Preinvexity of $n$-dimensional fuzzy number-valued functions: characterization, variational inequality and optimization problems

1.   Introduction

It is well known that the convex analysis is closely linked with the development of optimization theory. Meanwhile, there is often uncertainty of parameters or dates in the process of the mathematical modeling of the specific optimization problems. In order to describe these uncertain parameters or dates in a mathematical modeling, a straightforward and effective way to think about whether it can be represented as fuzzy number in some sense. Therefore, fuzzy convex analysis theory and its corresponding fuzzy optimization problems have been studied by many researchers. In 1992, Nanda and Kar ^[14] set up a mapping from a vector space to the space of fuzzy numbers, and they introduced the definitions of convex fuzzy mapping, strictly convex fuzzy mapping, quasiconvex fuzzy mapping, strictly quasiconvex fuzzy mapping and logarithmic convex fuzzy mapping, and then applied their results to the problems of fuzzy convex optimization. Moreover, Furukawa ^[5] proposed the concepts of convex fuzzy mapping and local Lischitz continuous fuzzy mapping by using "fuzzy-max" order, and the basis theorem of fuzzy mapping local Lischitz continuity is given. In 1999, based on the partial order relation of interval number, Syau ^[17,18] introduced the definitions of the convex fuzzy mapping and quasiconvex fuzzy mapping, and established characterization for convex fuzzy mapping. In 2000, Yang and Teo ^[27] investigated pseudoconvexity, invexity, pseudoinvexity for fuzzy mappings by considering the concept of ordering proposed by Goetsschel and Voxman ^[6], and discussed their interrelations. Meanwhile, based on the same order relation, Yan and Xu ^[28] introduced another concept of convex fuzzy mapping, and studied a kind of fuzzy convex optimization problems. In 2006, the operation of convex fuzzy mapping proposed by Nanda and Kar were investigated by Zhang and Yuan, and the important concepts of positive homogeneous fuzzy mapping, infimal convolution, convex hull were given, and the corresponding characterization theorem was presented by using the parameter of fuzzy number ^[31]. In 2008, based on the concept of differentiability of fuzzy mapping, Panigrahi and panda ^[16] gave the concepts of convexity, quasiconvexity, strictly quasiconvexity, strong quasiconvexity and pseudoconvexity of fuzzy mappings from $R^{n}$ to the set of fuzzy numbers, and derived the Karush-Kuhn-Tucker optimization condition for a constrained fuzzy optimization problem. In 2013, Li and Noor ^[9] discussed the properties of the convex fuzzy mappings based on a linear ordering of fuzzy numbers proposed by Goestschel and Voxman. Furthermore, they obtained the judgement theorems of convex, strictly convex and semi-strictly convex fuzzy mapping under lower and upper semicontinuity condition, respectively. In addition, convexity and other related problems of the fuzzy mapping have been studied extensively ^[4,10,23,25]. As a generalization of convex fuzzy mapping, in 1994, Noor ^[15]introduced the concepts of preinvex fuzzy mapping and invex set, and the minimization problem of preinvex fuzzy number-valued functions was described by using variational inequality. In 1999, Syau in ^[19] showed that the preinvexity given by Noor is too restrictive, redefined the preinvexity of $\eta$ vector-valued functions, established two characterizations for the preinvex fuzzy mappings, and applied their results to the optimization theory. After that several investigators ^{[1,2,3,11,12,20,21,24,30]} also proposed and studied different types of the preinvexity and the generalized preinvexity for fuzzy mappings. However, all of the above works are discussed for 1-dimensional fuzzy number-valued functions. The main reason is that the partial ordered relation in n-dimensional fuzzy number space, the difference between $n$-dimensional fuzzy numbers, and convex analysis of high-dimensional fuzzy mapping have not been discussed. Until 2016, Gong Zengtai et al.^[7] first introduced the partially ordered relation on $n$-dimensional fuzzy number space, the convexity of the $n$-dimensional fuzzy mapping, the differentiability, and the corresponding optimization theory. Based on the partially ordered relation in $n$-dimensional fuzzy number space, considering the convexity of vector-valued function, and combining with the characteristics of $n$-dimensional fuzzy mappings, they proposed and investigated the convexity of $n$-dimensional fuzzy number-valued functions, generalized convexity, upper semicontinuity, lower semicontinuity, and discussed their interrelations, and pointed out the local minimum point of convex fuzzy mapping is its global minimum point ^[8]. As a continuous research of ^[7,8], in this paper, we introduce the preinvexity of $n$-dimensional fuzzy number-valued functions based on the partial order relation in $n$-dimensional fuzzy number space and some properties of them are discussed. In addition, some counterexamples are given. Then we present criteria for $n$-dimensional preinvex fuzzy number-valued functions under upper or lower semicontinuity conditions, respectively. Furthermore, the two-parameter optimization problem, $n$-dimensional fuzzy variational-like inequality problem and the optimal conditions related to $n$-dimensional preinvex fuzzy number-valued function are discussed.

2.   Preliminaries

Let $R^n$ denote the $n$-dimensional Euclidean space and $F(R^n)$ denote the set of all fuzzy subset on $R^n$. Fuzzy set $u\in{F(R^n)}$ is called a fuzzy number if $u$ is a normal, convex fuzzy set, upper semi-continuous and $[u]^0 = \overline{\{x\in\; R^n, u(x) > 0\}}$ is compact. We denote $E^n$ as $n$-dimensional fuzzy number space ^[22,26].

Let $u\in{F(R^n)}$. For $r\in(0, 1]$, we denote $[u]^r = \{x\in\; R^n, u(x)\geq\; r\}.$ The addition and non-negative scalar multiplication are defined as follows for fuzzy number $u, v\in{E^n}, \alpha\in{R}$, $k, k_{1}, k_{2}\in R,$ according to the Zadeh's extension principle:

(1) $k(u+v) = ku+kv$;

(2) $k_1(k_2u) = (k_1k_2)u$;

(3)$(k_1+k_2)u = k_1u+k_2u$ when $k_1\geq0$ and $k_2\geq0$.

Given $u, v\in E^{n}$, the distance $D:E^{n}\times{E^{n}}\rightarrow [0, +\infty)$ between $u$ and $v$ is defined by the equation

where d is the Hausdorff metric

For $u\in E^{n},$ we denote the centroid of $[u]^r$, $r\in[0, 1]$ as

where $\int\cdots\int_{[u]^{r}}1dx_{1}dx_{2}\cdots dx_{n}$ is the solidity of $[u]^{r}$, $r\in[0, 1]$ and $\int\cdots\int_{[u]^{r}}x_{i}dx_{1}dx_{2}\cdots dx_{n}$ $(i = 1, 2, \cdots, n)$ is the multiple integral of $x_{i}$ on measurable sets $[u]^{r}$, $r\in[0, 1],$ refer to ^[7].

Let $u\in{E^{n}}$, $n$-dimensional vector-valued function $\tau$ denote the centroid of the fuzzy number,

where $\int_{0}^{1}r\frac{\int\cdots\int_{[u]^{r}}x_{i}dx_{1}dx_{2}\cdots dx_{n}}{\int\cdots\int_{[u]^{r}}1dx_{1}dx_{2}\cdots dx_{n}}dr$ $(i = 1, 2, \cdots, n)$ is the Lebesgue integral of $r\frac{\int\cdots\int_{[u]^{r}}x_{i}dx_{1}dx_{2}\cdots dx_{n}}{\int\cdots\int_{[u]^{r}}1dx_{1}dx_{2}\cdots dx_{n}}$ $(i = 1, 2, \cdots, n)$ on $[0, 1],$ refer to ^[7].

Definition 2.1 (see ^[7]) Let $u, v\in{E^{n}}, C\subseteq{R^n}$ be a closed convex cone with $0\in{C}$ and $C\neq{R^n}.$ We say that $u\preceq_{c}v$ ($u$ precedes $v$) if

The order relation $\preceq_{c}$ is reflexive and transitive, and $\preceq_{c}$ is a partially ordered relation on $E^{n}$. For $u, v\in{E^{n}}$, if either $u\preceq_cv,$ or $v\preceq_cu,$ then we say $u$ and $v$ are comparable; otherwise, they are non-comparable. If $u, v\in E^{1},$ $C = [0, +\infty)\subseteq R,$ then Definition 2.1 coincides with Definition 2.5 from ^[6].

Remark 2.1. (see ^[7].) Let $u, v\in E^{1}.$ If we write $\tau(u) = \frac{1}{2}\int_{0}^{1}r(u^+(r)+u^-(r))dr$, then $u\preceq_cv$ in the sense of Goetschel ^[6] if and only if $\tau(u)\leq{\tau}(v),$ i.e., $\tau(v)\in\tau(u)+[0, +\infty)$. Furthermore,

for $\lambda_1, \lambda_2 > 0$, where $[u]^r = [u^-(r), u^+(r)].$

Based on Definition 2.1 and the vector-valued function $\tau$, we say $u\prec_{c}v$ if $u\preceq_{c}v$ and $\tau(u)\neq\tau(v)$. Sometimes we may write $v\succeq_{c}u$(resp. $v\succ_{c}u)$ instead of $u\preceq_{c}v$(resp. $u\prec_{c}v)$.

Set-valued mappings $F_r:K\rightarrow {P_k(R^n)}$ are defined by $F_r(t) = [\widetilde{F}(t)]^{r}$, $r\in[0, 1],$ where $P_k(R^n)$ denotes the power set of $R^n$.

