# Univex Interval-Valued Mapping with Differentiability and Its Application in Nonlinear Programming

Authors: Lifeng Li; Sanyang Liu; Jianke Zhang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Interval-valued univex functions are introduced for differentiable programming problems. Optimality and duality results are derived for a class of generalized convex optimization problems with interval-valued univex functions.


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## Body

## 1. Introduction

Imposing the uncertainty upon the optimization problems is an interesting research topic. The uncertainty may be interpreted as randomness, fuzziness, or interval-valued fuzziness. The randomness occurring in the optimization problems is categorized as the stochastic optimization problems, and the imprecision (fuzziness) occurring in the optimization problems is categorized as the fuzzy optimization problems. In order to perfectly match the real situations, interval-valued optimization problems may provide an alternative choice for considering the uncertainty into the optimization problems. That is to say, the coefficients in the interval-valued optimization problems are assumed as closed intervals. Many approaches for interval-valued optimization problems have been explored in considerable details; see, for example, [1–3]. Recently, Wu has extended the concept of convexity for real-valued functions to LU-convexity for interval-valued functions, then he has established the Karush-Tucker conditions [4–6] for an optimization problem with interval-valued objective functions under the assumption of LU-convexity. Similar to the concept of nondominated solution in vector optimization problems, Wu has proposed a solution concept in optimization problems with interval-valued objective functions based on a partial ordering on the set of all closed intervals, then the interval-valued Wolfe duality theory [7] and Lagrangian duality theory [8] for interval-valued optimization problems have been proposed. Recently, Wu [9] has studied the duality theory for interval-valued linear programming problems.In 1981, Hanson [10] introduced the concept of invexity and established Karush-Tucker type sufficient optimality conditions for a nonlinear programming problem. In [11], Kaul et al. considered a differentiable multiobjective programming problem involving generalized type I functions. They investigated Karush-Tucker type necessary and sufficient conditions and obtained duality results under generalized type I functions. The class of B-vex functions has been introduced by Bector and Singh [12] as a generalization of convex functions, and duality results are established for vector valued B-invex programming in [13]. Bector et al. [14] introduced the concept of univex functions as a generalization of B-vex functions introduced by Bector et al. [15]. Combining the concepts of type I and univex functions, Rueda et al. [16] gave optimality conditions and duality results for several mathematical programming problems. Aghezzaf and Hachimi [17] introduced classes of generalized type I functions for a differentiable multiobjective programming problem and derived some Mond-Weir type duality results under the above generalized type I assumptions. Gulati et al. [18] introduced the concept of (F,α,ρ,d)-V-type I functions and also studied sufficiency optimality conditions and duality multiobjective programming problems.This paper aims at extending the Karush-Tucker optimality conditions to nonconvex optimization problem with interval-valued functions. First, we extend the concept of univexity for a real-valued function to an interval-valued function and present the concept of interval-valued univex functions. Then, the Karush-Tucker optimality conditions are proposed for an interval-valued function under the assumption of interval-valued univexity.


