)such that
∇fx∗L+ n
i=
vi∇gi
x∗= , ()
n
i=
vigix∗= , ()
vi≥, ()
then x∗is a non-dominated solution of problem(IVOP).
Proof For anyx∗∈P satisfyinggi(x∗)≤,i= , , . . . ,m, sincef(x) is gH-differentiable onX⊆Rnand interval-valued invex with respect toη, then [min{fL(x) –fL(x∗),fU(x) – fU(x∗)},max{fL(x) –fL(x∗),fU(x) –fU(x∗)}]η(x,x∗)t∇f(x∗).
(i) On the condition ofmin{fL(x) –fL(x∗),fU(x) –fU(x∗)}=fL(x) –fL(x∗),η(x,x∗)t×
∇f(x∗) = [η(x,x∗)t{∇f(x∗)}L,η(x,x∗)t{∇f(x∗)}U],
fL(x) –fLx∗≥ηx,x∗t ∇fx∗L
= –ηx,x∗t n
i=
vi∇gi
x∗
≥– m
i=
= m
i=
–vigi(x) +vigix∗
= m
i=
–vigi(x)
≥.
Thus,f(x)≺f(x∗) does not hold.
(ii) On the condition ofmin{fL(x) –fL(x∗),fU(x) –fU(x∗)}=fU(x) –fU(x∗),η(x,x∗)t×
∇f(x∗) = [η(x,x∗)t{∇f(x∗)}L,η(x,x∗)t{∇f(x∗)}U],
fU(x) –fUx∗≥ηx,x∗t ∇fx∗L
= –ηx,x∗t n
i=
vi∇gix∗
≥– m
i=
vigi(x) –gix∗
= m
i=
–vigi(x) +vigix∗
= m
i=
–vigi(x)
≥.
Thus,f(x)≺f(x∗) does not hold.
Similarly, we can prove the other two conditions.
The proof of the following theorem is similar to the one of Theorem .
Theorem Let f(x)be gH-differentiable on X⊆Rnand interval-valued invex with respect toη:X×X→Rn,and g
i(x) (i= , , . . . ,m)be invex with respect to the sameη.If there exist x∗∈P and vi(i= , , . . . ,m)such that
∇fx∗U+ n
i=
vi∇gix∗= , ()
n
i=
vigi
x∗= , ()
vi≥, ()
then x∗is a non-dominated solution of problem(IVOP).
Example
minimizef(x) =x+x+ ,x+
subject tog(x) =x– ≤,
From Theorem we have∇f(x) = [x, x+ ].
(i) It is easy to see that the problem satisfies the assumptions of Theorem . Then
x∗, x∗+ L+v–v= ,
v
x∗– = ,
v
–x∗– = .
We obtainv=v= , andx∗= is a non-dominated solution.
(ii) It is easy to see that the problem satisfies the assumptions of Theorem . Then
x∗, x∗+ U+v–v= ,
v
x∗– = ,
v
–x∗– = .
We obtainv=v= , andx∗= –is also a non-dominated solution.
Example Consider the following interval-valued programming problem:
minimizef(x) = [, ]sinx
subject tog(x) = (sinx– )– ≤,
x∈
,π
.
Note that functionsf andgare invex with respect to
η(x,y) =sinx–siny cosy .
And∇f(x) = [sinxcosx, sinxcosx],∇g(x) = (sinx– )cosx.
It is easy to see that the problem satisfies the assumptions of Theorem . Then
sinx∗cosx∗+ vsinx∗– cosx∗= ,
vsinx∗– –
= .
After some algebraic calculations, we obtainx∗=sin–( –
√),v= √
– . Therefore, x∗=sin–( –
√) is a non-dominated solution.
The following example also shows the advantages of our method in respect to [].
Example Consider the following interval-valued programming problem:
minimizef(x) = [x– sinx,x–sinx+ ]
g(x) = x+ x– ≤,
g(x) = –sinx≤,
g(x) = –sinx≤.
Then f is gH-differentiable and weakly differentiable. SincefLis not LU-convex and fL+fUis not LU-convex, methods in [] cannot be used.
Note that functionsf andgi(i= , , , ) are invex with respect to
η(x,y) =
sinx–siny
cosy
,sinx–siny cosy
t .
It is easy to see that the problem satisfies the assumptions of Theorem . Then
+v(cosx+ ) + v–vcosx= ,
–cosx+ vcosx+ v–vcosx= ,
v(sinx+ sinx+x– ) = ,
v(x+ x– ) = ,
v(–sinx) = ,
v(–sinx) = .
After some algebraic calculations, we obtainx∗= (,sin–
)
t,v= ( , ,
, )
t. Therefore,
x∗is a non-dominated solution.
5 Conclusion
The concept of convex interval-valued mappings has been studied in the literature by many researchers. The aim of this paper is to introduce the concept of invex interval-valued mappings with gH-differentiable functions. Then we discussed the relationships between interval-valued invex mappings and interval-valued weakly invex mappings. Fi-nally, the sufficient optimality condition for interval-valued objective functions has been derived under invexity.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have equal contributions. All authors read and approved the final manuscript.
Author details
1School of Science, Xi’an University of Posts and Telecommunications, Xi’an, China.2Department of Mathematics, School
of Science, Xidian University, Xi’an, China.
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 60974082, 11301415, 11401469), Foundation from Xi’an University of Posts and Telecommunications for Young Teachers (Grant No. ZL2014-34) and the Natural Science Foundation of Shaanxi Province (Grant No. 2013JQ1020).
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