A Class of Nonlinear Fuzzy Variational Inequality Problems

Abstract

In this paper, we consider nonlinear variational inequality problems with fuzzy variables. The fuzzy variables were introduced to deal with the variational inequality containing noise for which historical data is not available. The fuzzy expected residual minimization (FERM) problems were established. We discussed the $S{C}^1$ property of the FERM model. Furthermore, results of convergence analysis were obtained based on an approximation model of the FERM model. The convergence of global optimal solutions and the convergence of stationary points were analysed.

1. Introduction

In many real-world problems, examples contain some uncertain information, with examples including new stock, emergencies, military experiments, etc. For this uncertain information, often no historical data are available. In scientific fields, we also encountered similar uncertain situations, in which uncertainty contains noise that make measurements deviate from the correct, intended, or original values. Also, in these cases, experts cannot collect enough historical data to measure the noise. Zadeh [1] introduced fuzzy set theory, which provided efficacious ways to handle uncertain information. The uncertain information contained a belief degree, representing the strength with which we believe the event will happen. A fuzzy number is a kind of special fuzzy set and its operation was a key factor in processing fuzzy information. An interval-valued expectation of fuzzy numbers was given by Dubois and Prade [2,3] in 1987. The additive of the expectation was proved. They [4] also introduced possibility theory, about the introduction of qualitative and quantitative aspects, in 1998. In this paper, we define the possibility expectation values, possibility distribution function, and establish equivalence of fuzzy membership functions, which are consistent with the extension principle, and with the well-known definitions of expectation and variance in probability theory. The theory discussed in this paper was motivated by the principles in [2,3,4,5,6,7,8,9,10,11,12,13]. The concept of variational inequalities was introduced by Chang and Zhu [14] for fuzzy mappings in 1989. Soon afterwards, a new method was proposed by Huang [15] on a class of variational inequalities with fuzzy mappings. Huang [16] surveyed random generalized variational inclusions for random fuzzy mappings. In [17], the authors considered variational inequalities for fuzzy mappings. Moreover, an algorithm for fuzzy linear programming with piecewise linear membership functions was presented by Inuiguchi et al. [18]. A group of inequalities with fuzzy numbers, their membership functions are linear, were investigated by Hu and Fang [19]. They turned fuzzy inequalities into convex programming problems. Fang and Hu [20], and Hu [21] introduced and studied some fuzzy variational inequalities in a fuzzy environment, and proved the existence of the optimal solution for fuzzy variational inequalities by using the tolerance approach and the entropic regularization technique. In particular, in [22], Hu showed that solving the fuzzy variational inequalities are equivalent to solving a fuzzy generalized complementarity problem. Wang and Liao [23,24] established fuzzy variational inequalities based on fuzzy convex cone and fuzzy resolution. For some related works associated with variational inequalities, discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings, linear programming with fuzzy coefficients in constraints, maximum feasibility problems for continuous linear inequalities with applications to fuzzy linear programming, and bilevel decision with generalized semi-infinite optimization for fuzzy mappings as lower level problems, we refer the reader to [15,16,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]. In [35], the authors discussed the existence of a solution to the fuzzy variational-like inequality and some extended properties of the fuzzy variational-like inequality were found. Recently, Tang, Zhao, Wan and He [36] introduced two perturbation methods for a fuzzy mapping: in one method it is perturbed by a nonlinear mapping, and in the other it is perturbed by a vector. Under suitable coercivity conditions, they proved that the perturbed variational inequality with a fuzzy mapping has a solution. In this paper, we consider the variational inequality problem by using the expected value to solve fuzzy variables under possibility spaces. To begin with, let us recall that the fuzzy variational inequality problem is to find $\overline{x}\in S\subset R^n$ satisfying

$${(x-\overline{x})}^{T}F(\overline{x},\xi )\ge 0\forall x\in S,a.s.$$

(1)

Here, $F:R\to R^n$ is a mapping, $\xi$ is fuzzy variable defined on possibility space, and a.s. is “almost surely” in possibility measure.