## 2. Preliminaries

Let one denotes byℐ 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. For every a∈R, we denote a=[a,a].Definition 1.
LetA=[aL,aU] and B=[bL,bU] be in ℐ; one has(i)
A+B={a+b:a∈A and b∈B}=[aL+bL,aU+bU];(ii)
-A={-a:a∈A}=[-aU,-aL];(iii)
A×B={ab:a∈Aandb∈B}=[min⁡ab,max⁡ab], where min⁡ab=min⁡{aLbL,aLbU,aUbL,aUbU} and max⁡ab=max⁡{aLbL,aLbU,aUbL,aUbU}.
Then, we can see that(1)A-B=A+(-B)=[aL-bU,aU-bL],kA={ka:a∈A}={[kaL,kaU]ifk≥0,[kaU,kaL]ifk<0,
where k is a real number.
By usingHausdorff metric, Neumaier [19] has proposed Hausdorff metric between the two closed intervals A and B as follows:
(2)dH(A,B)=max⁡{|aL-bL|,|aU-bU|}.Definition 2.
LetA=[aL,aU] and B=[bL,bU] be two closed intervals in R. One writes A⪯B if and only if aL≤bL and aU≤bU, A≺B if and only if A⪯B and A≠B, that is, the following (a1), (a2), or (a3) is satisfied:(a1)
aL<bL and aU≤bU;(a2)
aL≤bL and aU<bU;(a3)
aL<bL and aU<bU.Definition 3 (see [20]).
LetA=[aL,aU] and B=[bL,bU] be two closed intervals, the gH-difference of A and B is defined by
(3)[aL,aU]⊖g[bL,bU]=[min⁡(aL-bL,aU-bU),max⁡(aL-bL,aU-bU)].
For example,[1,3]⊖g[0,3]=[0,1],[0,3]⊖g[1,3]=[-1,0]. And a-b=[a,a]⊖g[b,b]=[a-b,a-b]=a-b.Proposition 4.
(i) For everyA,B∈ℐ, A⊖gB always exists and A⊖gB∈ℐ.(ii)
A⊖gB⪯0  if and only if  A⪯B.