Definition 2.2. (see ^[7].) Let $\widetilde{F}:K\rightarrow {E^{n}}$, $\tau_F: K\rightarrow {R^n}$ is defined by $\tau_F(t) = \tau(\widetilde{F}(t))$ $\; \; \; \; = (2\int_{0}^{1}r\frac{\int\cdots\int_{F_r(t)}x_1dx_1dx_2\cdots\; dx_n}{\int\cdots\int_{F_r(t)}1dx_1dx_2\cdots\; dx_n}dr, 2\int_{0}^{1}r\frac{\int\cdots\int_{F_r(t)}x_2dx_1dx_2\cdots\; dx_n}{\int\cdots\int_{F_r(t)}1dx_1dx_2\cdots\; dx_n}dr \cdots, 2\int_{0}^{1}r\frac{\int\cdots\int_{F_r(t)}x_ndx_1dx_2\cdots\; dx_n}{\int\cdots\int_{F_r(t)}1dx_1dx_2\cdots\; dx_n}dr).$

Obviously, the fuzzy number-valued function $\widetilde{F}:E^{m}\rightarrow {E^{n}}$ is said to be increasing if $\widetilde{F}(u)\preceq_c\widetilde{F}(v)$, whenever $u, v\in{E^{m}}$, and $u\preceq_cv$.

In this article, the epigraph of $\widetilde{F}$, which is denoted by $epi(\widetilde{F}),$ is defined as

The generalized difference ($gH$-difference for short, and refer to ^[8]) of two fuzzy numbers $\widetilde{u}, \widetilde{v}\in E^{n}$ is given by its level sets as

where the $gH$-difference $\circleddash_{gH}$ is with interval operands $[\widetilde{u}]^{\beta}$ and $[\widetilde{v}]^{\beta}.$

Definition 2.3. Let $\widetilde{F}:K\rightarrow {E^{n}}$ be a fuzzy number-valued function on an invex set $K\subset{R}^{n}$, $K\neq\emptyset$, with respect to (w.r.t.) a function $\eta:K\times{K}\rightarrow {R^n}$. If for any $x, y \in{K}$, there exists a $\delta > 0$, such that the $H$-difference $\widetilde{F}(y+h\eta(x, y))-\widetilde{F}(y)$ exists for any real number $h\in(0, \delta)$, and $u_{\eta}^{i}\in{E}^{n}$, $i = 1, 2, \cdots{n},$ such that

then $\widetilde{F}$ is called fuzzy $\eta$-extended directionally differentiable at $y$. $\widetilde{\nabla}\widetilde{F}_{\eta}(y)\eta(x, y)$ is called the fuzzy $\eta$-extended directional derivative at $y$ in the direction $\eta(x, y)$(denoted $\widetilde{\nabla}\widetilde{F}_{\eta}(y) = (u_{\eta}^{1}, u_{\eta}^{2}, \cdots{u}_{\eta}^{n}$)).

The example 4.4 illustrates the notion of fuzzy $\eta$-extended directional differentiability.

3.   $n$-dimensional preinvex fuzzy number-valued functions

Since the space of the $n$-dimensional fuzzy numbers is a partially ordered set, two $n$-dimensional fuzzy numbers might not be comparable. For a fuzzy number-valued function $\widetilde{F}: K\rightarrow {E^{n}}$, $\widetilde{F}$ is said to be a comparable fuzzy number-valued function if for each pair $x, y\in{K}$ and $x\neq{y}$, $\widetilde{F}(x)$ and $\widetilde{F}(y)$ could be compared. In this paper, we assume that a fuzzy number-valued function $\widetilde{F}:K\rightarrow {E^{n}}$ involved is comparable.

Refer to the definition of ^[29], a set $K\subseteq{R^n}$ is said to be an invex set w.r.t. a function $\eta:R^n\times{R^n}\rightarrow {R^n}$ if $x, y\in{K}$ implies that $y+\lambda\eta(x, y)\in{K}$ for $\lambda\in[0, 1].$

Definition 3.1. Let K be an invex set of $R^n$ w.r.t. $\eta$, and $\widetilde{F}:K\rightarrow {E^{n}}$ be a fuzzy number-valued function.

(1) $\widetilde{F}$ is said to be preinvex $(p.)$ on $K$ if

for any $x, y\in{K}$ and $\lambda\in[0, 1].$

(2) $\widetilde{F}$ is said to be weakly preinvex $(w.p.)$ on $K$ if there exists a $\lambda\in(0, 1)$ such that

for any $x, y\in{K}.$

(3) $\widetilde{F}$ is said to be strictly preinvex $(s.p.)$ on $K$ if

for any $x, y\in{K}$ with $x\neq{y}$ and $\lambda\in(0, 1).$

(4) $\widetilde{F}$ is said to be weakly strictly preinvex $(w.s.p.)$ on $K$ if there exists a $\lambda\in(0, 1)$ such that

for any $x, y\in{K}$ with $x\neq{y}$.

(5) $\widetilde{F}$ is said to be prequasiinvex $(q.p.)$ on $K$ if

for any $x, y\in{K}$ and $\lambda\in[0, 1].$

(6) $\widetilde{F}$ is said to be weakly prequasiinvex $(w.q.p.)$ on $K$ if there exists a $\lambda\in(0, 1)$ such that

for any $x, y\in{K}.$

(7) $\widetilde{F}$ is said to be strictly prequasiinvex $(s.q.p.)$ on $K$ if

for any $x, y\in{K}$ with $x\neq{y}$ and $\lambda\in(0, 1).$

(8) $\widetilde{F}$ is said to be weakly strictly prequasiinvex $(w.s.q.p.)$ on $K$ if there exists a $\lambda\in(0, 1)$ such that

for any $x, y\in{K}$ with $x\neq{y}$.

Remark 3.1. Let $\widetilde{F}:K\rightarrow {E^{n}}$ be a preinvex fuzzy number-valued function, then $-\widetilde{F}$ is preincave on $K$.

Remark 3.2. Let $\widetilde{F}:K\rightarrow {E^{n}}$ be a strictly preinvex fuzzy number-valued function, then $-\widetilde{F}$ is strictly preincave on $K$.

Remark 3.3. In Definition 3.1, taking $\eta(x, y) = x-y,$ $\widetilde{F}$ is said to be convex, weakly convex, strictly convex, weakly strictly convex, quasiconvex, weakly quasiconvex, strictly quasiconvex, and weakly strictly quasiconvex on $K,$ respectively ^[7].

4.   The relationships of $n$-dimensional preinvex fuzzy number-valued functions

Theorem 4.1. If $\widetilde{F}:K\rightarrow {E^{n}}$ is a preinvex fuzzy number-valuedfunction, then $\widetilde{F}$ is prequasiinvex on $K$.

$Proof$. If $\widetilde{F}$ is preinvex on $K$, then we obtain

for any $x, y\in{K}$ and $\lambda\in[0, 1]$, This completes the proof.

Theorem 4.2. If $\widetilde{F}:K\rightarrow {E^{n}}$ is a strictly preinvex fuzzy number-valuedfunction, then $\widetilde{F}$ is strictly prequasiinvex on $K$.

The proof is similar to the proof of Theorem 4.1.

Example 4.1. Let $K = [1,2] \cup [3,4]$. It is easy to prove that $K$ is invex w.r.t. $\eta:R^{2}\times{R}^{2}\rightarrow {R}^{2}$ defined by

In fact, if $(x, y)\in [1,2]^{2}\cup [3,4]^{2}$, from the convexity of [1,2] and [3,4], we have

If $(x, y)\in [3,4]\times[1,2]$, $y+\lambda\eta(x, y) = y+\lambda(1-y) = (1-\lambda)y+\lambda,$ we can choose $\lambda = 1$ and $\lambda = 0$, it follows that $1\leq(1-\lambda)y+\lambda\leq 2,$ i.e., $y+\lambda\eta(x, y)\in [1,2]\subset{K}.$ Similarly, if $(x, y)\in [1,2]\times[3,4]$, we have $y+\lambda\eta(x, y)\in [3,4]\subset{K}.$

Let the fuzzy number-valued function $\widetilde{F}: K \rightarrow E^{2}$ be defined by

and $C = R^{2+}\subseteq{R^2}$, where $R^{2+} = \{(x_1, x_2)\in{R^2}:x_1\geq{0}, x_2\geq\; 0\}$. Then

For any $\xi \in [1,2]\cup [3,4],$ from Definition 2.2, it follows that

Therefore, for any $\xi_1, \xi_2\in [1,2]\cup [3,4]$ and $(\xi_1, \xi_2) \in [1,2]^{2}\cup [3,4]^{2}$, and for any $\lambda\in[0, 1]$, we have $\xi_2+\lambda\eta(\xi_1, \xi_2) = \xi_2 +\lambda(\xi_1-\xi_2).$ Thus,

In particular, from ^[7], we get $\lambda\tau(\widetilde{F}(\xi_1))+(1-\lambda)\tau(\widetilde{F}(\xi_2)) = \tau(\lambda\widetilde{F}(\xi_1)+(1-\lambda)\widetilde{F}(\xi_2)).$ We find that, for any $\xi_1, \xi_2 \in [1,2] \cup [3,4]$ and $(\xi_1, \xi_2) \in [1,2]^{2}\cup [3,4]^{2}$, $\lambda\in[0, 1]$,

i.e., $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\preceq_c\lambda\widetilde{F}(\xi_1)+(1-\lambda)\widetilde{F}(\xi_2).$

For any $\xi_1, \xi_2\in [1,2]\cup [3,4]$ and $(\xi_1, \xi_2)\in [3,4]\times [1,2]$,

and $(\xi_1, \xi_1)\in (1, 1)+C$, it follows that

i.e., $\widetilde{F}(t_2+\lambda\eta(t_1, t_2))\preceq_c\lambda\widetilde{F}(t_1)+(1-\lambda)\widetilde{F}(t_2).$

For any $\xi_1, \xi_2\in [1,2]\cup [3,4]$, and$(\xi_1, \xi_2)\in [1,2]\times [3,4]$,

and $(3, 3)\in(\xi_1, \xi_1)+C$, it follows that

i.e., $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\succeq_c\lambda\widetilde{F}(\xi_1)+(1-\lambda)\widetilde{F}(\xi_2).$ Above all, we denote $K_{1} = [1,2], K_{2} = [3,4], K = K_{1}\cup K_{2},$ then

(1) $\widetilde{F}$ is preinvex on $K_{1}$ w.r.t. $\eta$, but it is not strictly preinvex.

