# A Class of Fuzzy Variational Inequality Based on Monotonicity of Fuzzy Mappings

Authors: Zezhong Wu; Jiuping Xu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Invex monotonicity and pseudoinvex monotonicity of fuzzy mappings are introduced in this paper, and relations are discussed between invex monotonicity (pseudoinvex monotonicity) and invexity (pseudoinvexity) of fuzzy mappings. The existence of a solution to the fuzzy variational-like inequality is discussed, and the existence theorem can be achieved. Furthermore, some extended properties of the fuzzy variational-like inequality are researched. Finally, method of solution is discussed based on genetic algorithm.


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## Body

## 1. Introduction

In [1], Chang and Zadeh introduced the concept of fuzzy mapping. Since then, fuzzy mapping has been extensively studied by many authors. Nanda and Kar [2] proposed a concept of convex fuzzy mapping in 1992 and proved that a fuzzy mapping is convex if and only if its epigraph is a convex set. In recent years, there have been increasing attempts to weaken the convexity condition of fuzzy mapping, such as Yan and Xu [3], Panigrahi et al. [4], and Wu and Xu [5, 6]. It is well known that in classical (non-fuzzy) convex analysis, some properties are shared by different kinds of functions that are more general than convex functions, which has given rise to the study of generalized convexity and later to that of generalized monotonicity. Just as convex functions are characterized by a monotone gradient, different kinds of generalized convex functions give rise to gradient maps with certain generalized monotonicity properties which are inherited from generalized convexity of the underlying functions. At the same time, it is a useful method in researching variational inequality by means of monotonicity and generalized monotonicity of functions. For corresponding research of monotonicity and generalized monotonicity of functions, one can refer to [7–9]. Similarly, one can research some properties of fuzzy mappings and fuzzy variational inequality by studying their monotonicity. On the other hand, it is worth noting that monotonicity has played a very important role in the study of the existence and solution methods of variational inequality problems. Similarly, one can research fuzzy variational inequality by monotonicity of fuzzy mappings. However, very few investigations have appeared to study monotonicity of fuzzy mappings. Based on the above, we give concepts of monotonicity and generalized monotonicity of fuzzy mappings and discuss relations of generalized monotonicity and generalized convexity.Vector variational inequality was first introduced and studied by Cottle et al. [10] in finite-dimensional Euclidean spaces. This is a generalization of a scalar variational inequality to the vector case by virtue of multicriteria considering. Since then, the theory with applications for vector variational inequality and vector complementarity problems have been studied and generalized by many authors (see, e.g., [11–21] and the references therein). In particular, in [12], Guang-Ya and Xiao-Qi discussed the existence of a solution to the vector variational inequality and the existence of the solution of the vector complementary problem and obtained some valuable results. It offers some ideal for research of the existence of a solution of other variational inequality problems. However, facing uncertainty is a constant challenge for optimization and decision making. Treating uncertainty with fuzzy mathematics results in the study of fuzzy optimization and decision making. Recently, Chang and Zhu [22] introduced the concepts of the variational inequality problem for fuzzy mappings which were later developed by Noor [23–25]. For further research of variational inequalities of fuzzy mappings, one can refer to [26–30]. However, very few investigations have appeared to study fuzzy variational inequalities and the existence of a solution by means of fuzzy numbers. In [5], Wu and Xu discussed the relationship between the fuzzy prevariational inequality and the fuzzy vector optimization problem based on nonconvex fuzzy mappings. In [6], Wu and Xu discussed the relationship between the fuzzy variational-like inequality and the fuzzy vector optimization problem based on generalized convex fuzzy mappings and gave an application example of a fuzzy variational-like inequality: the fuzzy variational-like inequality representation of a fuzzy transportation equilibrium problem. On the other hand, the role of generalized monotonicity of the operator in variational inequality problems corresponds to the role of generalized convexity of the objective function in mathematical programming problems. Similarly, the role of monotonicity of fuzzy mappings should also be very important for research of fuzzy variational inequality problems, particularly research of the existence of a solution of a fuzzy variational inequality. Based on the above, we introduce the fuzzy variational-like inequality, discuss some properties of the fuzzy variational-like inequality, and study the existence of a solution of the fuzzy variational-like inequality by virtue of generalized monotonicity of fuzzy mappings. It is well known that we can solve variational inequality by transforming a variational inequality problem into an optimization problem, for example, [18, 31–33]. Similarly, fuzzy variational inequality also can be transformed into a fuzzy optimization problem. It is difficult to solve a fuzzy optimization problem by means of traditional optimization methods, but a special genetic algorithm can be used to solve fuzzy optimization problems. Therefore, in the sixth section, we discuss the solving of the fuzzy variational-like inequality by virtue of genetic algorithm.In this paper, we study invex monotonicity and pseudo-invex monotonicity of fuzzy mappings and discuss relations between invex monotonicity (pseudo-invex monotonicity) and invexity (pseudo-invexity) of fuzzy mappings. We discuss the existence of a solution to the fuzzy variational-like inequality. Furthermore, some extended properties of the fuzzy variational-like inequality are researched. Finally, method of solution is discussed based on genetic algorithm.This paper is organized as follows. Section2 recalls some definitions and results in reference to fuzzy numbers. Section 3 introduces invex monotonicity and pseudo-invex monotone of fuzzy mapping and discusses some properties. Section 4 defines a class of fuzzy variational inequality: fuzzy variational-like inequality, and discusses the solution existence for the fuzzy variational-like inequality. Section 5 discusses extended qualities of the fuzzy variational-like inequality. Section 6 discusses the method of solving for the fuzzy variational-like inequality and gives a genetic algorithm of a class of the fuzzy variational-like inequality.


