# On Interval-Valued Pseudolinear Functions and Interval-Valued Pseudolinear Optimization Problems

Authors: Jianke Zhang; Qinghua Zheng; Chang Zhou; Xiaojue Ma; Lifeng Li
Journal: Journal of Function Spaces
 (2015)

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## Abstract

Some basic characterizations of an interval-valued
pseudolinear function are derived. By means of the properties of interval-valued
pseudolinearity, a class of interval-valued pseudolinear programs is considered,
and the solution set of the interval-valued pseudolinear optimization problem
is characterized.


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## Body

## 1. Introduction

In recent years, many approaches for interval-valued optimization problems have been explored in considerable details; see examples in [1–3]. Several authors have been interested in the optimality conditions and duality results for the interval-valued optimization problems. Wu has extended the concept of convexity for real-valued functions to LU-convexity for interval-valued functions, and then he has established the Karush-Kuhn-Tucker (KKT) optimality conditions [4] and duality theory [5] for an optimization problem with an interval-valued objective function under the assumption of LU-convexity. Sun and Wang [6] have derived the Fritz John type and Kuhn-Tucker type necessary and sufficient optimality conditions for nondifferentiable interval-valued programming. Jayswal et al. [7] have discussed Mond-Weir and Wolfe type duality theorems for the interval-valued programming problems under the conditions of generalized convexity. Chalco-Cano et al. [8] have obtained KKT optimality conditions for interval-valued programming problems by using a more general concept of gH-derivative. Bhurjee and Panda [9, 10] have introduced the parametric form of interval-valued functions and studied the solution of convex interval-valued programming problems. Jana and Panda [11] have studied the preferable efficient solutions of the problem of interval-valued vector optimization. Zhang et al. [12] have extended the concepts of preinvexity and invexity to interval-valued functions and derived the KKT optimality conditions for LU-preinvex and invex optimization problems with an interval-valued objective function.Interval-valued linear optimization [13, 14] is a class of important and simple interval-valued optimization problems. Recently, Hladík [15] has proposed a method for testing basis stability of interval-valued linear optimization problems; Hladík’s studies have shown that if some basis stability criterion holds true, then the problems become much more easy to solve. In [16], Hladík has discussed lower and upper bound approximation for the best case optimal value of the interval-valued linear optimization problems and proposed an algorithm for computing the best case optimal value. Li et al. [17, 18] and Luo et al. [19] have established some necessary and sufficient conditions for checking weak and strong optimality of given feasible solutions for the interval-valued linear optimization problems.In this paper, we consider and give some characterizations for a class of generalized interval-valued linear function, which is interval-valued pseudolinear function. Then, by means of the basic properties of interval-valued pseudolinearity, a class of interval-valued pseudolinear programs is considered, and the solution set of the interval-valued pseudolinear optimization problem is characterized. It can be shown that the interval-valued linear optimization problems [13, 14] and the interval-valued linear fractional programs are included in interval-valued pseudolinear programs. These programs arise in many practical applications.