(2) $\widetilde{F}$ is preinvex on $K_{2}$ w.r.t. $\eta$, but it is not strictly preinvex.

(3) $\widetilde{F}$ is not preinvex on $K$ w.r.t. $\eta$. Since $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\preceq_c\lambda\widetilde{F}(\xi_1)+(1-\lambda)\widetilde{F}(\xi_2)$, for $\xi_1, \xi_2 \in K_{1},$ or $\xi_1, \xi_2 \in K_{2},$ or $\xi_1\in K_{2}, \xi_2\in K_{1}$. However, $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\succeq_c\lambda\widetilde{F}(\xi_1)+(1-\lambda)\widetilde{F}(\xi_2),$ for $\xi_1\in K_{1}, \xi_2\in K_{2}$. Also, it is not strictly preinvex.

Example 4.2. Let $K = [1,2]\cup [3,4]$. It is an invex set w.r.t. $\eta:R^{2}\times{R}^{2}\rightarrow {R}^{2},$ $\eta(x, y)$ is the same as Example 4.1. Let the fuzzy number-valued function $\widetilde{F}: K \rightarrow E^{2}$ be defined by

and $C = R^{2+}\subseteq{R^2}$. Then, $F_r(\xi) = \{(x_1, x_2):0\leq\; x_1\leq\ln2\xi\sqrt{1-r^2}, 0\leq\; x_2\leq3\}, r\in[0, 1]$. It is not difficult to calculate, for any $\xi\in [1,2]\cup[3,4],$

Thus, if for any $\xi_1, \xi_2\in[1,2]\cup[3,4]$ and $(\xi_1, \xi_2)\in [1,2]^{2}\cup [3,4]^{2}$, we have $\xi_2+\lambda\eta(\xi_1, \xi_2) = \xi_2+\lambda(\xi_1-\xi_2),$ In addition, for any $\lambda\in[0, 1],$

Without loss of generality, we assume that $\ln(2\xi_2+\lambda(2\xi_1-2\xi_2))\leq\max\{\ln2\xi_1, \ln2\xi_2\} = \ln2\xi_2,$ Thus

i.e., $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\preceq_c\max\{\widetilde{F}(\xi_1), \widetilde{F}(\xi_2)\}.$ However, for any $\xi_1, \xi_2\in[1,2]\cup[3,4]$ and $(\xi_1, \xi_2)\in [1,2]^{2}\cup [3,4]^{2}$, $\lambda\in[0, 1],$ we have $\ln(2\xi_2+\lambda(2\xi_1-2\xi_2))\geq \lambda\ln 2\xi_1+(1-\lambda)\ln2\xi_2.$ Taking $2\xi_1 = 2, 2\xi_2 = e, \lambda = \frac{1}{2}$, then

i.e., $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\succeq_c\lambda\; \widetilde{F}(\xi_1)+(1-\lambda)\widetilde{F}(\xi_2).$

If $\xi_1, \xi_2\in[1,2]\cup[3,4]$ and $(\xi_1, \xi_2)\in [3,4]\times[1,2]$, then $\xi_2+\lambda\eta(\xi_1, \xi_2) = \xi_2+\lambda(1-\xi_2)\leq \xi_2+\lambda(\xi_{1}-\xi_2).$ In addition, for any $\lambda\in[0, 1],$

Thus, $\max\{\tau(\widetilde{F}(\xi_1)), \tau(\widetilde{F}(\xi_2))\}\in\tau({\widetilde{F}}(\xi_2+\lambda\eta(\xi_1, \xi_2))+C.$ i.e., $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\preceq_c\max\{\widetilde{F}(\xi_1), \widetilde{F}(\xi_2)\}.$ For any $\xi_1, \xi_2\in[1,2]\cup[3,4]$ and $(\xi_1, \xi_2)\in [3,4]\times[1,2]$, for any $\lambda\in[0, 1],$ $\ln(2\xi_2+\lambda(2-2\xi_2))\leq\lambda\ln2\xi_1+(1-\lambda)\ln2\xi_2,$ it follows that,

i.e., $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\preceq_c\lambda\widetilde{F}(\xi_1)+(1-\lambda)\widetilde{F}(\xi_2).$

If $\xi_1, \xi_2\in[1,2]\cup[3,4]$ and $(\xi_1, \xi_2)\in [1,2]\times[3,4]$, then $\xi_2+\lambda\eta(\xi_1, \xi_2) = \xi_2+\lambda(3-\xi_2).$ It is easy to verify that $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\preceq_c\max\{\widetilde{F}(\xi_1), \widetilde{F}(\xi_2)\}.$ However, when $2\xi_1 = 2, 2\xi_2 = 7, \lambda = \frac{4}{5}$, we obtain

i.e. $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\succeq_c\lambda\; \widetilde{F}(\xi_1)+(1-\lambda)\widetilde{F}(\xi_2).$

Above all, we denote $K_{1} = [1,2], K_{2} = [3,4], K = K_{1}\cup K_{2},$ then $\widetilde{F}$ is prequasiinvex on $K$ w.r.t. $\eta$, but $\widetilde{F}$ is not preinvex on $K$ w.r.t. $\eta$. Since we have $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\preceq_c\lambda\widetilde{F}(\xi_1)+(1-\lambda)\widetilde{F}(\xi_2),$ for $\xi_1\in K_{2}, \xi_2\in K_{1}$ and $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\succeq_c\lambda\widetilde{F}(\xi_1)+(1-\lambda)\widetilde{F}(\xi_2),$ for $\xi_1, \xi_2 \in K_{1},$ or $\xi_1, \xi_2 \in K_{2},$ or $\xi_1\in K_{1}, \xi_2\in K_{2}.$

Example 4.3. Let $K = [-2, -1]\cup [1,3]$. It is easy to prove that $K$ is an invex set w.r.t. $\eta:R^{2}\times{R}^{2}\rightarrow {R}^{2}$ defined by

Let a fuzzy number-valued function $\widetilde{F}: K \rightarrow E^{2}$ be defined by

and $C = R^{2+}\subseteq{R}^2$. Then,

$F_r(\xi) = \{(x_1, x_2):-1\leq{x_1}\leq{0}, -2\leq{x_2}\leq{0}\},$ when $-2\leq\xi\leq-1$;

$F_r(\xi) = \{(x_1, x_2):{x_1}^2+{x_2}^2\leq{e^{2\xi}(1-r^2)}, x_1\geq{0}, x_2\geq{0}\},$ when $1\leq\xi\leq3$.

According to Definition 2.2, we obtain

$\tau(\widetilde{F}(\xi)) = (-\frac{1}{2}, -1),$ when $-2\leq\xi\leq-1$;

$\tau(\widetilde{F}(\xi)) = (\frac{8e^\xi}{9\pi}, \frac{8e^\xi}{9\pi}),$ when $1\leq\xi\leq3$.

Then, we have

for any $\xi_1, \xi_2\in{[-2, -1]\cup [1,3]}$ and for any $\lambda\in[0, 1]$. i.e., $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\preceq_c{\max}\{\widetilde{F}(\xi_1), \widetilde{F}(\xi_2)\}.$

Above all, we denote $K_{1} = [-2, -1], K_{2} = [1,3], K = K_{1}\cup K_{2},$ then $\widetilde{F}$ is prequasiinvex on $K$ w.r.t. $\eta$, but $\widetilde{F}$ is not strictly prequasiinvex on $K$ w.r.t. $\eta$. In fact, since we have $\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))\prec_c{\max}\{\widetilde{F}(\xi_1), \widetilde{F}(\xi_2)\}$, for any $\xi_1\in K_2, \xi_2\in K_1, \xi_2+\lambda\eta(\xi_1, \xi_2)\in K_{1}$ or $\xi_1, \xi_2\in K_{2}, \xi_2+\lambda\eta(\xi_1, \xi_2)\in K_{2}$ or $\xi_1\in K_{1},$ $\xi_2\in K_{2}, \xi_2+\lambda\eta(\xi_1, \xi_2)\in K_{2}$, with $\xi_1 \neq\xi_2$ and for any $\lambda\in(0, 1)$. However, $\tau(\widetilde{F}(\xi_2+\lambda\eta(\xi_1, \xi_2))) = \max\{\tau(\widetilde{F}(\xi_1)), \tau(\widetilde{F}(\xi_2))\}$, for $\xi_1, \xi_2\in K_{1}, \xi_2+\lambda\eta(\xi_1, \xi_2)\in K_{1}$ with $\xi_1\neq\xi_2$.

In order to discuss the relationships of preinvex and prequasiinvex fuzzy number-valued functions, we get the following special function $\eta$ and the fuzzy number-valued functions according to the discussion of ^[13].