## 2. Preliminaries

A fuzzy set ofRn is a mapping μ:Rn→[0,1]. For each such fuzzy set μ, its α-cut set is denoted by [μ]α={x∈Rn:μ(x)≥α} for all α∈(0,1]. The support of μ is denoted by suppμ; that is, {x∈Rn∣μ(x)>0}. The closure of suppμ is defined as [μ]0.Definition 1 (see Wu and Xu [6]).
A fuzzy numberμ is a fuzzy set with the following properties:(1)
μis normal; that is, there exists x0∈Rn such that μ(x0)=1;(2)
μ is convex fuzzy set; that is, μ(λx+(1-λ)y)≥min(μ(x),μ(y)), x,y∈Rn, λ∈[0,1];(3)
[μ]0 is compact.
LetE denote the family of fuzzy numbers; that is, E denotes the family of compact and convex fuzzy set on R1. Obviously, [μ]α is a nonempty compact convex subset of R1 (denoted by [μ*(α),μ*(α)]) for all μ∈E and for all α∈[0,1].
A precise numbera is a special case of fuzzy number encoded as
(1)a~(t)={1,ift=a,0,ift≠a.
However, a precise number will be denoted as usual, in particular, number 0. The fuzzy numbersμ,ν∈E are represented by (μ*(α),μ*(α)) and (ν*(α),ν*(α)), respectively. For each real number λ, the addition μ+~ν and scalar multiplication λμ are defined as follows:
(2)(μ+~ν)(x)=supy+z=xmin[μ(y),ν(z)],(λμ)(x)={μ(λ-1x)ifλ≠0,forμ,ν∈E,λ∈R.0ifλ=0,
It is well known that for allμ,ν∈E and λ∈R1(3)(μ+~ν)*(α)=μ*(α)+ν*(α),(μ+~ν)*(α)=μ*(α)+ν*(α),(λμ)*(α)={λμ*(α)ifλ≥0,λμ*(α)ifλ<0,(λμ)*(α)={λμ*(α)ifλ≥0,forgivenα∈[0,1].λμ*(α)ifλ<0,
Forx=(x1,x2,…,xn), y=(y1,y2,…,yn)∈Rn, x≤y if and only if xi≤yi  (i=1,2,…,n), and x<y if and only if x≤y and x≠y.Definition 2.
Forμ,ν∈E, μ⪯ν, if and only if for every α∈[0,1], μ*(α)≤ν*(α) and μ*(α)≤ν*(α). Ifμ⪯ν, ν⪯μ, then μ=ν.μ≺ν if and only if μ⪯ν and ∃α0∈[0,1], such that μ*(α0)<ν*(α0) or ν*(α0)<μ*(α0).
Forμ,ν∈E, if either μ⪯ν or ν⪯μ, then μ and ν are comparable; otherwise, they are noncomparable.
Ifμ,ν∈E, there exists ω∈E such that μ=ν+~ω, then we say the Hukuhara difference of μ and ν exists, call ω the H-difference of μ and ν, and denote μ-~ν=ω.
It is obvious that if the H-differenceμ-~ν exists, then (μ-~ν)*(α)=μ*(α)-ν*(α), (μ-~ν)*(α)=μ*(α)-ν*(α).Definition 3.
A mappingF:K(⊂Rn)→E is said to be a fuzzy mapping. Denote [F(x)](α)=[F(x)*(α),F(x)*(α)], for all α∈[0,1].Definition 4 (see Buckley and Feuring [34]).
LetF be a fuzzy mapping from the set of real numbers R to the set of all fuzzy numbers, and let [F(x)](α)=[F*(x)(α),F*(x)(α)]. Assume that the partial derivatives of F*(x)(α), F*(x)(α) with respect to x∈R for each α∈[0,1] exist and are denoted by F*′(x,α), F*′(x,α), respectively. Let Γ(x,α)=[F*′(x,α),F*′(x,α)] for x∈R, α∈[0,1]. If Γ(x,α) defines the α-cut of a fuzzy number for each x∈R, then F(x) is said to be differentiable and is written as (dF/dx)[α]=[F*′(x,α),F*′(x,α)], for all x∈R, α∈[0,1].Definition 5 (see Panigrahi et al. [4]).
LetF:K(⊂Rn)→E be a fuzzy mapping, where K⊂Rn is an open set. Let x=(x1,x2,…,xn)∈K. Let Dxi,  (i=1,2,…,n) stand for the “partial differentiation” with respect to the ith variable xi. Assume that, for all α∈[0,1], F*(x)(α), F*(x)(α) have continuous partial derivatives so that DxiF*(x,α), DxiF*(x,α) are continuous. Define
(4)DxiF(x,α)=[DxiF*(x,α),DxiF*(x,α)],fori=1,2,…,n,α∈[0,1].
If eachi=1,2,…,n, DxiF(x,α) defines the α-cut of a fuzzy number, then F is called differentiable at x, and it can be represented as
(5)∇~F(x)=(Dx1F(x,α),Dx2F(x,α),…,DxnF(x,α)).∇~F(x) is said to be the gradient of the fuzzy mapping F at x.
U is said to be an n-dimensional fuzzy vector if and only if the components of U are composed by n fuzzy numbers, denoted by U=(μ1,μ2,…,μn)T. The set of all n-dimensional fuzzy vectors is denoted by (E)n.
Aλ level vector of fuzzy vector U=(μ1,μ2,…,μn)T is defined as
(6)[U]λ=[[μ1]λ,[μ2]λ,…,[μn]λ]T,[U]*λ=[[μ1]*λ,[μ2]*λ,…,[μn]*λ]T,[U]λ*=[[μ1]λ*,[μ2]λ*,…,[μn]λ*]T.
The addition and the scalar multiplication of fuzzy vectors U=(μ1,μ2,…,μn)T and V=(ν1,ν2,…,νn)T are defined as
(7)U+~V=(μ1+~ν1,μ2+~ν2,…,μn+~νn)T,kU=(kμ1;kμ2;…;kμn)T,k∈R,kμi={(kμi*,kμi*,α):α∈[0,1]},fork>0.