The fuzzy variational inequality problem is as follows:

$$\mathcal{P}\{\gamma \in \mathrm{\Gamma }|{(x-\overline{x})}^{T}F(\overline{x},\xi \left(\gamma \right))\ge 0,\forall x\in S\}=1,\overline{x}\in S,$$

(2)

where $\xi \left(\gamma \right)$ is the fuzzy variable, $\mathrm{\Gamma }$ is a nonempty set and $F:R^n\times \mathrm{\Gamma }\to R^n$ is a mapping. The FERM method of the fuzzy variational inequality problem is to minimize an expected residual given by the regularized gap function

$$g(x,\xi ):=\underset{y\in S}{max}{\{(x-y)}^{T}F(x,\xi )-\frac{\alpha }{2}{\parallel x-y\parallel }_G^2\},$$

where $G_{n\times n}$ is symmetric and positive-definite, $\alpha$ is a positive, and ${\parallel .\parallel }_G$ is ${\parallel x\parallel }_G=\sqrt{x^{T}Gx}$ for $x\in R^n$. So the FERM model is

$$\begin{array}{c}min\theta \left(x\right),\hfill \\ \theta \left(x\right):=E\left[g(x,\xi )\right]={\int }_{T}g(x,t)d\mathcal{P}\left(t\right),\hfill \\ s.t.x\in S.\hfill \end{array}$$

(3)

Here, E is the expectation of the fuzzy variable $\xi$ and $\mathcal{P}\left(t\right)$ stands for the fuzzy distribution function with respect to the fuzzy variable $\xi$.

The remainder of this article is arranged as follows. Basic information about fuzzy set theory will be reviewed in Section 2. In Section 3, we introduce the $S{C}^1$ property of function $\theta \left(x\right)$. In Section 4, we discuss the FERM model based on uncertainty theory in the case that the fuzzy event space is compact. Finally, we give conclusions in Section 5.

2. Preliminaries

2.1. Fuzzy Set Theory

In this section, several basic results about fuzzy set theory are presented.

Definition 1.

let Θ be a nonempty set and $\mathcal{F}$ the power set of Θ. If the set function Pos: $\mathcal{F}\to [0,1]$ satisfies

(1)

normality, $\mathcal{P}os\left(\mathsf{\Theta }\right)=1$,

(2)

monotonicity, $\mathcal{P}os\left(A\right)\le \mathcal{P}os\left(B\right),A\subset B,a,b\in \mathcal{F}$,

(3)

for $i=1,2\dots ,A_i\in \mathcal{F}$, we have $\mathcal{P}os\left({\cup }_i{A}_i\right)=su{p}_i\mathcal{P}os\left(A_i\right)$.

The element in $\mathcal{F}$ is called an event; the mapping is called a possibility measure; and $(\mathsf{\Theta },\mathcal{F}(\mathsf{\Theta }),\mathcal{P}os)$ is called a possibility measure space.

Theorem 1.

The possibility measure is sub-additive. We have $\mathcal{P}os(A\cup B)\le \mathcal{P}os\left(A\right)+\mathcal{P}os\left(B\right)$.

Proof. 

By Definition 1, we get $\mathcal{P}os(A\cup B)=\mathcal{P}os\left(A\right)\vee \mathcal{P}os\left(B\right)\le \mathcal{P}os\left(A\right)+\mathcal{P}os\left(B\right).$ □

Definition 2.

Suppose ξ is a fuzzy variable defined in the possibility space. ξ is said to have a membership function ${\mu }_{\xi }\left(t\right)$, which is defined as follows:

$$u_{\xi }\left(t\right)=\mathcal{P}os\{\theta \in \mathsf{\Theta }|\xi \left(\theta \right)=t\},t\in \mathcal{R},$$

(denoted as $\mu \left(t\right)$ for convenience) with the premise that ambiguity does not occur.

Definition 3.

ξ is a fuzzy variable, its fuzzy distribution $\mathcal{P}\left(t\right)$ is defined by

$$\mathcal{P}\left(t\right)=\mathcal{P}os\{\xi \le t\},$$

where t is any real number.

Theorem 2.

Suppose ξ is a fuzzy variable defined in the possibility space. $\mu \left(t\right)$ is a membership function of ξ. For a real set B, we have

$$\mathcal{P}os(\xi \in B)=sup{\mu }_{\xi }\left(t\right).$$

Proof. 

From the definition of the axioms of the possibility measure, we have $\mathcal{P}os(\xi \in B)=\underset{\xi \in B}{sup}\mathcal{P}os\left(\xi \right)=\underset{\xi \in B}{sup}\mathcal{P}os(\xi =t)=sup{\mu }_{\xi }\left(t\right).$ □

Remark 1.

The fuzzy axiom system that we give and the fuzzy sets theory that Zadeh has put forward are consistent. In order to deal with fuzzy variables reasonably, we propose a fuzzy distribution function, which is consistent with membership function that Zadeh has given.