## 2. Preliminaries

In this section, we recall some basic concepts with regard to interval-valued functions.LetRn be n-dimensional Euclidean space, and let R+n be its nonnegative orthant. Let us denote by I the class of all closed intervals in R. A=[aL,aU]∈I denotes a closed interval, where aL and aU mean the lower and upper bounds of A, respectively. The closed interval A also can be expressed in terms of a parameter in several ways. Throughout this paper, we consider specific parametric representations of the interval as A=[aL,aU]={a(t)=aL+t(aU-aL)∣t∈[0,1]}.LetA=[aL,aU]={a(t1)∣t1∈[0,1]} and B=[bL,bU]={b(t2)∣t2∈[0,1]} be in I, and the interval operations can be performed with respect to parameters [9, 10] as follows:(a)
A⊕B=a(t1)+b(t2)∣t1,t2∈[0,1]};(b)
A⊖B=a(t1)-b(t2)∣t1,t2∈[0,1]};(c)
A⊙B={a(t1)·b(t2)∣t1,t2∈[0,1], and kA={ka(t1)∣t1∈[0,1]};(d)
A⊘B={a(t1)/b(t2)∣t1,t2∈[0,1],b(t2)≠0}.LetCvk be k dimensional column whose elements are intervals. That is, Cvk∈Ik, Cvk=C1,C2,…,CkT, and Cj=[cjL,cjU]=cjL+tj(cjU-cjL), tj∈[0,1], j=1,2,…,k. The interval-valued function in parametric form is as follows.Definition 1 (see [9]).
Forc(t)=c1t1,c2t2,…,cktkT∈Cvk, let fc(t):Rn→R. For a given interval vector Cvk, an interval-valued function FCvk:Rn→I is defined as(1)FCvkx=fctx∣fct:Rn⟶R,ct∈Cvk.For every fixedx, if fc(t)(x) is continuous in t, then mint∈[0,1]kfc(t)(x) and maxt∈[0,1]kfc(t)(x) exist. In that case(2)FCvkx=mint∈0,1k⁡fctx,maxt∈0,1k⁡fctx.Iffc(t)(x) is linear in t, then mint∈[0,1]kfc(t)(x) and maxt∈[0,1]kfc(t)(x) exist in the set of vertices of Cvk. If fc(t)(x) is monotonically increasing in t, then FCvk(x)=[fc(0)(x),fc(1)(x)].The interval-valued functionFCvk:Rn→I is differentiable at x∗ if fc(t)(x) is differentiable at x∗ for every t∈[0,1]k. The partial derivatives of FCvk at x∗ may be calculated as follows:(3)∂FCvkx∗∂xi=∂fctx∗∂xi∣for  everyt∈0,1k,ct∈Cvk.If fc(t)(x∗) is continuous in t, then(4)∂FCvkx∗∂xi=mint∈0,1k⁡∂fctx∗∂xi,maxt∈0,1k⁡∂fctx∗∂xi.The gradient of FCvk at x∗ is an interval vector:(5)∇FCvkx∗=∂FCvkx∗∂x1,∂FCvkx∗∂x2,…,∂FCvkx∗∂xnT.In this paper, we denote the pointλy+(1-λ)x, λ∈[0,1], by xλy. For the two interval-valued functions of FCvk(x)={fc(t1)(x)∣fc(t1):Rn→R,c(t1)∈Cvk} and GCvk(x)={gc(t2)(x)∣gc(t2):Rn→R,c(t2)∈Cvk}, we say that FCvk(x)=GCvk(x) or FCvk(x)⪯GCvk(x), which means(6)fct1x=gct2xorfct1x≤gct2xfor all t1, t2∈0,1k and t1 may or may not be equal to t2.However, we say thatFCvk(x)=wGCvk(x) or FCvk(x)⪯wGCvk(x), which means(7)fct1x=gct2xorfct1x≤gct2xfor all t1=t2∈0,1k; that is to say, t1 and t2 are the same in both sides.Definition 2 (see [9]).
Suppose thatX⊆Rn is a convex set and for given Cvk∈Ik, FCvk:Rn→I. For x1,x2∈X, 0≤λ≤1.(i)
FCvk is said to be convex with respect to ⪯ if(8)FCvkλx1+1-λx2⪯λFCvkx1⊕1-λFCvkx2,which means(9)fct′λx1+1-λx2≤λfct′′x1+1-λfct′′x2for allt′, t′′∈0,1k; t′ may or may not be equal to t′′.(ii)