## 3. The Fuzzy Invex Monotone Mapping

In this section, we put forward some definitions of invex monotonicity of fuzzy mappings and discuss the relationship between invexity and invex monotonicity of fuzzy mappings.Definition 6.
A differentiable comparable fuzzy mappingF:K→E is said to be(a)
fuzzy invex with respect toη:K×K→Rn, if and only if
(8)F(x)⪰η(x,y)T∇F(y)+~F(y),∀x,y∈K;(b)
fuzzy incave with respect toη:K×K→Rn, if and only if
(9)F(x)⪯η(x,y)T∇F(y)+~F(y),∀x,y∈K;(c)
fuzzy strictly invex with respect toη:K×K→Rn, if and only if
(10)F(x)≻η(x,y)T∇F(y)+~F(y),∀x,y∈K;(d)
fuzzy strictly incave with respect toη:K×K→Rn, if and only if
(11)F(x)≺η(x,y)T∇F(y)+~F(y),∀x,y∈K;(e)
fuzzy pseudo-invex with respect toη:K×K→Rn, if and only if
(12)η(x,y)T∇F(y)⪰0⟹F(x)⪰F(y),∀x,y∈K;(f)
fuzzy strictly pseudo-invex with respect toη:K×K→Rn, if and only if
(13)η(x,y)T∇F(y)⪰0⟹F(x)≻F(y),∀x,y∈K.Example 7 (see Wu and Xu [6]).
LetF:K(⊂R1)→E represent the reproduction rate of some germ:
(14)F(x)(t)={tx2t∈[0,x2],1-t-x2x2t∈(x2,2x2],0t∉[0,2x2].
So,[F(x)]α=[αx2,(2-α)x2], α∈[0,1]. Then there is η(x,y)=x-y, such that F(x) is a fuzzy invex mapping, where K=(0,∞), x,y∈K, x represents the predicted quantity, and t represents the actual reproduction quantity.Example 8.
Consider the fuzzy mapping[F(x)]α=[αx2,(2-α)x2], α∈[0,1]. Then, there is an η(x,y)=x5-y5 such that F(x) is a fuzzy pseudoinvex mapping, where K=(0,∞), x,y∈K.Remark 9.
For an invex fuzzy mapping, there must existη:K×K→Rn, such that
(15)F(x)⪰η(x,y)T∇F(y)+~F(y),∀x,y∈K,
holds.Proof.
SinceF is a comparable fuzzy mapping, then for all x,y∈Rn, there is
(16)F(x)⪰F(y)
or
(17)F(x)⪯F(y).
Without loss of generality, suppose that
(18)F(x)⪰F(y).
Thus, for all α∈[0,1],
(19)F*(x)(α)≥F*(y)(α),F*(x)(α)≥F*(y)(α).
If∇F(y)=0, for any η(x,y):K×K→Rn, the result holds.
If∇F(y)≠0, for given α∈[0,1].
(1) WhenF*(x)(α)-F*(y)(α)≥F*(x)(α)-F*(y)(α).
(i) If∇F*(y)(α)>0, then take
(20)η(x,y)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×∇F*(y)(α).
Thus,
(21)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)]T∇F*(y)(α)=F*(x)(α)-F*(y)(α)≤F*(x)(α)-F*(y)(α);
that is,
(22)F*(x)(α)≥η(x,y)T∇F*(y)(α)+F*(y)(α).
From ∇F*(y)(α)≥∇F*(y)(α)>0, there is
(23)[∇F*(y)(α)]T(∇F*(y)(α)-∇F*(y)(α))≥0;
so,
(24)[∇F*(y)(α)]T∇F*(y)(α)≥[∇F*(y)(α)]T∇F*(y)(α).
Hence,
(25)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)]T∇F*(y)(α)≤F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)]T∇F*(y)(α)=F*(x)(α)-F*(y)(α).
That is,
(26)F*(x)(α)≥η(x,y)T∇F*(y)(α)+F*(y)(α).
From (22) and (26), it follows that
(27)F(x)⪰η(x,y)T∇F(y)+~F(y),∀x,y∈K.
(ii) If∇F*(y)(α)<0, then take
(28)η(x,y)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×∇F*(y)(α).
Thus,
(29)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)]T∇F*(y)(α)=F*(x)(α)-F*(y)(α).
From ∇F*(y)(α)<0 and ∇F*(y)(α)≥∇F*(y)(α), there is
(30)[∇F*(y)(α)]T(∇F*(y)(α)-∇F*(y)(α))≤0;
so,
(31)[∇F*(y)(α)]T∇F*(y)(α)≤[∇F*(y)(α)]T∇F*(y)(α).
Hence,
(32)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)]T∇F*(y)(α)≤F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)]T∇F*(y)(α)=F*(x)(α)-F*(y)(α)≤F*(x)(α)-F*(y)(α).
That is,
(33)F*(x)(α)≥η(x,y)T∇F*(y)(α)+F*(y)(α).
From (29) and (33), it follows that
(34)F(x)⪰η(x,y)T∇F(y)+~F(y),∀x,y∈K.
(iii) If∇F*(y)(α) is indefinite, there is a vector C∈Rn, such that
(35)∇F*(y)(α)+C≤0,CT∇F*(y)(α)≤0
holds.
Take(36)η(x,y)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×(∇F*(y)(α)+C).
Thus,
(37)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)+C]T∇F*(y)(α)=F*(x)(α)-F*(y)(α)+F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×CT∇F*(y)(α)≤F*(x)(α)-F*(y)(α).
That is,
(38)F*(x)(α)≥η(x,y)T∇F*(y)(α)+F*(y)(α).
From ∇F*(y)(α)+C≤0 and ∇F*(y)(α)≥∇F*(y)(α), there is
(39)[∇F*(y)(α)+C]T(∇F*(y)(α)-∇F*(y)(α))≤0;
so,
(40)[∇F*(y)(α)+C]T∇F*(y)(α)≤[∇F*(y)(α)+C]T∇F*(y)(α).
Hence,
(41)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)+C]T∇F*(y)(α)≤F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)+C]T∇F*(y)(α)=F*(x)(α)-F*(y)(α)+F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×CT∇F*(y)(α)≤F*(x)(α)-F*(y)(α)≤F*(x)(α)-F*(y)(α).
That is,
(42)F*(x)(α)≥η(x,y)T∇F*(y)(α)+F*(y)(α).
From (38) and (42), it follows that
(43)F(x)⪰η(x,y)T∇F(y)+~F(y),∀x,y∈K.
(2) WhenF*(x)(α)-F*(y)(α)≤F*(x)(α)-F*(y)(α).
(i) If∇F*(y)(α)>0, then take
(44)η(x,y)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×∇F*(y)(α).