## 3. Interval-Valued Univex Functions

Definition 5 (interval-valued function).
The functionf:Ω→ℐ is called an interval-valued function, where Ω⊆Rn. Then, f(x)=f(x1,…,xn) is a closed interval in R for each x∈Rn, and f(x) can be also written as f(x)=[fL(x),fU(x)], where fL(x) and fU(x) are two real-valued functions defined on Rn and satisfy fL(x)≤fU(x) for every x∈Ω.Definition 6 (continuity of an interval-valued function).
The functionf:Ω⊆Rn→ℐ is said to be continuous at x∈Ω if both fL(x) and fU(x) are continuous functions of x.
The concept of gH-derivative of a functionf:(a,b)→ℐ is defined in [19].Definition 7.
Letx0∈(a,b) and h be such that x0+h∈(a,b), then the gH-derivative of a function f:(a,b)→ℐ at x0 is defined as
(4)f′(x0)=lim⁡x→0[f(x0+h)⊖gf(x0)].
If f′(x0)∈ℐ exists, then we say that f is generalized Hukuhara differentiable (gH-differentiable, for short) at x0.
Moreover, [21] also proved the following theorem.Theorem 8.
Letf:(a,b)→ℐ be such that f(x)=[fL(x),fU(x)]. The function f(x) is gH-differentiable if and only if fL(x) and fU(x) are differentiable real-valued functions. Furthermore,
(5)f′(x)=[min⁡{(fL)′(x),(fU)′(x)},max⁡{(fL)′(x),(fU)′(x)}].Definition 9 (gradient of an interval-valued function).
Letf(x) be an interval-valued function defined on Ω, where Ω is an open subset of Rn. Let Dxi(i=1,2,…,n) stand for the partial differentiation with respect to the ith variable xi. Assume that fL(x) and fU(x) have continuous partial derivatives so that DxifL(x) and DxifU(x) are continuous. For i=1,2,…,n, define
(6)Dxif(x)=[min⁡(DxifL(x),DxifU(x)),max⁡(DxifL(x),DxifU(x))].
We will say that f(x) is differentiable at x, and we write
(7)∇f(x)=(Dx1f(x),Dx2f(x),…,Dxnf(x))t.
We call ∇f(x) the gradient of the interval-valued univex function at x.Example 10.
Letf:R2→ℐ defined by f(x)=[x12+x22,2x12+2x22+3]. So fL(x)=x12+x22 and fU(x)=2x12+2x22+3. Dx1fL(x)=2x1,Dx2fL(x)=2x2,Dx1fU(x)=4x1,Dx2fU(x)=4x2. Thus,
(8)Dx1f(x)={[2x1,4x1]ifx1≥0,[4x1,2x1]ifx1<0,Dx2f(x)={[2x2,4x2]ifx2≥0,[4x2,2x2]ifx2<0.
Thus,(9)∇f(x)={([2x1,4x1],[2x2,4x2])tifx1≥0,x2≥0,([2x1,4x1],[4x2,2x2])tifx1≥0,x2<0,([4x1,2x1],[2x2,4x2])tifx1<0,x2≥0,([4x1,2x1],[4x2,4x2])tifx1<0,x2<0.
Further,
(10)∇Lf(x)={(2x1,2x2)tifx1≥0,x2≥0,(2x1,4x2)tifx1≥0,x2<0,(4x1,2x2)tifx1<0,x2≥0,(4x1,4x2)tifx1<0,x2<0.∇Uf(x)={(4x1,4x2)tifx1≥0,x2≥0,(4x1,2x2)tifx1≥0,x2<0,(2x1,4x2)tifx1<0,x2≥0,(2x1,4x2)tifx1<0,x2<0.Remark 11.
IffL=fU, then ∇f(x) of interval-valued functions is the extension of ∇f(x), where f:Ω→R.
The concept of convexity plays an important role in the optimization theory. In recent years, the concept of convexity has been generalized in several directions by using novel and innovative techniques. An important generalization of convex functions is the introduction of univex functions, which was introduced by Bector et al. [15].
LetK be a nonempty open set in Rn, and let f:K→R, η:K×K→Rn, Φ:R→R, and b:K×K×[0,1]→R+, b=b(x,y,λ). If the function f is differentiable, then b does not depend on λ; see [12] or [15].Definition 12.
A differentiable real-valued functionf is said to be univex at y∈K with respect to η,Φ,b if for all x∈K(11)b(x,y)Φ[f(x)-f(y)]≥ηt(x,y)∇f(y).
LetK be a nonempty open set in Rn, and let f:K→ℐ be an interval-valued function, η:K×K→Rn,Φ:ℐ→ℐ, and b:K×K×[0,1]→R+,b=b(x,y,λ).Definition 13 (interval-valued univex function).
A differentiable interval-valued functionf is said to be univex at y∈K with respect to η, Φ, b if for all x∈K(12)b(x,y)Φ[f(x)⊖gf(y)]≽ηt(x,y)∇f(y).Remark 14.
(i) An interval-valued univex function is the extension of a univex function byfL=fU.
(ii)Φ:ℐ→ℐ could be deduced from ϕ:R→R by Φ(A):={y:∃x∈A,ϕ(x)=y,y∈R}.Example 15.
Consider the real-valued functionϕ1 given by ϕ1(x)=x+1,x∈R, then we can obtain Φ1([aL,aU])=[aL+1,aU+1]. If ϕ2(x)=|x|,x∈R. Then
(13)Φ2([aL,aU])={[aL,aU]ifaL≥0,[-aU,-aL]ifaU≤0,[0,max⁡(-aL,aU)]ifaL<0,aU≥0.Example 16.
Letf(x)=[x2,2x2+3], x∈R,b=1, η(x,y)=x-y, Φ=Φ2, then f(x) is univex with respect to b, η, and Φ.Example 17.
Letf(x)=[x3,x3+1],x∈R,
(14)b(x,y)={y2x-yifx≥y,0ifx≤y,η(x,y)={x2+y2+xyifx≥y,x-yifx≤y.
Let Φ:ℐ→ℐ be defined by Φ([aL,aU])=3[aL,aU], then f(x) is univex with respect to b,η and Φ.