FCvk is said to be convex with respect to ⪯w if(10)FCvkλx1+1-λx2⪯wλFCvkx1⊕1-λFCvkx2,which means(11)fctλx1+1-λx2≤λfctx1+1-λfctx2for allt∈0,1k; t is same in both sides.It can be shown thatFCvk is convex [9] with respect to ⪯w if and only if fc(t)(x) is a convex function on X for every t. Similar result does not hold for convexity with respect to ⪯.Mangasarian [20] has introduced the following pseudoconvex real-valued functions.Definition 3 (see [20]).
The differentiable real-valued functionf:X→R on the open set X⊆Rn is pseudoconvex on X if(12)x,y∈X,x-yT∇fy≥0⟹fx-fy≥0;or equivalently,(13)x,y∈X,fx-fy<0⟹x-yT∇fy<0.Furthermore, we say that f is strictly pseudoconvex on X if(14)x,y∈X,fx-fy≤0⟹x-yT∇fy<0.We can also define the following pseudoconvex interval-valued function.Definition 4.
A differentiable interval-valued functionFCvk:X→I, on the convex set X⊆Rn for given Cvk∈Ik, is pseudoconvex with respect to ⪯w on X if(15)x,y∈X,x-yT∇FCvky⪰w0⟹FCvkx⪰wFCvky;or equivalently,(16)FCvkx≺wFCvky⟹x-yT∇FCvky≺w0.
An interval-valued functionFCvk is pseudoconcave if -FCvk is pseudoconvex with respect to ⪯w on X.Remark 5.
It is worth pointing out that the definition of pseudoconvex interval-valued function in [4] has some limitations, which define the pseudoconvex interval-valued function by using the endpoint functions fL and fU of the interval-valued function f(x)=[fL,fU]. However, in some case, if the two endpoint functions fL and fU are all real-valued pseudoconvex functions, the interval-valued function may be not a pseudoconvex interval-valued function, which can be seen in the following example.Example 6.
The interval-valued functionf(x)=[fL(x),fU(x)], x∈(0,1), fL(x)=min⁡{f1(x),f2(x)}, and fU(x)=max⁡{f1(x),f2(x)}, where f1(x)=log⁡x and f2(x)=-2x2.
It can be shown thatf1(x)=log⁡x is strictly increasing function, being f1x′=1/x>0, for all x∈(0,1); therefore, (y-x)f1x′≥0⇔y-x≥0. Thus, y-x≥0⇔y≥x⇒f1(y)≥f1(x). So f1(x) is pseudoconvex.
The functionf2(x)=-2x2 is strictly decreasing on (0,1), as f2x′=-4x<0. We have (y-x)f2x′≥0⇔y-x≤0; y-x≤0⇔y≤x⇒f2(x)≤f2(y), so f2(x) is pseudoconvex.
The sum isz(x)=λfL(x)+(1-λ)fU(x)λ=1/2∈[fL(x),fU(x)] and z(x)=1/2(log⁡x-2x2), 1>x>0. We have z′(x)=1/2log⁡x-2x2′=1-4x2/2x, z′(x)≥0⇔1-4x2≥0⇒x≤1/2. Therefore, z(x) has a maximum point at x=1/2, so it is not pseudoconvex.For a real-valued differentiable function, which is said to be a pseudolinear function [21–23], if it is both pseudoconvex and pseudoconcave, similar to the case of real-valued functions, we can also define the following interval-valued pseudolinear functions.Definition 7.
A differentiable interval-valued functionFCvk:X→I, on the convex set X⊆Rn for given Cvk∈Ik, is pseudolinear with respect to ⪯w on X if it is both pseudoconvex and pseudoconcave on X.This class of pseudolinear interval-valued functions includes many useful functions. For example, the following function is a pseudolinear interval-valued function.Example 8.
The interval-valued functionFCv2=1+2t1x+1+t2Inx:X→I, where t=t1,t2T, ti∈[0,1], i=1,2, and X={x∈R∣x>0}. It can be shown that this interval-valued function is a pseudolinear interval-valued function.