Thus,
(45)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α).
On the other hand,
(46)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)]TF*(y)(α)≤F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)]TF*(y)(α)=F*(x)(α)-F*(y)(α)≤F*(x)(α)-F*(y)(α).
That is,
(47)η(x,y)T∇F*(y)(α)≤F*(x)(α)-F*(y)(α).
From (45) and (47), it follows that
(48)F(x)⪰η(x,y)T∇F(y)+~F(y),∀x,y∈K.
(ii) If∇F*(y)(α)<0, then take
(49)η(x,y)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×∇F*(y)(α).
Thus,
(50)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)]T∇F*(y)(α)=F*(x)(α)-F*(y)(α)≤F*(x)(α)-F*(y)(α).
That is,
(51)η(x,y)T∇F*(y)(α)≤F*(x)(α)-F*(y)(α).
On the other hand,
(52)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)]T∇F*(y)(α)≤F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)]TF*(y)(α)=F*(x)(α)-F*(y)(α);
that is,
(53)η(x,y)T∇F*(y)(α)≤F*(x)(α)-F*(y)(α).
From (51) and (53), it follows that
(54)F(x)⪰η(x,y)T∇F(y)+~F(y),∀x,y∈K.
(iii) If∇F*(y)(α) is indefinite, there is a vector C∈Rn, such that
(55)∇F*(y)(α)+C≤0,CT∇F*(y)(α)≤0
holds.
Take(56)η(x,y)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×(∇F*(y)(α)+C).
Thus,
(57)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)+C]T∇F*(y)(α)=F*(x)(α)-F*(y)(α)+F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×CT∇F*(y)(α)≤F*(x)(α)-F*(y)(α)≤F*(x)(α)-F*(y)(α).
That is,
(58)η(x,y)T∇F*(y)(α)≤F*(x)(α)-F*(y)(α).
On the other hand,
(59)η(x,y)T∇F*(y)(α)=F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)+C]T∇F*(y)(α)≤F*(x)(α)-F*(y)(α)[∇F*(y)(α)]T∇F*(y)(α)×[∇F*(y)(α)+C]TF*(y)(α)<F*(x)(α)-F*(y)(α);
that is,
(60)η(x,y)T∇F*(y)(α)≤F*(x)(α)-F*(y)(α).
From (58) and (60), it follows that
(61)F(x)⪰η(x,y)T∇F(y)+~F(y),∀x,y∈K.
This completes the proof.Definition 10.
A comparable fuzzy mappingF:K→(E)n is said to be(a)
fuzzy invex monotone onK, if ∃η:K×K→Rn such that for any x,y∈K,
(62)η(y,x)TF(y)⪰η(y,x)TF(x);(b)
fuzzy pseudo-invex monotone onK, if ∃η:K×K→Rn such that for any x,y∈K,
(63)η(y,x)TF(y)⪰0⟹η(y,x)TF(x)⪰0;(c)
fuzzy strictly invex monotone onK, if ∃η:K×K→Rn such that for any x,y∈K, x≠y,
(64)η(y,x)TF(y)≻η(y,x)TF(x);(d)
fuzzy strictly pseudo-invex monotone if and only if∃η:K×K→Rn such that for any x,y∈K, x≠y,
(65)η(y,x)TF(y)⪰0⟹η(y,x)TF(x)≻0.Definition 11.
The functionη:K×K→Rn is said to be a skew function if
(66)η(x,y)=-η(y,x),∀x,y∈K.Definition 12.
Lety∈K, K is said to be invex at y with respect to η:K×K→Rn if, for each x∈K, λ∈[0,1],
(67)y+λη(x,y)∈K.K is said to be an invex set with respect to η if K is invex at each y∈K.Theorem 13.
If a differentiable fuzzy mappingF:K→E is invex on K with respect to η:K×K→Rn and η is a skew function. Then, ∇F:K→(E)n is fuzzy invex monotone with respect to the same η.Proof.
LetF be invex on K, then there exists η(x,y)∈Rn, such that
(68)F(x)⪰η(x,y)T∇F(y)+~F(y),∀x,y∈K.
That is, there are
(69)F*(x)(α)≥η(x,y)T∇F*(y)(α)+F*(y)(α),F*(x)(α)≥η(x,y)T∇F*(y)(α)+F*(y)(α),
for all α∈[0,1].
By changingx for y,
(70)F(y)⪰η(y,x)T∇F(x)+~F(x).
That is, there are
(71)F*(y)(α)≥η(y,x)T∇F*(x)(α)+F*(x)(α),F*(y)(α)≥η(y,x)T∇F*(x)(α)+F*(x)(α),
for all α∈[0,1].
From (69) and (71), it follows that
(72)η(x,y)T∇F*(y)(α)+η(y,x)T∇F*(x)(α)≤0,η(x,y)T∇F*(y)(α)+η(y,x)T∇F*(x)(α)≤0.
As η(x,y)+η(y,x)=0, then
(73)η(y,x)T∇F*(y)(α)≥η(y,x)T∇F*(x)(α),η(y,x)T∇F*(y)(α)≥η(y,x)T∇F*(x)(α).
Therefore, from (73), there is
(74)η(y,x)T∇F(y)⪰η(y,x)T∇F(x).Corollary 14.
If a differentiable fuzzy mappingF:K→E is invex on K with respect to η:K×K→Rn and η is a skew function. Then, ∇F:K→(E)n is fuzzy pseudo-invex monotone with respect to the same η.Proof.
From Theorem13, it follows that
(75)η(y,x)T∇F(y)⪰η(y,x)T∇F(x),∀x,y∈K.
Thus,
(76)η(y,x)T∇F*(y)(α)≥η(y,x)T∇F*(x)(α),η(y,x)T∇F*(y)(α)≥η(y,x)T∇F*(x)(α),
for all α∈[0,1].
Ifη(y,x)T∇F(x)⪰0,
(77)η(y,x)T∇F*(x)(α)≥0,η(y,x)T∇F*(x)(α)≥0,
for all α∈[0,1].
Thus, from (76) and (77),
(78)η(y,x)T∇F*(y)(α)≥0,η(y,x)T∇F*(y)(α)≥0
holds. Therefore, η(y,x)T∇F(y)⪰0.Theorem 15.
If a differentiable fuzzy mappingF:K→E is strictly invex on K with respect to η:K×K→Rn and η is a skew function. Then, ∇F:K→(E)n is fuzzy strictly invex monotone on K with respect to the same η.Proof.
Assume thatF is strictly invex on K, then there exists η(x,y)∈Rn, such that, for any x,y∈K, x≠y,
(79)F(x)≻η(x,y)∇F(y)+~F(y).
Thus, there exists some α0∈[0,1], such that