Let $\eta:R^n\times{R^n}\rightarrow {R^n}$, we say that a function $\eta$ satisfies the condition C if

for any $x, y\in{R^n}$, $\lambda\in[0, 1]$ (refer to ^[13]).

A fuzzy number-valued function $\widetilde{F}:K\rightarrow {E^{n}}$ satisfies Condition D, if $K\subset{R^n}$ is an invex set w.r.t. $\eta:R^n\times{R^n}\rightarrow {R^n}$, for any $x, y \in{K},$ we have

In order to include singletons in $R^n$ as an invex sets, we assume that for all $x\in{R^n}$,

where $O$ being the origin of $R^n$.

Example 4.4. Let $K = [-1, \frac{1}{2}]\cup [\frac{1}{4}, 1]$. It is easy to prove that $K$ is invex w.r.t. $\eta:R^{2}\times{R}^{2}\rightarrow {R}^{2}$ defined by

In fact, if $(x, y)\in [-1, \frac{1}{2}]^{2}\cup [\frac{1}{4}, 1]^{2}$, from the convexity of $[-1, \frac{1}{2}]$ and $[\frac{1}{4}, 1]$, we have

If $(x, y)\in [\frac{1}{4}, 1] \times [-1, \frac{1}{2}]$, $y+\lambda\eta(x, y) = y+\lambda(1-y) = (1-\lambda)y+\lambda,$ we can choose $\lambda = 1$ and $\lambda = 0)$, it follows that $-1 \leq(1-\lambda)y+\lambda\leq \frac{1}{2},$ i.e., $y+\lambda\eta(x, y)\in [-1, \frac{1}{2}] \subset{K}.$ Similarly, if $(x, y)\in [-1, \frac{1}{2}]\times [\frac{1}{4}, 1]$, we have $y+\lambda\eta(x, y)\in [\frac{1}{4}, 1] \subset{K}.$

Let the fuzzy-number-valued function $\widetilde{F}:K \rightarrow E^{2}$ be defined as

Then, for any $r\in[0, 1],$

Since

Thus, $[\frac{\widetilde{F}(0+h\eta(x, y))\ominus_{g}\widetilde{F}(0)}{h}]^{r} = \frac{[\widetilde{F}(h\eta(x, y))\ominus_{g}\widetilde{F}(0)]^{r}}{h} = [-1+r, 1-r]|\eta(x, y)|\times[-1, 1]|\eta(x, y)|$ for any $r\in[0, 1].$ Assume that

we have $[\widetilde{u}]^{r} = [-1+r, 1-r]\times[-1, 1]$ for any $r\in[0, 1].$ Then,

$\widetilde{F}$ is fuzzy $\eta$-extended directionally differentiable at $0$, and $\widetilde{\nabla}\widetilde{F}_{\eta}(0)$ is the fuzzy $\eta$-extended directional derivative at $0$ in the direction $\eta(x, y)$(denoted $\widetilde{\nabla}\widetilde{F}_{\eta}(0) = (-1) \widetilde{u}$.

Theorem 4.3. Let $K\subset{R^n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, and $\widetilde{F}:K\rightarrow {E^{n}}$ be a preinvex fuzzy number-valued function. If $\widetilde{F}$ is weakly strictly preinvex on $K$, i.e., there exists a $\lambda_0\in(0, 1)$ such that

for any $x, y\in{K},$ with $x\neq{y}$, then $\widetilde{F}$ is strictly preinvex on $K$.

$Proof$. Assume that $\widetilde{F}$ is not strictly preinvex on $K$, then $x, y\in{K}$ with $x\neq{y}$ and for any $\lambda\in(0, 1)$ such that

Choose $\lambda_1, \lambda_2\in(0, 1)$ such that $\lambda = \lambda_0\lambda_1+(1-\lambda_0)\lambda_2$ and by taking $\overline{x} = y+\lambda_1\eta(x, y)$, $\overline{y} = y+\lambda_2\eta(x, y)$, from Condition C, we have

From the preinvexity of $\widetilde{F}$, we find that

From (4.1), it follows that

It is a contradiction to (4.2), i.e., $\widetilde{F}$ is strictly preinvex on $K$.

Lemma 4.1. Let $K\subset{R^n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, and $\widetilde{F}:K\rightarrow {E^{n}}$ satisfy Condition $\textrm{D}$. If there exists a $\alpha\in(0, 1)$ such that

for any $x, y\in{K}$, then the set

is dense in $[0, 1]$.

$Proof$. It is obvious that $0\in{A}$, from Condition D, it follows that $1\in{A}$, i.e., $A\neq\emptyset$ and $A$ is not a single point set. Suppose that $A$ is not dense in $[0, 1]$, then there exists a $\lambda_0\in(0, 1)$ such that $U\cap{A} = \emptyset,$ and where $U$ is a $\delta$-neighborhood $N_{\delta}(\lambda_0)$ of $\lambda_0$. Now, we denote

then, we have $0\leq\lambda_2\leq\lambda_1\leq\; 1$. Due to $\alpha, (1-\alpha)\in(0, 1)$, we can choose $u_1, u_2\in{A}$ such that

and take $u_1\geq\lambda_1,$ $u_2\leq\lambda_2.$ Let $\overline{\lambda} = \alpha\; u_1+(1-\alpha)u_2$, from Condition C, for any $x, y\in{K}$, we have

According to the fact that $u_1, u_2\in{A}$, we find that

Then, it follows that $\overline{\lambda}\in{A}.$ If $\overline{\lambda}\geq\lambda_0$, from the definition of $\lambda_1$, we get $\lambda_1\leq\overline{\lambda}$. In addition, we have

moreover,

It is a contradiction. Similar to that $\overline{\lambda}\leq\lambda_0$. Thus, $A$ is dense in $[0, 1]$.

Theorem 4.4. Let $K\subset{R^n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, $\widetilde{F}:K\rightarrow {E^{n}}$ satisfy Condition $\textrm{D}$, and $\widetilde{F}$ be a prequasiinvex fuzzy number-valued function. If $\widetilde{F}$ is weakly preinvex on $K$, i.e., there exists a $\lambda_0\in(0, 1)$ such that

for any $x, y\in{K},$ then $\widetilde{F}$ is preinvex on $K$.

$Proof$. Assume that $\widetilde{F}$ is not preinvex on $K$, then for $x, y\in{K}$, there exists a $\lambda\in[0, 1]$ such that

If $\widetilde{F}(x) = \widetilde{F}(y)$, then we have $\widetilde{F}(y+\lambda\eta(x, y))\succ_c{\widetilde{F}(x)}$. Choose $\lambda_1, \lambda_2\in[0, 1]$, such that $\lambda = \lambda_0\lambda_1+(1-\lambda_0)\lambda_2$ and by taking $\overline{x} = y+\lambda_1\eta(x, y), \overline{y} = y+\lambda_2\eta(x, y)$, from Condition C, we get $y+\lambda\eta(x, y) = \overline{y}+\lambda_0\eta(\overline{x}, \overline{y}).$ From the prequasiinvexity of $\widetilde{F}$, it follows that

From (4.3), we find that

which is a contradiction to (4.4), i.e., $\widetilde{F}$ is preinvex on $K$.

Otherwise, let $\widetilde{F}(x)\prec_c{\widetilde{F}}(y)$. Since $\widetilde{F}$ is weakly preinvex on $K$, then, according to Lemma 4.1, there exists a $\lambda_1\in{A}$ with $\lambda_1 < \lambda$, such that

Thus,

Choose $\lambda_2 = \frac{\lambda-\lambda_1}{1-\lambda_1}$ and by taking $\overline{x} = x, \overline{y} = y+\lambda_1\eta(x, y)$, then from Condition C, we have

If $\widetilde{F}(\overline{x})\preceq_c{\widetilde{F}}(\overline{y})$, then from the prequasiinvexity of $\widetilde{F}$, we obtain

which is a contradiction to (4.5), i.e., $\widetilde{F}$ is preinvex on $K$.

If $\widetilde{F}(\overline{x})\succ_c{\widetilde{F}}(\overline{y})$, then from the prequasiinvexity of $\widetilde{F}$, we obtain

which is a contradiction to (4.4), i.e., $\widetilde{F}$ is preinvex on $K$.

According to Theorem 4.3 and Theorem 4.4, we have the following conclusion.

Theorem 4.5. Let $K\subset{R^n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, $\widetilde{F}:K\rightarrow {E^{n}}$ satisfy Condition $\textrm{D}$, and $\widetilde{F}$ be a prequasiinvex fuzzy number-valued function. If $\widetilde{F}$ is weakly strictly preinvex on $K$, i.e., there exists a $\lambda_0\in(0, 1)$ such that

for any $x, y\in{K}$ with $x\neq{y}$, then $\widetilde{F}$ is strictly preinvex on $K$.

Theorem 4.6. Let $K\subset{R^n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, and $\widetilde{F}:K\rightarrow {E^{n}}$ be a prequasiinvex fuzzy number-valued function. If $\widetilde{F}$ is weakly strictly prequasiinvex on $K$, i.e., there exists a $\lambda_0\in(0, 1)$ such that

for any $x, y\in{K}$ with $x\neq{y}$, then $\widetilde{F}$ is strictly prequasiinvex on $K$.