## 3. Characterizations of PseudolinearInterval-Valued Functions

In this section, we provide some characterizations of the pseudolinear interval-valued function.Theorem 9.
Let a differentiable interval-valued functionFCvk:X→I, on the convex set X⊆Rn for given Cvk∈Ik. Then, the following statements (i)–(iii) are equivalent.(i)
FCvk is pseudolinear over X.(ii)
For anyx and y in X,y-xT∇FCvk(x)=w0 if and only if FCvk(y)=wFCvk(x).(iii)
There exists an interval-valued functionPCvk defined on X×X such that PCvk(x,y)≻w0 for given Cvk∈Ik and(17)FCvky=wFCvkx⊕PCvkx,yy-xT∇FCvkxfor any x and y in X.Proof.
(i)⇒(ii) Suppose thatFCvk is pseudolinear interval-valued function for given Cvk∈Ik on X. It can be shown that FCvk(y)=wFCvk(x) implies y-xT∇FCvk(x)=w0.
Letx and y be two points of X such that FCvk(y)=wFCvk(x). For any λ∈(0,1), we denote the point λy+(1-λ)x by xλy, and it can be shown that FCvk(xλy)=wFCvk(x).
IfFCvk(xλy)≻wFCvk(x), then x-xλyT∇FCvk(xλy)≺w0 since FCvk is pseudoconvex for given Cvk∈Ik. But y-xλy=-(1-λ)(x-xλy)/λ, and then y-xλyT∇FCvk(xλy)≻w0. According to the pseudoconvexity of FCvk, then FCvk(y)⪰wFCvk(xλy), which contradicts the assumption of FCvk(xλy)≻wFCvk(x)=wFCvk(y). So the assumption of FCvk(xλy)≻wFCvk(x) is false. Similarly, it can be shown that FCvk(xλy)≺wFCvk(x) is false by the pseudoconcavity of the interval-valued function FCvk. Then, we can show that FCvk(xλy)=wFCvk(x) for given Cvk∈Ik on X and all λ in (0,1). So(18)limλ→0+⁡FCvkxλy⊖FCvkxλ=w∇FCvkxTy-x=w0.
(ii)⇒(iii) Let the interval-valued functionPCvkx,y: X×Y→I defined on X×X, if ∇FCvkxT(y-x)=w0 for x, y in X and given Cvk∈Ik on X, and we define PCvk(x,y)=w1 for given Cvk∈Ik on X; if ∇FCvkxT(y-x)≠w0 for x, y in X and given Cvk∈Ik on X, we define(19)PCvkx,y=wFCvky⊖FCvkx∇FCvkxTy-x=fcty-fctx∇fctxTy-x∣t∈0,1k,ct∈Cvk.We can show that PCvk(x,y)≻w0 for given Cvk∈Ik.
Suppose thatFCvk(y)≻wFCvk(x) for given Cvk∈Ik. For some λ∈(0,1), if FCvk(xλy)⪯wFCvk(x), using the continuity of FCvk, there exists r∈(0,1) such that λ≤r and FCvk(xry)=wFCvk(x). By (ii),(20)limλ→0+⁡FCvkxry⊖FCvkxr=w∇FCvkxTy-x=w0,which contradicts the assumption of ∇FCvkxT(y-x)≠w0 for x,y in X and given Cvk∈Ik on X. So FCvk(xλy)≻wFCvk(x) for given Cvk∈Ik and all λ∈(0,1); we also have(21)limλ→0+⁡FCvkxλy⊖FCvkxλ=w∇FCvkxTy-x⪰w0,and since ∇FCvkxT(y-x)≠w0 for x,y in X and given Cvk∈Ik on X, ∇FCvkxT(y-x)≻w0 holds. It can be shown that(22)PCvkx,y=wFCvky⊖FCvkx∇FCvkxTy-x≻w0for given Cvk∈Ik on X.
Similarly, we can show thatPCvk(x,y)≻w0 if FCvk(y)≺wFCvk(x) for given Cvk∈Ik.