## 4. Optimality Criteria

Letf(x),g1(x),…,gm(x) be differentiable interval-valued functions defined on a nonempty open set X⊆Rn. Throughout this paper we consider the following primal problem (P):
(P)min⁡f(x)s.t.g(x)⪯0,i=1,2,…,m.LetP:={x∈X:g(x)⪯0,i=1,2,…,m}. We say x* is an optimal solution of (P) if f(x)≽f(x*) for all P-feasible x. In this section, we obtain sufficient optimality conditions for a feasible solution x* to be efficient or properly efficient for (P) in the form of the following theorems.Theorem 18.
Letx* be P-feasible. Suppose that(i)
there existη, Φ0, b0, Φi, bi, i=1,2,…,m such that
(15)b0(x,x*)Φ0[f(x)⊖gf(x*)]≽ηt(x,x*)∇f(x*),(16)-bi(x,x*)Φi[gi(x*)]≽ηt(x,x*)∇gi(x*)for all feasiblex;(ii)
there existy*=(y1,y2,…,ym)t∈Rm such that
(17)∇f(x*)+∑i=1m‍yi∇g(x*)=0,(18)y*≥0.Further, suppose that
(19)Φ0(A)≽0⇒A≽0,(20)A⪯0⇒Φi(A)≽0,(21)b0(x,x*)>0,bi(x,x*)>0
for all feasible x. Then, x* is an optimal solution of (P).Proof.
Letx be P-feasible. Then,
(22)gi(x)⪯0.
From (20), we conclude that
(23)Φi[gi(x)]≽0.
Thus,
(24)ΦiL[gi(x)]≥0,ΦiU[gi(x)]≥0.
By (15) and Definition 2, we have
(25){b0(x,x*)Φ0[f(x)⊖gf(x*)]}L≥{ηt(x,x*)∇f(x*)}L,{b0(x,x*)Φ0[f(x)⊖gf(x*)]}U≥{ηt(x,x*)∇f(x*)}U.
From (17),
(26)ηt(x,x*)∇f(x*)+ηt(x,x*)∑i=1m‍yi∇g(x*)=0.
It follows from Definition 2 that
(27){ηt(x,x*)∇f(x*)}L+{ηt(x,x*)∑i=1m‍yi∇g(x*)}L=0,{ηt(x,x*)∇f(x*)}U+{ηt(x,x*)∑i=1m‍yi∇g(x*)}U=0.
It is equivalent to
(28){ηt(x,x*)∇f(x*)}L=-{ηt(x,x*)∑i=1m‍yi∇g(x*)}L,{ηt(x,x*)∇f(x*)}U=-{ηt(x,x*)∑i=1m‍yi∇g(x*)}U.
From (16), we have
(29){-bi(x,x*)Φi[gi(x*)]}L≥{ηt(x,x*)∇gi(x*)}L,{-bi(x,x*)Φi[gi(x*)]}U≥{ηt(x,x*)∇gi(x*)}U.
From Definition 1, we have
(30)-{bi(x,x*)Φi[gi(x*)]}U≥{ηt(x,x*)∇gi(x*)}L,-{bi(x,x*)Φi[gi(x*)]}L≥{ηt(x,x*)∇gi(x*)}U.