$Proof$. Assume that $\widetilde{F}$ is not strictly prequasiinvex on $K$. Then for $x, y\in{K}$ with $x\neq{y}$, there exists a $\lambda\in(0, 1)$ such that

Choose $\lambda_1, \lambda_2\in(0, 1)$, such that $\lambda = \lambda_0\lambda_1+(1-\lambda_0)\lambda_2$ and by taking $\overline{x} = y+\lambda_1\eta(x, y)$, $\overline{y} = y+\lambda_2\eta(x, y),$ then, using Condition C, we have $\overline{y}+\lambda_0\eta(\overline{x}, \overline{y}) = y+\lambda\eta(x, y)$. According to the prequasiinvex of $\widetilde{F}$, it follows that

From (4.6), we get

which is a contradiction to (4.7). i.e., $\widetilde{F}$ is strictly prequasiinvex on $K$.

It is similar to Theorem 4.5, we have the following result.

Theorem 4.7. Let $K\subset{R^n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, $\widetilde{F}:K\rightarrow {E^{n}}$ satisfy Condition $\textrm{D}$, and $\widetilde{F}$ be a strictly prequasiinvex fuzzy number-valued function. If $\widetilde{F}$ is weakly strictly preinvex on $K$, i.e., there exists a $\lambda_0\in(0, 1)$ such that

for any $x, y\in{K}$ with $x\neq{y}$, then $\widetilde{F}$ is strictly preinvex on $K$.

The above relationships of preinvexity, weak preinvexity, strict preinvexity, weakly strict preinvexity, prequasiinvexity, weak prequasiinvexity, strict prequasiinvexity, weakly strict prequasiinvexity can be summarized in the following diagram (D refers to Condition $\textrm{D}$, C refers to Condition $\textrm{C}$).

5.   Properties of $n$-dimensional preinvex, prequasiinvex fuzzy number-valued functions

In this section, we introduce the properties of $n$-dimensional preinvex, prequasiinvex fuzzy number-valued functions, and their applications in the fuzzy optimization problems.

Theorem 5.1. Let $K$ be an invex set of $R^{n}$ w.r.t. $\eta$, and $\widetilde{F}:K\rightarrow {E^{n}}$ be a preinvex fuzzy number-valued function. Then the epigraph

of $\widetilde{F}$ is an invex set of $K\times{E}^{n}$ w.r.t. the function

defined by

for $(x, u), (y, v)\in{epi}(\widetilde{F})$ with $x, y\in{K}$ and $u, v\in{E}^{n}$. Here $epi(\widetilde{F})$ is an invex set of $K\times{E}^{n}$ means that $(y, v)+\lambda\eta'((x, u), (y, v)) \in{epi}(\widetilde{F})$ for any $(x, u), (y, v)\in{epi}(\widetilde{F})$ with $x, y\in{K}.$

$Proof$. If $epi(\widetilde{F})$ is the empty set or a singleton, then it is obvious that it is an invex set w.r.t. $\eta'$. Let $(x, u)$, $(y, v)\in{epi}(\widetilde{F})$, where $x, y\in{K}$ and $u, v\in{E}^{n}$. Then, from (5.1), we have

From the preinvexity of $\widetilde{F}$, for any $\lambda\in[0, 1]$, we have

which implies that for any $\lambda\in[0, 1]$,

This proves that ${epi(\widetilde{F})}$ is an invex set of $K\times{E}^{n}$ w.r.t. the function $\eta'$ defined by (5.2).

Theorem 5.2. Let $\widetilde{F}:K\rightarrow {E^{l}}$ be a preinvex fuzzy number-valued function.

(1) If $\widetilde{G}:E^{l}\rightarrow {E^n}$ is convex and increasing, then $\widetilde{G}\circ{\widetilde{F}}:K\rightarrow {E^{n}}$ is a preinvex fuzzy number-valued function;

(2) If $\widetilde{G}:E^{l}\rightarrow {E^n}$ is a positively homogeneous, increasing and sub-addition, then $\widetilde{G}\circ{\widetilde{F}}:K\rightarrow {E^{n}}$ is a preinvex fuzzy number-valued function.

$Proof$. Let $x, y\in{K}, \lambda\in[0, 1],$ since $\widetilde{F}:K\rightarrow {E^{l}}$ is a preinvex, we have

(1) Since $\widetilde{G}:E^{l}\rightarrow {E^n}$ is an increasing, it follows that

In addition, since $\widetilde{G}$ is a convex fuzzy mapping, it follows that

From the above arguments, we have for $x, y\in{K}$ and $\lambda\in[0, 1],$

which proves that $\widetilde{G}\circ{\widetilde{F}}:K\rightarrow {E^{n}}$ is a preinvex mapping on $K$.

(2) Since $\widetilde{G}:E^{l}\rightarrow {E^n}$ is a positively homogeneous, increasing and sub-addition fuzzy mapping, it follows that

Theorem 5.3. Let $\widetilde{F}_j:K\rightarrow {E^{n}}, j = 1, 2\cdots{l}$ be preinvex fuzzy number-valued functions. For$k_1, k_2, \cdots, k_l > 0$, The fuzzy mapping $\widetilde{F}:K\rightarrow {E^{n}}$ defined by

is a preinvex fuzzy number-valued function.

$Proof$. Since $\widetilde{F}_j:K\rightarrow {E^{n}}, j = 1, 2\cdots{l}$ is preinvex for each $j = 1, 2\cdots{l}$, we have for $x, y\in{K}$ and $\lambda\in[0, 1],$

From (5.3), it follows that for $x, y\in{K}$ and $\lambda\in[0, 1],$

which proves that $\widetilde{F}:K\rightarrow {E^{n}}$ is a preinvex fuzzy number-valued function.

Theorem 5.4. Let $\widetilde{F}:K\rightarrow {E^{n}}$ be a fuzzy number-valued function. Then $\widetilde{F}$ is preinvex w.r.t. $\eta$if and only if $\widetilde{F}(y+\lambda\eta(x, y))\prec_{c}\lambda{u}+(1-\lambda)v$ for any $x, y\in{K}$ satisfying $\widetilde{F}(x)\prec_{c}u$, $\widetilde{F}(y)\prec_{c}v$ and $\lambda\in [0, 1].$

$Proof$. Necessity is easy to prove.

Conversely, let for any $\widetilde{\varepsilon}\succ_c\widetilde{0}$, we have $\widetilde{F}(x)\prec_c{\widetilde{F}(x)}+\widetilde{\varepsilon}$, $\widetilde{F}(y)\prec_c{\widetilde{F}(y)}+\widetilde{\varepsilon}$, such that

Since $\widetilde{\varepsilon}$ is an arbitrary positive fuzzy number, then for any $\lambda\in[0, 1],$ we obtain

This completes the proof.

Theorem 5.5. Let $\widetilde{F}:K\rightarrow {E^{n}}$ be a preinvex fuzzy number-valued function w.r.t. $\eta.$Then for $u\in{E}^{n}$, the lower $u$-level set

of $\widetilde{F}$ is an invex set.

$Proof$. For any $x, y\in{K_u(\widetilde{F})}$, we have $\widetilde{F}(x)\preceq_c{u}$ and $\widetilde{F}(y)\preceq_c{u}$. Then, by the preinvexity of $\widetilde{F}$, it follows that, for any $\lambda\in[0, 1],$

which implies that

Definition 5.1. Let $S\subset{R}^{n}\times{E}^{n}$, $S$ is said to be $G$-invex set, if there exists a function $\eta:{R}^{n}\times{R}^{n}\rightarrow {R}^{n}$, for any $(x, u)$, $(y, v)\in{S}$, $(y+\lambda\eta(x, y), \lambda{u}+(1-\lambda)v)\in{S}, \; \; \; 0\leq\lambda\leq1.$

Theorem 5.6. Assume $K$ is an invex set, then $\widetilde{F}:K\rightarrow {E^{n}}$ is a preinvex fuzzy number-valued functionon $K$ if and only if ${epi(\widetilde{F})}$ is $G$-invex set of ${R}^{n}\times{E}^{n}$.

$Proof$. Assume that $\widetilde{F}$ is a preinvex fuzzy number-valued function on $K.$ For $(x, u), (y, v)\in{epi(\widetilde{F})},$ $\lambda\in[0, 1],$ it follows that $y+\lambda\eta(x, y)\in{K}$ and

Thus,

which implies that $epi(\widetilde{F})$ is $G$-invex set of ${R}^{n}\times{E}^{n}$ w.r.t. a given function $\eta\times\eta_{o}$, where $\eta_{o}:{E}^{n}\times{E}^{n}\rightarrow {E}^{n}$, $(u, v)\rightarrow {u}-v$.

Conversely, since ${epi(\widetilde{F})}$ is a $G$-invex set of ${R}^{n}\times{E}^{n}$, $(x, \widetilde{F}(x))\in epi(\widetilde{F})$ and $(y, \widetilde{F}(y))\in epi(\widetilde{F})$ for any $x, y\in{K}, \lambda\in[0, 1].$ Thus we have

which implies that

This shows that $\widetilde{F}$ is preinvex fuzzy number-valued function on $K$.

Theorem 5.7. Let $\{S_i\}_{i\in{I}}$ be a finite or infinite collection of $G$-invex sets of ${R}^{n}\times{E}^{n}$, where $I$ is index set. Then $S = \bigcap_{i\in{I}}S_i$ is a $G$-invex set.

$Proof$. Let for any $(x, u)$, $(y, v)\in{S}$, we have, for any $i\in{I}$, $(x, u), (y, v)\in{S_i}$, then

Therefore, we obtain

That is, $S = \bigcap_{i\in{I}}S_i$ is a $G$-invex set.