(iii)⇒(i) It is obvious that(23)FCvky=wFCvkx⊕PCvkx,yy-xT∇FCvkxfor any x and y in X, which implies y-xT∇FCvk(x)=w0 if and only if FCvk(y)=wFCvk(x).Theorem 10.
LetFCvk:X→I be an interval-valued function defined on the convex set X⊆Rn for given Cvk∈Ik, which is once continuously differentiable on X. Then FCvk is pseudolinear on X if and only if y-xT∇FCvk(x)=w0 implies FCvk(x)=wFCvk(xλy) for any x and y in X and any λ∈[0,1].Proof.
Suppose thatFCvk is pseudolinear on X. Let x and y in X be such that y-xT∇FCvk(x)=w0 for given Cvk∈Ik. Then, for any λ∈[0,1],(24)xλy-xT∇FCvkx=wλy-xT∇FCvkx=w0.
By (ii) of Theorem9, FCvk(x)=wFCvk(xλy) for any x and y in X and any λ∈[0,1].
Conversely, suppose thaty-xT∇FCvk(x)=w0 implies FCvk(x)=wFCvk(xλy) for any x and y in X and any λ∈[0,1]. If FCvk is not pseudolinear on X, then there exist x and y in X such that FCvk(x)=wFCvk(y), but y-xT∇FCvk(x)≠w0, which means that y-xT∇FCvk(x)=w0 and FCvk(x)=wFCvk(y) cannot occur. Suppose that y-xT∇FCvk(x)≻w0, which means y-xT∇fc(t)(x)>0 for c(t)∈Cvk. We define f(λ)=FCvk(xλy), which is once continuously differentiable function. According to the assumption, f(0)=f(1) and f′0={y-xT∇fc(t)(x)>0∣∀c(t)∈Cvk}. So f(λ) assumes a local maximum at some point λ0∈(0,1). Then(25)f′λ0=y-xT∇FCvkxλ0y=0.So xλ0y-xT∇FCvk(xλ0y)=λ0y-xT∇FCvk(xλ0y)=0. By the assumption, FCvk(x)=wFCvk(xλy) for all 0≤λ≤λ0, and then y-xT∇FCvk(x)=0, which is a contradiction.How to check for interval pseudolinearity can be shown by the following example.Example 11.
Consider the interval-valued function onX={π/2>x>0}:(26)FCv2x=fctx=2+t1x+1+t2sin⁡x∣∀t1,t2∈0,1.
From (ii) of Theorem9, for any x and y in X, y-xT∇FCv2(x)=w0 means y-xT∇fct(x)=(y-x)[(2+t1)+1+t2cos⁡x]=0 for all t1,t2∈[0,1]. Since ∇fct(x)=(2+t1)+1+t2cos⁡x≠0, if and only if x=y, so, for fc(t)(x)=(2+t1)x+(1+t2)sinx and fcty=2+t1y+1+t2sin⁡y, we have fc(t)(x)=fc(t)(y) for all t1, t2∈[0,1], which implies FCv2(y)=wFCv2(x).
Therefore, the interval-valued function is a pseudolinear interval-valued function.The following interval-valued linear fractional function is a class of interval-valued pseudolinear functions:(27)FCvkx=ATx+CBTx+D,where x∈Rn, A and B are all in In, and C and D are interval number. We can show that FCvk(x) is pseudolinear on SF={x∈Rn∣BTx+D≻w0},(28)FCvky⊖FCvkx=wBTx+DBTy+Dy-xT∇FCvkx,where PCvk(x,y)=(BTx+D)/(BTy+D)≻w0 if x and y are in SF.For interval-valued linear fractional function, we have(29)PCvkx,yPCvky,z=PCvkx,zfor any x, y, and z. It can be shown that (29) implies PCvk(x,x)=1 and PCvk(x,y)=1/PCvk(y,x) for any x and y in SF. We can show the following conclusion.Theorem 12.