(80)F*(x)(α0)>η(x,y)T∇F*(y)(α0)+F*(y)(α0),
or
(81)F*(x)(α0)>η(x,y)T∇F*(y)(α0)+F*(y)(α0).
For α0∈[0,1], without loss of generality, suppose that
(82)F*(x)(α0)>η(x,y)T∇F*(y)(α0)+F*(y)(α0).
By changing x for y,
(83)F*(y)(α0)>η(y,x)T∇F*(x)(α0)+F*(x)(α0).
By (82), (83) and since η is a skew function, we have
(84)η(y,x)T∇F*(y)(α0)>η(y,x)T∇F*(x)(α0).
On the other hand, for other α∈[0,1],
(85)η(y,x)T∇F(y)⪰η(y,x)T∇F(x).
Therefore,
(86)η(y,x)T∇F(y)≻η(y,x)T∇F(x).Theorem 16.
If a differentiable fuzzy mappingF:K→E is strictly pseudo-invex on K with respect to η:K×K→Rn and η is a skew function. Then, ∇F:K→(E)n is fuzzy strictly pseudo-invex monotone on K with respect to the same η.Proof.
LetF be a fuzzy strictly pseudo-invex, then there exists η(x,y)∈Rn, such that for any x,y∈K, x≠y,
(87)η(x,y)T∇F(y)⪰0⟹F(x)≻F(y).
We need to show that there exists η(x,y)∈Rn, such that
(88)η(x,y)T∇F(y)⪰0⟹η(x,y)T∇F(x)≻0,
for all x,y∈K, x≠y.
By contradiction, suppose thatη(x,y)T∇F(x)⪯0, then there exists some α0∈[0,1], such that
(89)η(x,y)T∇F*(x)(α0)≤0,
or
(90)η(x,y)T∇F*(x)(α0)≤0.
Without loss of generality, assume that
(91)η(x,y)T∇F*(x)(α0)≤0.
As η(x,y)+η(y,x)=0, then
(92)η(y,x)T∇F*(x)(α0)≥0.
Since F is strictly pseudo-invex on K, then for α0∈[0,1], there is
(93)η(y,x)T∇F*(x)(α0)≥0⟹F*(x)(α0)≤F*(y)(α0),
which is a contradiction.Theorem 17.
LetF:K→(E)n be a fuzzy strictly pseudo monotone mapping on K with respect to η:K×K→Rn; then, F is a fuzzy pseudo monotone with respect to η on K.Proof.
AsF is a strictly pseudo monotone with respect to η on K, then for any α∈[0,1],
(94)η(y,x)T∇F*(x)(α)≥0⟹F*(x)(α)≤F*(y)(α),η(y,x)T∇F*(x)(α)≥0⟹F*(x)(α)≤F*(y)(α).F*(x)(α), F*(x)(α) is strictly pseudo monotone with respect to η on K. Thus, F*(x)(α), F*(x)(α) also is pseudo monotone with respect to η on K; that is, F is pseudo monotone with respect to η on K.Theorem 18.
LetF:K→E be a differentiable mapping, and suppose that (i)
η satisfies the following conditions:
(a)
η(y,y+tη(x,y))=-tη(x,y),(b)
η(x,y+tη(x,y))=(1-t)η(x,y);(ii)
K is an invex set with respect to η;(iii)
for  eachx≠y, some α0∈[0,1],
(a)
F*(y)(α0)>F*(x)(α0) implies η(x,y)T∇F*(y+λ~η(x,y))(α0)<0,or
(b)
F*(y)(α0)>F*(x)(α0) implies η(x,y)T∇F*(y+λ~η(x,y))(α0)<0;(iv)
∇F:K→(E)n is fuzzy pseudo-invex monotone with respect to η on K.Then, F is a fuzzy pseudo-invex mapping η on K.Proof.
Setx,y∈K, x≠y, and η(x,y)T∇F(y)⪰0 holds. Thus, we need to show that F(x)⪰F(y); that is, F*(x)(α)≥F*(y)(α) and F*(x)(α)≥F*(y)(α), for all α∈[0,1].
Assume the contrary, that is,F(x)F(y). Thus, there exists some α0∈[0,1], such that
(95)F*(x)(α0)<F*(y)(α0)
or
(96)F*(x)(α0)<F*(y)(α0).
Without loss of generality, suppose that
(97)F*(x)(α0)<F*(y)(α0).
By hypothesis (iii),
(98)η(x,y)T∇F*(y+λ~η(x,y))(α0)<0,
for some λ~∈(0,1).
It follows from (98) and (i) that
(99)η(y+λ~η(x,y),y)T×∇F*(y+λ~η(x,y))(α0)<0,
for some λ~∈(0,1).
Since∇F is a pseudo-invex monotone with respect to η, thus
(100)η(y+λ~η(x,y),y)T∇F*(y)(α0)<0.
From η(y+λ~η(x,y),y)=λ~η(x,y) and λ~∈(0,1), (100) becomes
(101)η(x,y)T∇F*(y)(α0)<0.
This contradicts η(x,y)T∇F*(y)(α)⪰0.Theorem 19.
LetF:K→E be a differentiable mapping, and suppose that (i)
η satisfies the following conditions:
(a)
η(y,y+tη(x,y))=-tη(x,y),(b)
η(x,y+tη(x,y))=(1-t)η(x,y);(a)
η(y,y+tη(x,y))=-tη(x,y),(b)
η(x,y+tη(x,y))=(1-t)η(x,y);(ii)
K is an invex set with respect to η;(iii)
for eachx≠y, some α0∈[0,1],
(a)
F*(y)(α0)>F*(x)(α0) implies η(x,y)T∇F*(y+λ~η(x,y))(α0)<0,or
(b)
F*(y)(α0)>F*(x)(α0) implies η(x,y)T∇F*(y+λ~η(x,y))(α0)<0;(iv)
∇F:K→(E)n is fuzzy strictly pseudo-invex monotone with respect to η on k.Then, F is a fuzzy strictly pseudoinvex mapping η on k.Proof.
Letx,y∈K, x≠y, such that η(x,y)T∇F(y)⪰0. Thus, we need to show that F(x)≻F(y).
By contradiction, suppose thatF(x)≻~F(y); then, there exists some α0∈[0,1], such that
(102)F*(x)(α0)≤F*(y)(α0)
or
(103)F*(x)(α0)≤F*(y)(α0).
Without loss of generality, suppose that
(104)F*(x)(α0)≤F*(y)(α0).
By hypothesis (iii),
(105)η(x,y)T∇F*(y+λ~η(x,y))(α0)≤0,
for some λ~∈(0,1).
It follows from (i) and above inequity (105) that
(106)η(y,y+λ~η(x,y))T×∇F*(y+λ~η(x,y))(α0)≥0.
Since ∇F is a strictly pseudo-invex monotone with respect to η, thus
(107)η(y,y+λ~η(x,y))T∇F*(y)(α0)>0.
From η(y,y+λ~η(x,y))=-λ~η(x,y) and λ~∈(0,1), it follows that
(108)η(x,y)T∇F*(y)(α0)<0.
This contradicts η(x,y)T∇F*(y)(α)≥0.