Theorem 5.8. Let $K\subseteq{R^n}$ be an invex set w.r.t. $\eta$, $\{\widetilde{F}_i\}_{i\in{I}}$be a set of $n$-dimensional preinvex fuzzy number-valued functions on $K.$ If $\sup\{\widetilde{F}_i(x)|i\in{I}\}$ exists in ${E}^{n}$ for any $x\in{K}$, then $\widetilde{F}(x) = \sup\{\widetilde{F}_i(x)|i\in{I}\}$ is a $n$-dimensional preinvex fuzzy number-valued function on $K$.

$Proof$. Since each $\widetilde{F}_i(i\in{I})$ is $n$-dimensional preinvex fuzzy number-valued function on $K$, then by Theorem 5.6, we know that

is a $G$-invex set of ${R}^{n}\times{E}^{n}$. By Theorem 5.7, we have

is a $G$-invex set of ${R}^{n}\times{E}^{n}$. It is an easy matter to verify that

Thus, $epi(\widetilde{F})$ is a $G$-invex set of ${R}^{n}\times{E}^{n}$. By Theorem 5.6, we find that $\widetilde{F}$ is a $n$-dimensional preinvex fuzzy number-valued function on $K$.

Theorem 5.9. Let $K\subseteq{R^n}$ be an invex set w.r.t. $\eta$. Then $\widetilde{F}:K\rightarrow {E^{n}}$ is a prequasiinvex fuzzy number-valued function on $K$ if and only if the lower $u$-level set

of $\widetilde{F}$ is an invex set w.r.t. $\eta$ for each $u\in{E^n}$.

$Proof$. Necessity is easy to prove.

Conversely, assume that $L_u(\widetilde{F})$ is an invex set for each $u\in{E}^{n}$. Let $x, y\in{K}$, without loss of generality, we may assume that $\widetilde{F}(x)\preceq_c{\widetilde{F}(y)}$. Let $u = \widetilde{F}(y)$, since $\preceq_c$ is reflexive and transitive, we have

which implies that

We have $y+\lambda\eta(x, y)\in{L}_u(\widetilde{F})$, which implies that

which completes the proof.

Theorem 5.10. Let $\widetilde{F}:K\rightarrow {E^{n}}$ be a prequasiinvex fuzzy number-valued function. Then $t\widetilde{F}$ is prequasiinvex fuzzy number-valued function on $K$ for any $t > 0$.

$Proof$.

Theorem 5.11. Let $\widetilde{F}:K\rightarrow {E^{n}}$ be prequasiinvex fuzzy number-valued function w.r.t. $\eta$, and$\overline{x}\in{K}$ be the global minimizer of $\widetilde{F}$ on $K.$ Then, the set

is an invex set w.r.t. $\eta$.

$Proof$. If $\Omega$ is the empty set or singleton, then it is obvious an invex set. Assume that $x, y\in{\Omega}$, then

Since $\widetilde{F}:K\rightarrow {E^{n}}$ is a prequasiinvex fuzzy number-valued function w.r.t. $\eta$, we have

for any $\lambda\in[0, 1].$ Consider $\overline{x}\in{K}$ is a global minimizer of $\widetilde{F}$, it follows that

for $\lambda\in[0, 1].$ It implies that $y+\lambda\eta(x, y)\in\Omega$ for $\lambda\in[0, 1]$. Thus, $\Omega$ is an invex set w.r.t. $\eta$. This completes the proof.

Theorem 5.12. Let $\widetilde{F}:K\rightarrow {E^{n}}$ be preinvex fuzzy number-valued function w.r.t. $\eta,$ and$\overline{x}\in{K}$ satisfying $\widetilde{F}(\overline{x}) = \min_{x\in{K}}\widetilde{F}(x).$ If $u = \min_{x\in{K}}\widetilde{F}(x)$, then the set

is an invex set w.r.t. $\eta$.

6.   Semicontinuity and preinvexity of fuzzy number-valued functions

In this section, we present several practical criteria for preinvex fuzzy number-valued functions under the lower or upper semicontinuity conditions.

Definition 6.1. (see ^[7].) Let $\widetilde{F}:K\rightarrow {E^{n}}$ be a fuzzy number-valued function

(1) $\widetilde{F}$ is said to be lower semicontinuous(l.c.) at $x_{0}\in{K}$ if for any $\widetilde{\epsilon}\succ_c0$, a neighborhood $U$ of $x_{0}$ exists when $x\in{K}$, and we have

(2) $\widetilde{F}$ is said to be upper semicontinuous(u.c.) at $x_{0}\in{K}$ if for any $\widetilde{\epsilon}\succ_c0$, a neighborhood $U$ of $x_{0}$ exists when $x\in{K}$, and we have

A fuzzy number-valued function $\widetilde{F}:K\rightarrow {E^{n}}$ is continuous at $x_{0}\in{K}$ if it is both l.c. and u.c. at $x_{0}$, and that it is continuous at every point of $K$.

Theorem 6.1. Let $K\subset{R}^{n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, $\widetilde{F}:K\rightarrow {E^{n}}$be a lower semicontinuous fuzzy number-valued function, and $\widetilde{F}$ satisfy Condition $\textrm{D}$. If $\widetilde{F}$ is weakly preinvex on $K$, i.e., there exists a $\lambda_{0}\in(0, 1)$ such that

for any $x, y\in{K}$, then $\widetilde{F}$ is preinvex on $K$.

$Proof$. Assume that $\widetilde{F}$ is not preinvex on $K$, then, $x, y\in{K}$ and there exists a $\lambda\in[0, 1]$ such that

By the weak preinvexity of $\widetilde{F}$ and Lemma 4.1, we can choose a sequence $\lambda_{n}\in{A}(n = 1, 2, \cdots)$ with $\lambda_n\rightarrow \lambda(n\rightarrow \infty)$ and

From the lower semicontinuity of $\widetilde{F}$, for any $\widetilde{\varepsilon}\succ_{c}\widetilde{0}$, an $N > 0$ exists when $n > N$ and we have

Since $\widetilde{\varepsilon}$ is an arbitrary positive fuzzy number, by taking the limit as $n\rightarrow \infty$, and by combining with (6.2) and (6.3), we have

This contradicts the fact that (6.1), i.e., $\widetilde{F}$ is preinvex on $K$.

Theorem 6.2. Let $K\subset{R}^{n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, $\widetilde{F}:K\rightarrow {E^{n}}$ be a upper semicontinuous fuzzy number-valued function, and $\widetilde{F}$ satisfy Condition $\textrm{D}$. If $\widetilde{F}$ is weakly preinvex on $K$, i.e., there exists a $\lambda_{0}\in(0, 1)$ such that

for any $x, y\in{K}$, then $\widetilde{F}$ is preinvex on $K$.

$Proof$. Assume that $\widetilde{F}$ is not preinvex on $K$, then, $x, y\in{K}$ and there exists a $\lambda\in[0, 1]$ such that

By the weak preinvexity of $\widetilde{F}$ and Lemma 4.1, we can choose a sequence $\lambda_{n}\in{A}(n = 1, 2, \cdots)$ with $\lambda_n\rightarrow \lambda(n\rightarrow \infty)$ and

for any $\overline{x}, \overline{y}\in{K}$, by taking $\overline{x} = x\in{K}$, and $\overline{y} = y+\frac{\lambda-\lambda_n}{1-\lambda_n}\eta(x, y)\in{K}$, using Condition C

and $\overline{y}\rightarrow {y}(n\rightarrow \infty)$. According to the upper semicontinuity of $\widetilde{F}$, for any $\widetilde{\varepsilon}\succ_{c}\widetilde{0}$, there exists an $N > 0$ when $n > N$ and we get

Since $\widetilde{\varepsilon}$ is an arbitrary positive fuzzy number, and by combining with (6.5) and (6.6), we have

By taking the limit as $n\rightarrow \infty$, we have

This contradicts the fact that (6.4), i.e., $\widetilde{F}$ is preinvex on $K$.

By combining Theorem 6.1 and Theorem 6.2, we have the following result.

Corollary 6.1. Let $K\subset{R}^{n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, $\widetilde{F}:K\rightarrow {E^{n}}$ satisfy Condition $\textrm{D}$, and $\widetilde{F}$ be a lower semicontinuous or upper semicontinuous fuzzy number-valued function. Then $\widetilde{F}$ is preinvex on $K$ if and only if $\widetilde{F}$ is weakly preinvex on $K$.

Theorem 6.3. Let $K\subset{R}^{n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, $\widetilde{F}:K\rightarrow {E^{n}}$ satisfy Condition $\textrm{D}$, and $\widetilde{F}$ be a lower (resp. an upper) semicontinuous fuzzy number-valued function. If $\widetilde{F}$ is weakly strictly preinvex on $K$, i.e., there exists a $\lambda_{0}\in(0, 1)$ such that

for any $x, y\in{K}$ with $x\neq{y}$, then $\widetilde{F}$ is strictly preinvex on $K$.

By combining Definition 3.1 and Theorem 6.3, we have the following result.

Corollary 6.2. Let $K\subset{R}^{n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, $\widetilde{F}:K\rightarrow {E^{n}}$ satisfy Condition $\textrm{D}$, and $\widetilde{F}$ be a lower semicontinuous or upper semicontinuous fuzzy number-valued function. Then $\widetilde{F}$ is strictly preinvex on $K$ if and only if $\widetilde{F}$ is weakly strictly preinvex on $K$.

Lemma 6.1. Let $K\subset{R}^{n}$ be an invex set w.r.t. $\eta$, $\eta$ satisfy Condition $\textrm{C}$, and $\widetilde{F}:K\rightarrow {E^{n}}$ satisfy Condition $\textrm{D}$. If there exists a $\alpha\in(0, 1)$ such that

for any $x, y\in{K}$, then the set

is dense in $[0, 1]$.