Thus,
(31){b0(x,x*)Φ0[f(x)⊖gf(x*)]}L≥{ηt(x,x*)∇f(x*)}L=-{ηt(x,x*)∑i=1m‍yi∇g(x*)}L≥∑i=1m‍{bi(x,x*)Φi[gi(x*)]}U≥∑i=1m‍{bi(x,x*)Φi[gi(x*)]}L≥0,{b0(x,x*)Φ0[f(x)⊖gf(x*)]}U≥{b0(x,x*)Φ0[f(x)⊖gf(x*)]}L≥0.
So,
(32)b0(x,x*)Φ0[f(x)⊖gf(x*)]≽0.
From (21), it follows that
(33)Φ0[f(x)⊖gf(x*)]≽0.
By (19),
(34)f(x)⊖gf(x*)≽0.
From Proposition 4, it follows that
(35)f(x)≽f(x*).
Therefore, x* is an optimal solution of (P).Theorem 19.
Letx* be P-feasible. Suppose that(i)
there existη, Φ0, b0, Φi, bi, i=1,2,…,m such that
(36)ηt(x,x*)∇f(x*)≽0⇒b0(x,x*)Φ0[f(x)⊖gf(x*)]≽0,(37)-bi(x,x*)Φi[gi(x*)]⪯0⇒ηt(x,x*)∇gi(x*)⪯0for all feasiblex;(ii)
there existy*=(y1,y2,…,ym)t∈Rm such that
(38)∇f(x*)+∑i=1m‍yi∇g(x*)=0,(39)y*≥0.Further, suppose that(40)Φ0(A)≽0⇒A≽0,(41)A⪯0⇒Φi(A)≽0,(42)b0(x,x*)>0,bi(x,x*)>0for all feasiblex. Then, x* is an optimal solution of (P).Proof.
Letx be P-feasible. Then, gi(x*)⪯0, from (41), we obtain that
(43)Φi[gi(x*)]≽0.
So,
(44)-bi(x,x*)Φi[gi(x*)]⪯0.
By (37),
(45)ηt(x,x*)∇gi(x*)⪯0.
Thus,
(46)-∑i=1m‍yiηt(x,x*)∇gi(x*)≽0.
Then, we have
(47)-{∑i=1m‍yiηt(x,x*)∇gi(x*)}U={-∑i=1m‍yiηt(x,x*)∇gi(x*)}L≥0,-{∑i=1m‍yiηt(x,x*)∇gi(x*)}L={-∑i=1m‍yiηt(x,x*)∇gi(x*)}U≥0.
From (38) and Definition 2, it follows that
(48){ηt(x,x*)∇f(x*)}L+{ηt(x,x*)∑i=1m‍yi∇g(x*)}L=0,{ηt(x,x*)∇f(x*)}U+{ηt(x,x*)∑i=1m‍yi∇g(x*)}U=0.
It is equivalent to
(49){ηt(x,x*)∇f(x*)}L=-{ηt(x,x*)∑i=1m‍yi∇g(x*)}L,{ηt(x,x*)∇f(x*)}U=-{ηt(x,x*)∑i=1m‍yi∇g(x*)}U.
Therefore,
(50){ηt(x,x*)∇f(x*)}L≥0,{ηt(x,x*)∇f(x*)}U≥0.
From Definition 2, we obtain that
(51)ηt(x,x*)∇f(x*)≽0.
By (36),
(52)b0(x,x*)Φ0[f(x)⊖gf(x*)]≽0.
Then, from (40) and (42), we have
(53)f(x)⊖gf(x*)≽0.
From Proposition 4, it follows that
(54)f(x)≽f(x*).
Therefore, x* is an optimal solution of (P).