## 4. The Existence of a Solution to the Fuzzy Variational-Like Inequality

LetK⊂Rn, η(x,x*):K×K→Rn, F:K→(E)n, the fuzzy variational-like inequality problem be: find x*∈K, F(x*)∈(E)n (denoted by F(x*)=(F1(x*),F2(x*),…,Fn(x*))n), such that
(FVLI)η(x,x*)TF(x*)≺-0,∀x∈K.Definition 20.
LetK be an invex set with respect to η(x,x*):K×K→Rn. A fuzzy mapping F:K→(E)n is called η-hemicontinuous, if for x,y∈K, for all α∈[0,1], the mappings t→F*(y+tη(x,y))(α) and t→F*(y+tη(x,y))(α) are continuous at 0+, with t∈[0,1].Lemma 21.
LetK be a nonempty convex set in Rn, and suppose that(i)
F:K→(E)n is a fuzzy pseudo-invex monotone with respect to η and η-hemicontinuous on K;(ii)
η:K×K→Rn satisfies
(a)
η(y,y+λη(x,y))=-λη(x,y), for all x,y∈K, λ∈[0,1],(b)
η(x,y+λη(x,y))=(1-λ)η(x,y), for all x,y∈K, λ∈[0,1];(iii)
for any fixedy∈K, x→η(x,y) is linear; that is, for x(i)∈K, i=1,2,…,n, ti∈[0,1], ∑i=1nti=1, with η(∑i=1ntix(i),y)=∑i=1ntiη(x(i),y).Then, for y∈K, η(x,y)TF(y)≺-0, for all x∈K if and only if η(x,y)TF(x)≺-0, for all x∈K.Proof.
⇒ By contradiction, suppose that there exists a x-∈K, such that η(x-,y)TF(x-)≺0. Thus, there exists some α0∈[0,1],
(109)η(x-,y)TF*(x-)(α0)<0,
or
(110)η(x-,y)TF*(x-)(α0)<0.
Without loss of generality, assume that
(111)η(x-,y)TF*(x-)(α0)<0.
Since F:K→(E)n is a fuzzy pseudo-invex monotone with respect to η, thus
(112)η(x-,y)TF(y)⪰0⟹η(x-,y)TF(x-)⪰0.
That is, there are
(113)η(x-,y)TF*(y)(α)≥0⟹η(x-,y)TF*(x-)(α)≥0,η(x-,y)TF*(y)(α)≥0⟹η(x-,y)TF*(x-)(α)≥0,
for all α∈[0,1].
Therefore,(114)η(x-,y)TF*(x-)(α)<0⟹η(x-,y)TF*(y)(α)<0,η(x-,y)TF*(x-)(α)<0⟹η(x-,y)TF*(y)(α)<0.
In particular, for α0∈[0,1],
(115)η(x-,y)TF*(x-)(α0)<0⟹η(x-,y)TF*(y)(α0)<0.
This contradicts η(x,y)TF(y)≺-0.
⇐ By contradiction, suppose that there exists a x-∈K, such that η(x-,y)TF(y-)≺0. Thus, there exists some α0∈[0,1],
(116)η(x-,y)TF*(y)(α0)<0,
or
(117)η(x-,y)TF*(y)(α0)<0.
Without loss of generality, assume that
(118)η(x-,y)TF*(y)(α0)<0.
Since K is an invex set and by condition (a), we knowthat
(119)y+t(x--y)∈K,t∈[0,1],η(y,y)=0.
When η satisfies (iii), there is
(120)η(y+t(x--y),y)=tη(x-,y)+(1-t)η(y,y)=tη(x-,y),t∈[0,1].
Since F is η-hemicontinuous on K, it follows that
(121)t⟶0+,F*(y+t(x--y))(α0)⟶F*(y)(α0).
Therefore,
(122)η(x-,y)TF*(y)(α0)<0⟹η(y+t(x--y),y)T×F*(y+t(x--y))(α0)<0.
Let x^=y+t(x--y), then
(123)η(x^,y)TF*(x^)(α0)<0.
This contradicts η(x,y)TF(x)≺-0, for all x∈K.
By the hypothesis of the pseudo-invex monotonicity ofF and the linearity of η, the existence theorem can be obtained.Theorem 22.
LetK be a nonempty convex set in Rn, suppose that (i)
F:K→(E)n is fuzzy pseudo-invex monotone with respect to η and η-hemicontinuous on K;(ii)
η:K×K→Rn satisfies
(a)
η(y,y+λη(x,y))=-λη(x,y), for all x,y∈K, λ∈[0,1],(b)
η(x,y+λη(x,y))=(1-λ)η(x,y), for all x,y∈K, λ∈[0,1];(iii)
for any fixedy∈K, x→η(x,y) is linear; that is, for x(i)∈K, i=1,2,…,n, ti∈[0,1], i=1,2,…,n, for ∑i=1nti=1, with η(∑i=1ntix(i),y)=∑i=1ntiη(x(i),y).Then, there exists x*∈K, such that η(x,x*)TF(x*)≺-0, for all x∈K.Proof.
Let(124)Y(y)={x∈K∣η(y,x)TF(x)≺-0},∀y∈K.
That is, for all α∈[0,1], there are
(125)Y*(y)(α)={x∈K∣η(y,x)TF*(x)(α)<-0},∀y∈K,Y*(y)(α)={x∈K∣η(y,x)TF*(x)(α)<-0},∀y∈K.
Let x(i)∈K, i=1,2,…,n, ti∈[0,1], ∑i=1nti=1. Suppose that
(126)x=∑i=1n‍tix(i)∈-⋃‍Y*(x(i))(α).
Then, there exists some α0∈[0,1], such that
(127)η(x(i),x)TF*(x)(α)<0,i=1,2,…,n.
Therefore,
(128)∑i=1n‍tiη(x(i),x)TF*(x)(α)<0.
From (iii), for fixed x∈K,
(129)η(∑i=1n‍tix(i),x)TF*(x)(α)<0.
That is,
(130)η(x,x)TF*(x)(α)<0.
From η(x,x)=0, it follows that 0<0, which is absurd. So x=∑i=1ntix(i)∈⋃‍Y*(x(i))(α).
Similarly,x=∑i=1ntix(i)∈⋃‍Y*(x(i))(α) can be proofed. Thus,
(131)conv({x(1),x(2),…,x(n)})⊂⋃‍Y*(x(i))(α),conv({x(1),x(2),…,x(n)})⊂⋃‍Y*(x(i))(α)
can be obtained. Therefore, Y*(y)(α), Y*(y)(α) are KKM mappings.
Let(132)Y-(y)={x∈K∣η(y,x)TF(y)≺-0},∀y∈K.
That is, for all α∈[0,1],
(133)Y-*(y)={x∈K∣η(y,x)TF*(y)(α)<-0},∀y∈K,Y-*(y)={x∈K∣η(y,x)TF*(y)(α)<-0},∀y∈K.
Let x∈Y(y); that is, η(y,x)TF(x)≺-0. By Lemma 21, η(y,x)TF(y)≺-0 holds, that is, x∈Y-(y). Thus,
(134)Y*(y)(α)⊂Y-*(y)(α),Y*(y)(α)⊂Y-*(y)(α).
As Y*(y)(α), Y*(y)(α) are KKM mappings, thus Y-*(y)(α), Y-*(y)(α) are also KKM mappings. By Lemma 21, we have
(135)⋂y∈KY*(y)(α)=⋂y∈KY-*(y)(α),⋂y∈KY*(y)(α)=⋂y∈KY-*(y)(α).
As Y-*(y)(α), Y-*(y)(α) are closed for every y∈K and K is a bounded set, then Y-*(y)(α), Y-*(y)(α) are bounded; hence, Y-*(y)(α), Y-*(y)(α) are compact. Therefore,
(136)⋂y∈KY*(y)(α)=⋂y∈KY-*(y)(α)≠∅,⋂y∈KY*(y)(α)=⋂y∈KY-*(y)(α)≠∅.
Hence, there exists x*∈K, such that
(137)η(y,x*)TF*(x*)(α)<-0,∀y∈K,η(y,x*)TF*(x*)(α)<-0,∀y∈K.
That is,
(138)η(y,x*)TF(x*)≺-0,∀y∈K.
The pseudo-invex monotonicity ofF assures us of the existence of a solution to (FVLI), but not the uniqueness of such a solution. To achieve this, we assume the strictly pseudo-invex monotonicity of F.Theorem 23.
LetK be a nonempty convex set in Rn, and suppose that (i)
F:K→(E)n is strictly fuzzy pseudo-invex monotone with respect to η and η-hemicontinuous on K;(ii)
η:K×K→Rn is a skew function and satisfies
(a)
η(y,y+λη(x,y))=-λη(x,y), for all x,y∈K, λ∈[0,1],(b)
η(x,y+λη(x,y))=(1-λ)η(x,y), for all x,y∈K, λ∈[0,1];(iii)
for any fixedy∈K, x→η(x,y) is linear; that is, for x(i)∈K, i=1,2,…,n, ti∈[0,1], ∑i=1nti=1, with η(∑i=1ntix(i),y)=∑i=1ntiη(x(i),y).Then, there exists unique x*∈K, such that η(x,x*)TF(x*)≺-0, for all x∈K.Proof.
From Theorem17, F is a fuzzy pseudo-invex monotone. Also from Theorem 22, there exists a solution for problem (FVLI).
Suppose that(FVLI) has two distinct solutions x*, x-. Then,
(139)η(x-,x*)TF(x*)≺-0,η(x*,x-)TF(x-)≺-0.
Since F is a fuzzy strictly pseudo-invex monotone on K, then
(140)η(x-,x*)TF(x*)⪰0⟹η(x-,x*)TF(x-)≻0.
That is, there are
(141)η(x-,x*)TF*(x*)(α)≥0⟹η(x-,x*)TF*(x-)(α)>0,η(x-,x*)TF*(x*)(α)≥0⟹η(x-,x*)TF*(x-)(α)>0,
for all α∈[0,1].
Sinceη is a skew function, there is η(x-,x*)+η(x*,x-)=0. Thus
(142)η(x-,x*)TF*(x*)(α)≥0⟹η(x*,x-)TF*(x-)(α)<0,η(x-,x*)TF*(x*)(α)≥0⟹η(x*,x-)TF*(x-)(α)<0.
That is,
(143)η(x*,x-)TF(x-)≺0,
which contradicts η(x*,x-)TF(x-)≺-0.