This proof is similar to the proof of Lemma 4.1.

Theorem 6.4. Let $K\subset{R}^{n}$ be an invex set w.r.t. $\eta$, and $\eta$ satisfy Condition $\textrm{C}$, $\widetilde{F}:K\rightarrow {E^{n}}$ satisfy Condition $\textrm{D}$, and $\widetilde{F}$ be an upper semicontinuous fuzzy number-valued function. If $\widetilde{F}$ is weakly prequasiinvex on $K$, i.e., thereexists a $\lambda_{0}\in(0, 1)$ such that

for any $x, y\in{K}$, then $\widetilde{F}$ is prequasiinvex on $K$.

$Proof$. Assume that $\widetilde{F}$ is not prequasiinvex on $K.$ Then, for any $x, y\in{K}$ and there exists a $\lambda\in[0, 1]$ such that

By the weak prequasiinvexity of $\widetilde{F}$ and Lemma 6.1, we can choose a sequence $\lambda_{n}\in{A}(n = 1, 2, \cdots)$ with $\lambda_n\rightarrow \lambda(n\rightarrow \infty)$ and

for any $\overline{x}, \overline{y}\in{K}$, by taking $\overline{x} = x\in{K}$, and $\overline{y} = y+\frac{\lambda-\lambda_n}{1-\lambda_n}\eta(x, y)\in{K}$, using Condition C, we have $y+\lambda\eta(x, y) = \overline{y}+\lambda_n\eta(\overline{x}, \overline{y})$, and $\overline{y}\rightarrow {y}(n\rightarrow \infty)$. From the upper semicontinuity of $\widetilde{F}$, for any $\widetilde{\varepsilon}\succ_{c}\widetilde{0}$, there exists an $N > 0$ when $n > N$ and we obtain

Since $\widetilde{\varepsilon}$ is an arbitrary positive fuzzy number, and by combining with (6.8) and (6.9), it follows that

This contradicts to the fact that (6.7), i.e., $\widetilde{F}$ is prequasiinvex on $K$.

According to Theorem 4.6 and Theorem 6.4, we have the following result.

Theorem 6.5. Let $K\subset{R}^{n}$ be an invex set w.r.t. $\eta$, and $\eta$ satisfy Condition $\textrm{C}$, $\widetilde{F}:K\rightarrow {E^{n}}$ satisfy Condition $\textrm{D}$, and $\widetilde{F}$ be an upper semicontinuous fuzzy number-valued function. If $\widetilde{F}$ is weakly strictly prequasiinvex on $K$, i.e., there exists a $\lambda_{0}\in(0, 1)$ such that

for any $x, y\in{K}$ with $x\neq{y}$, then $\widetilde{F}$ is strictly prequasiinvex on $K$.

7.   Two-parameter optimization problem

In this section, two types of the parameter optimization problems are investigated. They are widely applied in the optimization theory of consumers and producers. In which the optimal value of objective function depends on the values of the parameters. Therefore, the optimal solution and optimal values are all functions of parameters. The central task of economic analysis is to clarify the character of these functions. Two-parameters optimization problems is shown as follows.

where $X, A, B$ $\subset{R^n}$ are invex sets w.r.t. $\eta:{R^n}\times{R^n}\rightarrow {R^n}$, $\widetilde{F}:X\rightarrow {E^{n}}$, $\widetilde{G}:X\rightarrow {E^{n}}$. In the problem $P(\alpha)$, the parameter appear in the fuzzy constraint function, and the parameter appear in the fuzzy objective function in the problem $P(\beta)$. We always assume that the problems $P(\alpha)$ and $P(\beta)$ have the optimal solution for any fixed parameters $\alpha$, $\beta$ respectively, and write $\widetilde{Z}(\alpha)$ and $\widetilde{\psi}(\beta)$ as the optimal objective values for $P(\alpha)$ and $P(\beta)$ respectively.

Theorem 7.1. Consider the problem $P(\alpha)$, if $\widetilde{G}(x, \alpha)$ is a $n$-dimensional preincave fuzzy number-valued function on $A$w.r.t. $\eta:{R^n}\times{R^n}\rightarrow {R^n}$, then $\widetilde{Z}(\alpha)$ is a $n$-dimensional prequasiinvex fuzzy number-valued function on $A$ w.r.t. the same function $\eta$.

$Proof$. For $\alpha_{1}, \alpha_{2}\in{A}$ and for any $\lambda\in[0, 1]$, let $x_{\lambda}$ be a optimal solution for $P(\alpha_{2}+\lambda\eta(\alpha_{1}, \alpha_{2}))$. From the preincavity of $\widetilde{G}$ w.r.t. $\alpha$, we obtain

Since $\lambda$ and $(1-\lambda)$ are all non-negative, it follows that, $\widetilde{G}(x_{\lambda}, \alpha_{1})$ and $\widetilde{G}(x_{\lambda}, \alpha_{2})$ at least one non-positive. Without loss of generality, we assume that

it follows that $x_{\lambda}$ is a feasible solution for $P(\alpha_{1})$ and $\widetilde{Z}(\alpha_{1})\succeq_{c}\widetilde{F}(x_{\lambda})$. Then, we have, for any $\lambda\in[0, 1]$,

i.e., $\widetilde{Z}(\alpha)$ is a $n$-dimensional prequasiinvex fuzzy number-valued function on $A$ w.r.t. the same function $\eta$.

Theorem 7.2. Consider $P(\beta)$, if $\widetilde{F}(x, \beta)$ is a $n$-dimensional preincave fuzzy number-valued function on $B$w.r.t. parameter $\beta$ and $\eta:{R^n}\times{R^n}\rightarrow {R^n}$, then $\widetilde{\psi}(\beta)$ is a $n$-dimensional preincave fuzzy number-valued function on $B$ w.r.t. the same function $\eta$.

$Proof$. For $\alpha_{1}, \alpha_{2}\in{B}$ and for any $\lambda\in[0, 1]$, let $x_{\lambda}$ be a optimal solution for $P(\beta_{2}+\lambda\eta(\beta_{1}, \beta_{2}))$. Since $\widetilde{G}(x_{\lambda})\preceq_{c}\widetilde{0}$, it follows that, $x_{\lambda}$ is the feasible solution to $P(\beta_{1})$ and $P(\beta_{2})$, which implies

From the preincavity of $\widetilde{F}$ w.r.t. $\beta$, we obtain

i.e., $\widetilde{\psi}(\beta)$ is a $n$-dimensional preincave fuzzy number-valued function on $B$ w.r.t. the same function $\eta$.

Example 7.1. The optimization problem in consumer theory. A consumer is an economic entity that uses available resources (income) to purchase goods and obtains satisfaction from the consumption of goods. The problem of the consumer is how to select the consumption bundle so that the consumer can get the maximum satisfaction from his consumption under the constraint that the total expenditure is not greater than the income of the consumer. Let $\widetilde{p} = (\widetilde{p}_{1}, \cdots, \widetilde{p}_{n})^{T}$ be the price vector, where $\widetilde{p}_{i}\succ_{c}0 (i = 1, \cdots, n)$ is the price of the good $i$, and let $x = (x_{1}, \cdots, x_{n})^{T}$ be the consumption bundle, where $x_{i}\geqslant0 (i = 1, \cdots, n)$ is the quantity of the good $i$ consumed. If the total expenditure $p^{T}x$ is not greater than the consumer's income $m$, there must be a constraint $p^{T}x\leqslant m$, which is the consumer's budget constraint. The set $S = \{x\in R_{+}^{n}: \widetilde{p}^{T}x\preceq_{c} m\}$ of all feasible consumption bundles is called the budget set. The consumer has different satisfaction for different consumption bundles, which is called consumer preference. We use utility functions to describe this preference. The quantity is only an estimated quantity, then using a fuzzy-valued function to express the quantity is more appropriate than using a crisp quantity. Specifically, a utility function $\widetilde{U}:R_{+}^{n}\rightarrow E_{+}^{n}$ is a nonnegative fuzzy-valued function satisfying the following specification: $\widetilde{U}(x)\succ_{c}\widetilde{U}(y)$ means that consumption bundle $x$ is better than consumption bundle $y$; $\widetilde{U}(x) = \widetilde{U}(y)$ means that consumption bundle $x$ equals consumption bundle $y$; $\widetilde{U}(x)\succeq_{c}\widetilde{U}(y)$ means that consumption bundle $x$ is not worse than consumption bundle $y$.

The utility maximization problem of consumers can be formalized into the following optimization model:

where $\widetilde{p}\succ_{c}0$, $m > 0$. Obviously, this is a $P(\alpha)$ type optimization problem, where the parameters $(\widetilde{p}, m)$ appear in the fuzzy constraint function. Let $v(\widetilde{p}, m) = \widetilde{U}(x(\widetilde{p}, m))$ for every $\widetilde{p}\succ+{c}0$ and $m > 0$, and $v(\widetilde{p}, m)$ is said to be an indirect utility function. Notice that the function $g(x; \widetilde{p}, m) = \widetilde{p}^{T}x-m$ is a preincave fuzzy number-valued function w.r.t. $(\widetilde{p}, m)$ and $\eta = x-y$. According to Theorem 7.1, $v(\widetilde{p}, m) = \widetilde{U}(x(\widetilde{p}, m))$ a $n$-dimensional prequasiinvex fuzzy number-valued function on $R_{+}^{n}$ w.r.t. the same function $\eta$.

8.   $n$-dimensional fuzzy variational-like inequality

Here, we present the fuzzy variational-like inequality and discuss the relationships between the fuzzy variational-like inequality problem and the unconstrained fuzzy vector optimization problem.