## 5. Duality

Consider the following:(D)max⁡f(u)s.t.∇f(u)+∑i=1m‍yi∇gi(u)=0,yi∇gi(u)≽0yi≥0.Theorem 20 (Weak duality).
Letx be P-feasible, and let (u,y) be D-feasible. Assume that there exist η, Φ0, b0, Φi, bi, i=1,2,…,m such that
(55)b0(x,u)Φ0[f(x)⊖gf(u)]≽ηt(x,u)∇f(u),-bi(x,u)Φi[gi(u)]≽ηt(x,u)∇gi(u)
at u;
(56)Φ0(A)≽0⇒A≽0,b0(x,u)>0,bi(x,u)≥0
and ∑i=1m‍bi(x,u)yiΦi(gi(u))≽0. Then, f(x)≽f(u).Proof.
It is similar to the proof of Theorem18.Theorem 21 (weak duality).
Letx be P-feasible, and let (u,y) be D-feasible. Assume that there exist η, Φ0, b0, Φ1, b1 such that
(57)ηt(x,u)∇f(u)≽0⇒b0(x,u)Φ0[f(x)⊖gf(u)]≽0,(58)-b1(x,u)Φ1[∑i=1m‍yigi(u)]⪯0⇒∑i=1m‍yiηt(x,u)∇gi(u)⪯0
at u;
(59)Φ0(A)≽0⇒A≽0,(60)A≽0⇒Φ1(A)≽0,(61)b0(x,u)>0,b1(x,u)≥0.
Then, f(x)≽f(u).Proof.
Since(u,y) is D-feasible, then yit∇gi(u)≽0, from (60) and (61),
(62)-b1(x,u)Φ1[∑i=1m‍yigi(u)]⪯0.
Then, we have
(63)∑i=1myiηt(x,u)∇gi(u)⪯0.
Thus,
(64)-∑i=1m‍yiηt(x,u)∇gi(u)≽0,-{∑i=1m‍yiηt(x,u)∇gi(u)}U={-∑i=1m‍yiηt(x,u)∇gi(u)}L≥0,-{∑i=1m‍yiηt(x,u)∇gi(u)}L={-∑i=1m‍yiηt(x,u)∇gi(u)}U≥0.
Since (u,y) is D-feasible we can obtain that,
(65)∇f(u)+∑i=1myi∇gi(u)=0.
So,
(66)ηt(x,u)∇f(u)+ηt(x,u)∑i=1m‍yi∇gi(u)=0.
By Definition2, it follows that
(67){ηt(x,u)∇f(u)}L+{ηt(x,u)∑i=1m‍yi∇gi(u)}L=0,{ηt(x,u)∇f(u)}U+{ηt(x,u)∑i=1m‍yi∇gi(u)}U=0.
Therefore,
(68){ηt(x,u)∇f(u)}L=-{ηt(x,u)∑i=1m‍yi∇gi(u)}L≥0,{ηt(x,u)∇f(u)}U=-{ηt(x,u)∑i=1m‍yi∇gi(u)}U≥0.
Then,
(69)ηt(x,u)∇f(u)≽0.
By (57),
(70)b0(x,u)Φ0[f(x)⊖gf(u)]≽0.
From (59) and (61),
(71)f(x)⊖gf(u)≽0.
thus,
(72)f(x)≽f(u).Theorem 22 (strong duality).
Ifx* is P-optimal and a constraint qualification is satisfied at x*, then there exists y*=(y1,y2,…,ym)t∈Rm such that (x*,y*) is D-feasible and the values of the objective functions for (P) and (D) are equal at x* and (x*,y*), respectively. Furthermore, if for all P-feasible x and D-feasible (u,y), the hypotheses of Theorem 19 are satisfied, then (x*,y*) is D-optimal.Proof.
Since a constraint qualification is satisfied atx*, there exists y*∈Rm such that the following Kuhn-Tucker conditions are satisfied:
(73)∇f(x*)+∑i=1m‍yi∇gi(x*)=0,∑i=1m‍yi∇gi(x*)=0,yi≥0.
Therefore,(x*,y*) is D-feasible.
Suppose that(x*,y*) is not D-optimal. Then, there exists a D-feasible (u,y) such that f(u)≻f(x*). This contradicts Theorem 20. Therefore, (x*,y*) is D-optimal.


## 6. Numerical Example

Consider the following example:(74)minimizef(x)=[x1-sin(x2)+1,x1-sin(x2)+3]subjecttog1(x)=[sin(x1)-4sin(x2)-2,sin(x1)-4sin(x2)]⪯0,g2(x)=[2sin(x1)+7sin(x2)+x1-7,2sin(x1)+7sin(x2)+x1-6]⪯0,g3(x)=[2x1+2x2-5,2x1+2x2-3]⪯0,g4(x)=[4x12+4x22-12,4x12+4x22-9]⪯0,g5(x)=[-sin(x1)-1,-sin(x1)]⪯0,g6(x)=[-sin(x2)-1,-sin(x2)]⪯0.Note that the interval-valued objective function is univex with respect tob=1, η(x,u)=x-u, Φ([aL,aU])=[aL,aU], and every gi(i=1,2,…,6) is univex with respect to b=1, Φ([aL,aU])=[aL,aU](75)η(x,u)=(sinx1-sinu1cos⁡u1,sinx2-sinu2cos⁡u2)T,
where x=(x1,x2)T and u=(u1,u2)T.It is easy to see that the problem satisfies the assumptions of Theorem18. Then,
(76)(1,-cos⁡x2)T+μ1(cos⁡x1,-4cos⁡x2)T+μ2(2cos⁡x1+1,7cos⁡x2)T+μ3(2,2)T+μ4(8x1,8x2)T+μ5(-cos⁡x1,0)T+μ6(0,-cos⁡x2)T=(0,0)T.After some algebraic calculations, we obtain thatx*=(0,sin-1(6/7))T and u*=(0,1/7,0,0,10/7,0)T. Therefore, x* is a solution.



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