## 5. Qualities of the Fuzzy Variational-Like Inequality

A well-known fact in mathematical programming is that the variational inequality problem has a close relationship with the optimization problem. Similarly, the fuzzy variational inequality problem also has a close relationship with the fuzzy optimization problem.Consider the unconstrained fuzzy vector optimization problem:(FP)minx∈KF(x),
where K is a subset of n dimension Euclidean space Rn, F:K→E is a fuzzy mapping.A pointx∈K is called a feasible point. If x*∈K and no x∈K, F(x)≺F(x*), then x* is called an optimal solution, a global optimal solution, or simply a solution to the problem (FP). If x*∈K and there exists an δ-neighborhood Nδ(x*) around x*, such that for no x(≠x*)∈K∩Nδ(x*), F(x)≺F(x*), then x* is called a local optimal solution. Similarly, if x*∈K and there exists an δ-neighborhood Nδ(x*) around x*, such that for no x∈K∩Nδ(x*), F(x)⪯F(x*), then x* is called a strict local optimal solution.The following lemmas and theorems discuss the properties of fuzzy variational inequality.Lemma 24 (see Wu and Xu [6]).
LetF:K→E be a fuzzy differentiable pseudoinvex mapping. If (x*,∇F(x*)) is a solution of (FVLI), then x* is a local optimal solution of (FP).Lemma 25 (see Wu and Xu [6]).
LetF:K→E be a fuzzy differentiable strictly pseudoinvex mapping. If (x*,∇F(x*)) is a solution of (FVLI), then x* is a strictly local optimal solution of (FP).Theorem 26.
LetF:K→E be a differentiable fuzzy mapping, and suppose that (i)
η satisfies the following conditions:
(a)
η(y,y+tη(x,y))=-tη(x,y),(b)
η(x,y+tη(x,y))=(1-t)η(x,y);(ii)
K is an invex set with respect to η;(iii)
for eachx≠y, some α0∈[0,1],
(a)
F*(y)(α0)>F*(x)(α0) implies η(x,y)T∇F*(y+λ~η(x,y))(α0)<0,or
(b)
F*(y)(α0)>F*(x)(α0) implies η(x,y)T∇F*(y+λ~η(x,y))(α0)<0;(iv)
∇F:K→(E)n is fuzzy pseudo-invex monotone with respect to η on k.If (x*,∇F(x*)) is a solution of (FVLI), then x* is a local optimal solution of (FP).Proof.
From Theorem18, we know that F is a fuzzy pseudo invex. By Lemma 24, we can show it.Theorem 27.
LetF:K→E be a differentiable fuzzy mapping, and suppose that(i)
η satisfies the following conditions:
(a)
η(y,y+tη(x,y))=-tη(x,y),(b)
η(x,y+tη(x,y))=(1-t)η(x,y);(ii)
K is an invex set with respect to η;(iii)
for eachx≠y, some α0∈[0,1],
(a)
F*(y)(α0)>F*(x)(α0) implies  η(x,y)T∇F*(y+λ~η(x,y))(α0)<0,or
(b)
F*(y)(α0)>F*(x)(α0) implies η(x,y)T∇F*(y+λ~η(x,y))(α0)<0;(iv)
∇F:K→(E)n is fuzzy strictly pseudo-invex monotone with respect to η on k.If (x*,∇F(x*)) is a solution of (FVLI), then x* is a strictly local optimal solution of (FP).Proof.
From Theorem19, we know that F is a fuzzy strictly pseudo-invex. By Lemma 25, we can show it.Theorem28 gives the equivalent relationship of two fuzzy variational-like inequalities.Theorem 28.
LetK be an invex set with respect to η:K×K→Rn. Suppose that (i)
F:K→(E)n is fuzzy pseudo-invex monotone with respect to η and η-hemicontinuous on K;(ii)
η satisfies the following conditions:
(a)
η(y,y+λη(x,y))=-λη(x,y),(b)
η(x,y+λη(x,y))=(1-λ)η(x,y).Then, x*∈K satisfies η(x,x*)TF(x*)≺-0, for all x∈K if and only if it satisfies η(x,x*)TF(x)≺-0, for all x∈K.Proof.
⇒ By contradiction, suppose that there exists an x-∈K, such that η(x-,x*)TF(x-)≺0. Thus, there exists some α0∈[0,1],
(144)η(x-,x*)TF*(x-)(α0)<0,
or
(145)η(x-,x*)TF*(x-)(α0)<0.
Without loss of generality, assume that
(146)η(x-,x*)TF*(x-)(α0)<0.
Since F:K→(E)n is fuzzy pseudo-invex monotone with respect to η, thus
(147)η(x-,x*)TF(x*)⪰0⟹η(x-,x*)TF(x-)⪰0.
That is, for any α∈[0,1],
(148)η(x-,x*)TF*(x*)(α)≥0⟹η(x-,x*)TF*(x-)(α)≥0,η(x-,x*)TF*(x*)(α)≥0⟹η(x-,x*)TF*(x-)(α)≥0.
Therefore,
(149)η(x-,x*)TF*(x-)(α)<0⟹η(x-,x*)TF*(x*)(α)<0,η(x-,x*)TF*(x-)(α)<0⟹η(x-,x*)TF*(x*)(α)<0.
In particularly, for α0∈[0,1], there is
(150)η(x-,x*)TF*(x-)(α0)<0⟹η(x-,x*)TF*(x*)(α0)<0.
This contradicts η(x-,x*)TF(x*)≺-0.
⇐ By contradiction, suppose that there exists an x-∈K, such that η(x-,x*)TF(x*)≺0. Thus, there exists some α0∈[0,1],
(151)η(x-,x*)TF*(x*)(α0)<0,
or
(152)η(x-,x*)TF*(x*)(α0)<0.
Without loss of generality, assume that
(153)η(x-,x*)TF*(x*)(α0)<0.
Since K is invex set with respect to η and assumption (ii), it follows that
(154)x*+λη(x-,x*)∈K,η(x*+λη(x-,x*),x*)=λη(x-,x*),λ∈[0,1].
Again, since F is η-hemicontinuous on K, there is
(155)λ⟶0+,F*(x*+λη(x-,x*))(α0)⟶F*(x*)(α0).
Therefore,
(156)η(x-,x*)TF*(x*)(α0)<0⟹η(x*+λη(x-,x*))T×F*(x*+λη(x-,x*))(α0)<0.
Set x~=x*+λη(x-,x*), then
(157)η(x~,x*)TF*(x~)(α0)<0,x~∈K.
This contradicts η(x,x*)TF(x)≺-0, for all x∈K.