Let $K\subseteq{R^n}$, $\eta(x, \overline{x}):K\times{K}\rightarrow {R^n}$, $\widetilde{F}:K\rightarrow {E^{n}}$ be a $n$-dimensional fuzzy number-valued function. Then the fuzzy variational-like inequality problem is to be found $\overline{x}\in{K}$, $u\in{E^n}$, such that

Consider the unconstrained fuzzy vector optimization problem

where $K\subseteq{R^n}$ is an invex set w.r.t. $\eta$, $\widetilde{F}:K\rightarrow {E^{n}}$ is a $n$-dimensional fuzzy number-valued function.

A point $x_{0}\in{K}$ is called a local minimum of $\widetilde{F}$, if $x_{0}\in{K}$ and there exists a $\delta$-neighborhood $N_{\delta}(x_{0})$ around $x_{0}$, such that for any $x\in{K}\cap{N}_{\delta}(x_{0})$, $\widetilde{F}(x_{0})\preceq_{c}\widetilde{F}(x)$. Similarly, if $x_{0}\in{K}$ and there exists a $\delta$-neighborhood $N_{\delta}(x_{0})$ around $x_{0}$, such that for any $x\in{K}\cap{N}_{\delta}(x_{0})$, with $x\neq{x}_{0}$, $\widetilde{F}(x_{0})\prec_{c}\widetilde{F}(x)$, then $x_{0}$ is called a strict local minimum point of $\widetilde{F}$.

Theorems 8.1-8.3 show the relationship between the fuzzy variational-like inequality problem and the preinvexity of $n$-dimensional fuzzy number-valued function.

Theorem 8.1. Let $K$ be an invex set of $R^n$ w.r.t. $\eta$, $\overline{x}\in{K}$, $\widetilde{F}:K\rightarrow {E^{n}}$ a preinvex fuzzy number-valued function w.r.t. $\eta$ and $\widetilde{F}$ fuzzy $\eta$-extended directionally differentiable on $K$. If $(\overline{x}, \widetilde{\nabla}\widetilde{F}_{\eta}(\overline{x}))$ is a solution of $(FVLI)$, then $\overline{x}$ is a strict local optimal solution of $(P)$.

$Proof$. Let $(\overline{x}, \widetilde{\nabla}\widetilde{F}_{\eta}(\overline{x}))$ be a solution of $(FVLI)$. Suppose that there exists an $x^{*}\in{K}\cap{N}_{\delta}(\overline{x})$, such that

Since $\widetilde{F}$ is a preinvex fuzzy number-valued function, it follows that

From the $\eta$-extended directionally differentiability of $\widetilde{F}$, and taking the limit as $\lambda\rightarrow {0}^{+}$, we find that

According to (8.1) and (8.2), we obtain

This contradicts the fact that $(\overline{x}, \widetilde{\nabla}\widetilde{F}_{\eta}(\overline{x}))$ is a solution of $(FVLI)$.

Theorem 8.2. Let $K$ be an invex set of $R^n$ w.r.t. $\eta$, $\overline{x}\in{K}$, $\widetilde{F}:K\rightarrow {E^{n}}$ a preincave fuzzy number-valued function w.r.t. $\eta$ and $\widetilde{F}$ fuzzy $\eta$-extended directionally differentiable on $K$. If $\overline{x}$ be a strict local optimal solution of $(P)$, then $(\overline{x}, \widetilde{\nabla}\widetilde{F}_{\eta}(\overline{x}))$ is a solution of $(FVLI)$.

$Proof$. Let $\overline{x}$ be a strict local optimal solution of $(P)$. Suppose that there exists an $x^{*}\in{K}$, such that

Since $\widetilde{F}$ is a preincave fuzzy number-valued function, it follows that

By the $\eta$-extended directionally differentiability of $\widetilde{F}$, and taking the limit as $\lambda\rightarrow {0}^{+}$, we obtain

Therefore, we have

This contradicts the fact that $\overline{x}$ is a strict local optimal solution of $(P)$.

Theorem 8.3. Let $K$ be an invex set of $R^n$ w.r.t. $\eta$, $\overline{x}\in{K}$, $\widetilde{F}:K\rightarrow {E^{n}}$ be a strictly preincave fuzzy number-valued function w.r.t. $\eta$ and $\widetilde{F}$ be fuzzy $\eta$-extended directionally differentiable on $K$. If $\overline{x}$ be an optimal solution of $(P)$, then $(\overline{x}, \widetilde{\nabla}\widetilde{F}_{\eta}(\overline{x}))$ is a solution of $(FVLI)$.

$Proof$. Let $\overline{x}$ be an optimal solution of $(P)$. Suppose that there exists an $x^{*}\in{K}$, such that

Since $\widetilde{F}$ is a strictly preincave fuzzy number-valued function, it follows that

From the $\eta$-extended directionally differentiability of $\widetilde{F}$. Taking the limit as $\lambda\rightarrow {0}^{+}$, we obtain

Therefore, we have

This contradicts the fact that $\overline{x}$ is a optimal solution of $(P)$.

9.   Kuhn-Tucker conditions for a multiobjective fuzzy programming problem

Consider a multiobjective fuzzy programming problem,

where $\widetilde{F}:K\rightarrow {E^{n}}$, $\widetilde{G}_{i}:K\rightarrow {E^{n}}$, $(i = 1, 2, \cdots{m})$, $X\subset{R^{n}}$ is an invex set w.r.t. $\eta$.

Let $S$ be the set of the feasible solution for $(P)$, $x_{0}$ is a feasible point for $(P)$. For a feasible point $x_{0}$, we denote

Then,

Let $\widetilde{F}$ be a $n$-dimensional fuzzy number-valued function defined on $X$ and $\widetilde{G}$ be a $m$-dimensional fuzzy number-valued function defined on $X$, $X\subset{R^{n}}$ is an invex set w.r.t. $\eta$. ($\widetilde{G} = ({\widetilde{G_{i}}})_{i = 1}^{m}$, means $\widetilde{G_{i}}$ is a $n$-dimensional fuzzy number-valued function, for each $i = 1, 2, \cdots{m}$).

Define the $n$-dimensional lagrangian fuzzy function as

The Kuhn-Tucker stationary point of a $n$-dimensional fuzzy optimal problem is to find a $x\in{X}$, $\lambda = (\lambda_{1}, \lambda_{2}, \cdots, {\lambda_{m}})\in{R}^{m}$, if they exists, such that

In what follows, we show the connection between the Kuhn-Tucker stationary point for a $n$-dimensional fuzzy optimal problem and the optimal solution of $(P)$.

Theorem 9.1. Let $x_{0}$ be a feasible solution of $(P)$, and let $\widetilde{F}:K\rightarrow {E^{n}}$ be a preinvex fuzzy number-valued function at $x_{0}$ w.r.t. $\eta$ and $\widetilde{F}$ be fuzzy $\eta$-extended directionally differentiable at $x_{0}$ on $K$. Let $\widetilde{G}_{i}:K\rightarrow {E^{n}}(1, 2, \cdots{m})$ be aprequasiinvex fuzzy number-valued function at $x_{0}$ w.r.t. the same function $\eta$ and $\widetilde{G}_{i}(1, 2, \cdots{m})$ be fuzzy $\eta$-extended directionally differentiable at $x_{0}$ on $K$. Moreover, if there exist $\lambda_{i}\geq{0}(1, 2, \cdots{m})$, such that

Then, $x_{0}$ is the global minimum point of $(P)$.

$Proof$. Assume that $x_{0}$ is not a global minimum point of $(P)$. Then, there exists $\overline{x}\in{S}$ such that

Since $\widetilde{F}$ is a preinvex fuzzy number- valued function at $x_{0}$, and $\widetilde{F}$ is $\eta$-extended directionally differentiable at $x_{0}$, we have

Also, since $\widetilde{G}_{i}(\overline{x})\preceq_{c}\widetilde{0} = \widetilde{G}_{i}(x_{0}), i\in{I}(x_{0})$ and the $\eta$-extended directionally differentiability of $\widetilde{G}_{i}$ at $x_{0}$, for any $\lambda\in[0, 1]$, we get

$\widetilde{G}_{i}$ is a prequasiinvex fuzzy number-valued function at $x_{0}$ w.r.t. the same function $\eta$, then

By combining (9.1) and (9.2), we find that

According to $\lambda_{i}\widetilde{G}_{i}(x_{0}) = \widetilde{0}$, it follows that, $\forall{i}\in{I}'(x_{0}), \lambda_{i} = 0.$

From above discussion, we have

which is a contradiction to the condition

It completes the proof.

10.   Conclusions

In this paper, we first introduce the concept of the preinvexity of $n$-dimensional fuzzy number-valued functions based on the partial order relation in $n$-dimensional fuzzy number space and their properties are discussed. In addition, some counterexamples are given to show the proposed concepts and their relationships. Then we present the criteria theorems for $n$-dimensional preinvex fuzzy number-valued functions under the upper or lower semicontinuity conditions, respectively. Furthermore, the two-parameter optimization problem, $n$-dimensional fuzzy variational-like inequality problem, and the optimality conditions related to $n$-dimensional preinvex fuzzy number-valued function are discussed. These results can be applied in many fields, such as fuzzy optimization, fuzzy control, engineering science, fuzzy-making problems and so on.

Acknowledgments

The authors would like to thank the anonymous referees and the editor. This work is supported by the National Natural Science Foundation of China (12061067, 11901265).

Conflict of interest

The authors declare that they have no conflict of interest.