Let a pseudolinear interval-valued functionFCvk:X→I, on the convex set X⊆Rn with proportional functional PCvk(x,y) for given Cvk∈Ik. If PCvk(x,y) satisfies condition (29) and PCvk(x,·) is differentiable for each x in X, then there exists an interval-valued linear fractional function GCvk such that X⊂SG and GCvk(x)=wFCvk(x) for all x in X.Proof.
Suppose thatFCvk is not constant on X and X contains more than one point. Take y and z in X such that FCvk(z)≠FCvk(y). Then(30)FCvkx=wFCvky⊕PCvky,xx-yT∇FCvkyfor any x and y in X.
SincePCvk(x,·) is differentiable for each x in X, then(31)∇FCvkx=wPCvky,x∇FCvky⊕x-yT∇FCvky∇2PCvky,xfor any x and y in X, where ∇2P denotes the gradient of P with respect to the second variable.
From (30), we have(32)FCvkx⊖FCvkz=wPCvkz,x∇FCvkzTx-z,FCvkx⊖FCvkz=wFCvky⊕PCvky,x∇FCvkyTx-y⊖FCvky⊕PCvky,z∇FCvkyTz-y=wPCvky,x∇FCvkyTx-y⊖PCvky,z∇FCvkyTz-y.
By (31), we have(33)PCvkz,x∇FCvkzTx-z=wPCvkz,x·PCvky,z∇FCvky⊕z-yT∇FCvky∇2PCvky,zTx-z,
By (32)-(33), we have(34)PCvky,x⊖PCvky,z∇FCvkyTz-y=wPCvky,x∇FCvkyTx-y-x-z⊖PCvky,z∇FCvkyTz-y=wPCvkz,x∇FCvkyTz-y∇2PCvky,zT·x-z.
SinceFCvk(z)≠FCvk(y), we have ∇FCvkyT(z-y)≠0. From (34), we have(35)PCvky,x⊖PCvky,z=wPCvkz,x∇2PCvky,zTx-z.
Multiplying (35) by PCvk(x,y), we obtain(36)1⊖PCvkx,z=wPCvkz,y∇2PCvky,zTx-z.
So(37)PCvkz,x=w1PCvkx,z=w1⊖PCvkz,y∇2PCvky,zTx-z-1.
Substituting (37) into (30) and replacing y by z, it can be shown that(38)FCvkx=wFCvkz⊕x-zT∇FCvkz·1⊖PCvkz,y∇2PCvky,zTx-z-1is an interval-valued linear fractional function GCvk.The result of Theorem12 and (iii) of Theorem 10 can be shown by the following example.Example 13.
Consider the interval-valued linear fractional function(39)FCv1x=ATx+CBTx+D,where x∈R+, AT=[2,3]=2+t1, and BT=[3,4]=3+t2 are all in I2 and C=[0,1]=t3 and D=[1,2]=1+t4 are interval number, where ti∈[0,1], i=1,2,3,4.
We denote that(40)PCv1x,y=BTx+DBTy+D=3+t2x+1+t43+t2y+1+t4>0∣∀ti∈0,1.
It can be shown thatPCv1(x,y) satisfies (29), and the following is satisfied:(41)FCv1y=wFCv1x⊕PCv1x,yy-xT∇FCv1xfor any x and y in R+. From (iii) of Theorem 9, the interval-valued function FCv1(x) is a pseudolinear interval-valued function.
According to Theorem12, there exists an interval-valued linear fractional function GCv1(x)=wFCv1(z)⊕x-zT∇FCv1(z)1⊖PCv1z,y∇2PCv1y,zTx-z-1, such that GCv1(x)=wFCv1(x).
For allti∈[0,1], i=1,2,3,4, we have(42)1⊖PCv1z,y∇2PCv1y,zTx-z-1=1-3+t2z+1+t43+t2y+1+t4·-3+t23+t2y+1+t43+t2z+1+t42x-z∣∀ti∈0,1=3+t2x+1+t43+t2z+1+t4∣∀ti∈0,1=wPCv1x,z.
Then, we get(43)GCv1x=wFCv1z⊕x-zT∇FCv1zPCv1x,z.
By (iii) of Theorem9, we have GCv1(x)=wFCv1(x).