## 6. Solution to the Fuzzy Variational-Like Inequality

In order to solve the fuzzy variational-like inequality, it is important find an equivalent fuzzy problem. Next, we discuss the equivalent fuzzy generalized complementarity problem of(FVLI), where η(y,x)=y-x.Denote(FVLI): find x∈K⊂Rn, F(x)∈(E)n, such that
(FVLI)F(x)T(y-x)≺-0,∀y∈K,
where F:K→(E)n.It is well known that variational inequalities are equivalent to the generalized complementary problem over a convex cone [35]. Similarly, we consider solving the fuzzy variational-like inequality in view of the following fuzzy generalized complementarity problem.Findx∈K⊂Rn, F(x)∈(E)n, such that
(FCP)F(x)Tx=0,F(x)∈K*,K*={z∈(E)n∣zTy≺-0,∀y∈K}.The following theorem shows the equivalence between the fuzzy variational-like inequality and the fuzzy generalized complementarity problem.Theorem 29.
The fuzzy variational-like inequality(FVLI) is equivalent to the fuzzy generalized complementarity problem (FCP), when K is a convex cone.Proof.
At first, we show(FCP)⊂(FVLI). Suppose that (x*,F(x*)) is a solution of (FCP), then
(158)F(x*)Tx*=0,(159)F(x*)Ty≺-0,∀y∈K.
Combining (158) and (159), we have
(160)F(x*)T(y-x*)≺-0,∀y∈K.
Therefore, (x*,F(x*)) is also a solution of (FVLI).
Next, we show(FVLI)⊂(FCP). Let (x*,F(x*)) be a solution of (FVLI) which the degree of membership α∈[0,1], then
(161)F(x*)T(y-x*)⪰μCy-1(α)⪰-p~,∀y∈K,
where p~⪰0 is the tolerance level which a decision maker can tolerate in the accomplishment of the fuzzy variation-like inequality F(x)T(y-x)≺-0. By contradiction, suppose that F(x*)Tx*≠0, then there exists some α0∈[0,1], q~1⪰0, such that
(162)[F(x*)Tx*]*(α0)>q~1*(α0)
or
(163)[F(x*)Tx*]*(α0)>q~1*(α0).
Or there exists some α1∈[0,1], q~2≻0, such that
(164)[F(x*)Tx*]*(α1)<q~2*(α1)
or
(165)[F(x*)Tx*]*(α1)>q~2*(α1).
Without loss of generality, suppose that
(166)[F(x*)Tx*]*(α0)>q~1*(α0),[F(x*)Tx*]*(α1)<q~2*(α1).
Since K is a convex cone, then when y=λx* with λ>1, there is
(167)F(x*)Tx*⪰-p~λ-1,p~⪰0.
When y=0,
(168)F(x*)Tx*⪯p~.
If [F(x*)Tx*]*(α0)>q~1*(α0), then
(169)[-p~λ-1]*(α0)>p~1*(α0).
This leads to a contradiction.
If[F(x*)Tx*]*(α1)<q~2*(α1), then
(170)p~*(α1)<q~2*(α1).
This also leads to a contradiction. Therefore, F(x*)Tx*=0. Furthermore, from (161), it follows that
(171)F(x*)T(y-x*)⪰-p~⟹F(x*)Ty⪰-p~,∀y∈K.
Therefore,
(172)F(x*)Ty≺-0.
This shows that (x*,F(x*)) is a solution of (FCP).To solve(FVLI) with K={x∈Rn∣Dx⪰o,D∈Rm×n,andD=[di],foralli=1,2,…,m} being a convex cone, we consider the problem (FCP) with dix⪰0, for all i=1,2,…,m.That is,(173)dix⪰0,F(x)Tx=0,F(x)∈K*,∀i=1,2,…,m,
where K*={z∈(E)n∣zTy≺-0,forally∈K}.Similarly [26], it can be shown that (173) can be rewritten as follows: find x∈K, F(x)∈(E)n such that
(174)dix⪰0,F(x)Tx=0,di′F(x)⪰0,∀i=1,2,…,m,
where di′ is a normal to di. Of course, (174) can be also rewritten as follows: find x∈K, F(x)∈(E)n such that
(175)dix⪰0,F(x)Tx⪰0,-F(x)Tx⪰0,di′F(x)⪰0,∀i=1,2,…,m.
In (174), each fuzzy inequality can be represented by a fuzzy set Sj with the corresponding membership function μsj(x), for j=1,2,…,2m+2; that is
(176)μj(x)={1,ifdjx⪰0,μj(djx),if-tj≺djx⪯0,∀j=1,2,…,m,0,ifdjx⪯-tj,μsm+1(x)={1,ifF(x)Tx⪰0,μm+1(F(x)Tx),if-tm+1≺F(x)Tx⪯0,0,ifF(x)Tx⪯-tm+1,μsm+2(x)={1,if-F(x)Tx⪰0,μm+2(-F(x)Tx),if-tm+1≺-F(x)Tx⪯0,0,if-F(x)Tx⪯-tm+2,μj(x)={1,ifdj-m-2′x⪰0,μj(dj-m-2′x),if-tj≺dj-m-2′⪯0,∀j=m+3,m+4,…,2m+2,0,ifdj-m-2′x⪯-tj,
where tj⪰0, for all j=1,2,…,m are the tolerance level of djx⪰0, j=1,2,…,m. tj⪰0, for all j=m+1,m+2,…,2m+2 other fuzzy inequalities in which tj are fuzzy numbers.To find a solution to problem (175), we define a fuzzy decision D~ of (175) as the fuzzy set resulting from the intersection of fuzzy set sj, j=1,2,…,2m+2. By choosing the commonly used “minimum operator” for the fuzzy set intersections, we can define the membership function for D~ as
(177)μj=minj{μsj(x)}.
Therefore, a solution to problem (175) can be taken as the solution with the highest membership in the fuzzy decision set D~ and obtained by solving the following problem:
(178)maxxμj=maxxminj=1,2,…,2m+2{μsj(x)}.Problem (178) is an unconstrained optimization, but its objective function is not continuous and derivable. It cannot be solved by traditional optimization methods, but it may be solved by genetic algorithms [36, 37]. Next, we discuss a special genetic algorithm with mutation along the weighted gradient direction developed by Wang and Tang [37–39].For individualx, let μmin(x)=min{μs1(x),μs2(x),…,μs2m+2(x)}. If μmin(x)≤μsj(x)<1, then move along the gradient direction of μsj(x), and the value of μsj(x) may be improved. The smaller μsj(x) is, the greater the improvement in μsj(x) that may be achieved. Based on the above idea, construct the direction as follows:
(179)G(x)=∑j=12m+2‍wj∇μsj(x),
where 0<μsj(x)≤1.G
(
x
) is called the weighted gradient direction of μD~, and wj is the gradient direction weight defined as follows:
(180)wj={0,ifμsj(x)=1,1μsj-μmin+ϵ,ifμmin≤μsj<1,
where ϵ is a sufficiently small positive number and 1/ϵ is the largest weight.The childxjk+1 generated from xik by mutation along the weighted gradient direction G(x) can be described as follows:
(181)xjk+1=xik+βkG(xik),
where βk is a random step length of the Erlang distribution generated by a random number generator with declining means. The degree of membership is calculated as follows:
(182)μsj(x(j))={μmin(x(j)),ifμmin>α0,ϵμmin(x(j)),else,
where α0 is the acceptable satisfaction degree preferred by the decision maker and ϵ∈U(0,1).For individualj  (j=1,2,…,pop-size), calculate its fitness function F(j) and selected probability P(j) by
(183)F(j)=μsj(x(j))+ϵ~,P(j)=F(j)∑j=1pop-sizeF(j),
where ϵ~ is a small positive number. It is used to guarantee a nonzero denominator. The procedure can be written as follows.Step 1.
Transforming(FVLI) into problem (175).Step 2.
Initialize.(i)
Input an acceptable satisfaction degreeα0 and the largest max-gen and pop-size.(ii)
Input the criteria set:CI=1,2,…,n,n+1,…,n+2m+2,n+2m+3, where j=1,2,…,n stand for the variables, j=n+1,…,n+2m+2 stand for inequality constraints to (175), and j=n+2m+3 stands for the sum of weighted satisfaction degree, respectively. Give the initial values and the upper and the lower values of criteria r,ri∈CI.(iii)
Input the types of membership function that describes the fuzzy constraints.(iv)
Input the weightswj.Step 3.
Randomly produce the initial population and calculate their membership degrees byxi(j)=ξxiup, where ξ∈U(0,1), i=1,2,…,n; j=1,2,…,pop-size and xiup is the upper bound of the ith element of variable x. The membership function μSj(x(j)) is calculated with (182).Step 4.
Set iteration indexk=1.Step 5.
Calculate the fitness functionF(j) and selection probabilities P(j) by means of (183).Step 6.
Produce new individualx(j) with the parent x(i) as x(j)k=x(i)(k-1)+βkG(x(i)(k-1)). G(x) is defined by (179).Step 7.
For individualj, calculate the membership function μSj(x(j)) with (182), and update optimal degree of membership μmax and the upper and the lower values of criteria r.Step 8.
Setk+1=k; if k≤max-gen, go to Step 5; otherwise, go to Step 9.Step 9.
Output the optimal membership degreeμmax and the upper and the lower values of criteria preferred by the decision maker, then stop.


## 7. Conclusion

In this paper, we have introduced the concepts of invex monotonicity and pseudoinvex monotonicity of fuzzy mapping and discussed the relationship between invex monotonicity(pseudoinvex monotonicity) and invexity (pseudoinvexity) of fuzzy mapping. We have put forward the fuzzy variational-like inequality and also discussed extended qualities of it. Finally, we discussed the existence of a theorem for a solution and solving method to the fuzzy variational-like inequality. We got some useful results and gave a method of solving the fuzzy variational-like inequality of a genetic algorithm.



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*Source: 854751-2013-11-11.xml*