## 4. The Solution Sets of Interval-Valued Pseudolinear Programs

In this section, we consider the following interval-valued programming:(44)IVOPmin⁡FCvkxsubject  tox=x1,…,xn∈X⊆Rn,where FCvk(x):X→I is an interval-valued function for given Cvk∈Ik and the feasible set X is a convex subset of Rn. We denote the solution set of (IVOP) as(45)S¯=argminx∈X⁡FCvkx,which is nonempty. If FCvk(x) is pseudolinear, then the solution set S¯ of (IVOP) is a convex set.The following characterization of the solution set of the problem of (IVOP) can be obtained.Theorem 14.
LetFCvk(x):X→I be a pseudolinear interval-valued function for given Cvk∈Ik and x¯∈S¯. Then S¯=S~=S^=S˘, where(46)S~=x∈X∣x¯-xT∇FCvkx=w0,(47)S^=x∈X∣x¯-xT∇FCvkx¯=w0,(48)S˘=x∈X∣x¯-xT∇FCvkxλx¯=w0,∀λ∈0,1.Proof.
It is obvious thatx∈S¯ if and only if FCvk(x)=wFCvk(x¯), which implies fc(t)(x)=fc(t)(x¯) for t∈0,1k and c(t)∈Cvk. From (ii) of Theorem 9, FCvk(x)=wFCvk(x¯) if and only if(49)x¯-xT∇FCvkx=w0=wx¯-xT∇FCvkx¯.Therefore, S¯=S~=S^.
It can be seen thatS˘⊂S~=S¯. Let x∈S¯=S~, and then(50)x¯-xT∇FCvkx=w0.
For eachλ∈(0,1), we get(51)x¯-xλx¯T∇FCvkx=w1-λx¯-xT∇FCvkx=w0.
Therefore, from (ii) of Theorem9, we have(52)FCvkx=wFCvkxλx¯.
By the pseudolinearity ofFCvk(x) and (52), we obtain(53)x¯-xT∇FCvkxλx¯=w0,∀λ∈0,1.
Sox∈S˘ and S¯⊂S˘. Thus, S¯=S˘.Corollary 15.
LetFCvk(x):X→I be a pseudolinear interval-valued function for given Cvk∈Ik and x¯∈S¯. Then S¯=S~∗=S^∗=S˘∗, where(54)S~∗=x∈X∣x¯-xT∇FCvkx⪰w0,S^∗=x∈X∣x¯-xT∇FCvkx¯⪰w0,S˘∗=x∈X∣x¯-xT∇FCvkxλx¯⪰w0,∀λ∈0,1.Proof.
From (46), it is clear that S¯⊂S~∗. Let x∈S~∗; that is, for x∈X,(55)x¯-xT∇FCvkx⪰w0.
Then, from (iii) of Theorem9 and the pseudolinearity of FCvk(x), we have(56)FCvkx¯=wFCvkx⊕PCvkx,x¯x¯-xT∇FCvkx⪰wFCvkx.
Thenx∈S¯, which implies S¯=S~∗. It can be also similarly proved that S¯=S^∗.
We can also prove thatS¯=S˘∗ as follows. In fact, it is clear that(57)S¯=S˘⊂S˘∗⊂S~∗=S¯,where S˘ is defined by (48). Thus, the conclusion follows.Example 16.
Consider the following pseudolinear interval-valued program:(58)min⁡FCv2x=2+t1x+1+t2sin⁡x∣∀ti∈0,1,i=1,2subject  tox∈X⊆R,where FCv2(x):X→I is an interval-valued function, t=t1,t2T, ti∈[0,1], i=1,2, and X={x∈R∣π/2>x>0}. By Example 11, FCv2(x) is a pseudolinear interval-valued function and S¯={0}.
According to Theorem14, we get S¯=S~=S^=S˘.


## 5. Conclusions

In this paper, we introduced the concept of the pseudolinear interval-valued functions. We have obtained some basic characterizations of an interval-valued pseudolinear function. By means of the properties of interval-valued pseudolinearity, a class of interval-valued pseudolinear programs was considered, and the solution set of the interval-valued pseudolinear optimization problem was characterized.



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*Source: 610848-2015-06-02.xml*