Let $\xi$ be a fuzzy variable defined in the possibility space and $\mu \left(t\right)$ be a membership function of $\xi$. The following equations are established

(1)

$\mathcal{P}os\{\xi \le t\}=\underset{y\le t}{sup}\mu \left(y\right),t\in \mathcal{R},$

(2)

$\mathcal{P}os\{\xi \ge t\}=\underset{y\ge t}{sup}\mu \left(y\right),t\in \mathcal{R}.$

Next, we propose the concept of expected value.

Definition 4.

ξ is a fuzzy variable defined in possibility space. If one of the following two integrals is finite, the expectation of ξ is given by

$$E\left[\xi \right]={\int }_0^{+\infty }\{1-\mathcal{P}os\{\xi \le t\}\}dt-{\int }_{-\infty }^0\mathcal{P}os\{\xi \le t\}dt.$$

Theorem 3.

Let ξ be a fuzzy variable with possibility distribution in $\mathcal{P}$. If its expectation exists, then

$$E\left[\xi \right]={\int }_{-\infty }^{+\infty }td\mathcal{P}\left(t\right),$$

where $\mathcal{P}\left(t\right)$ means $\mathcal{P}os\{\xi \le t\}$.

Proof. 

In fact, we can obtain

$$\begin{array}{cc}\hfill E\left[\xi \right]& ={\int }_0^{+\infty }\{1-\mathcal{P}os\{\xi \le t\}\}dt-{\int }_{-\infty }^0\mathcal{P}os\{\xi \le t\}dt\hfill \\ & ={\int }_0^{+\infty }\{1-\mathcal{P}\left(t\right)\}dt-{\int }_{-\infty }^0\mathcal{P}\left(t\right)dt\hfill \\ & {=t(1-\mathcal{P}\left(t\right))|}_0^{+\infty }-{\int }_0^{+\infty }{td(1-\mathcal{P}\left(t\right))-t\mathcal{P}\left(t\right)|}_{-\infty }^0+{\int }_{-\infty }^{0}td\mathcal{P}\left(t\right)\hfill \\ & ={\int }_0^{+\infty }td\mathcal{P}\left(t\right)+{\int }_{-\infty }^{0}td\mathcal{P}\left(t\right)\hfill \\ & ={\int }_{-\infty }^{+\infty }td\mathcal{P}\left(t\right).\hfill \end{array}$$

Therefore the conclusion is reached. □

Corollary 1.

Let ξ be an fuzzy variable and $\mathcal{P}\left(t\right)$ is its possibility distribution. If $f\left(t\right)$ is strictly monotone, then we have

$$E\left[f\left(\xi \right)\right]={\int }_{-\infty }^{+\infty }f\left(t\right)d\mathcal{P}\left(t\right).$$

Proof. 

Substituting $\mathcal{P}\left(t\right)$ with $\alpha$, by Theorem 3 we have

$$E\left[\xi \right]={\int }_{-\infty }^{+\infty }td\mathcal{P}\left(t\right)={\int }_0^1{\mathcal{P}}^{-1}\left(\alpha \right)d\alpha .$$

Let $\omega =f\left(\xi \right)$ with possibility distribution $P\left(t\right)$. Then we have

$$E\left[\omega \right]={\int }_{-\infty }^{+\infty }td\mathcal{P}\left(t\right)={\int }_0^1{\mathcal{P}}^{-1}\left(\alpha \right)d\alpha .$$

Thus,

$$E\left[f\left(\xi \right)\right]={\int }_0^1{\mathcal{P}}^{-1}\left(\alpha \right)d\alpha ={\int }_0^{1}f\left({\mathcal{P}}^{-1}\left(\alpha \right)\right)d\alpha ={\int }_{-\infty }^{+\infty }f\left(t\right)d\mathcal{P}\left(t\right).$$

The corollary is proved. □

2.2. Approximation Method

Let a fuzzy event space T be a nonempty compact set. From the continuity of $(F,{\nabla }_{x}F)$ and Theorem 4 of [13], we easily see that the function $\theta$ is continuously differentiable in S, and its derivative is as follows:

$$\nabla \theta \left(x\right)=E\left[{\nabla }_{x}g(x,\xi )\right],\forall x\in S.$$

(4)

Since a fuzzy variable usually has no density function, $\mathcal{P}\left(x\right)$ is not usually differentiable in fuzzy theory. However, we can use the differentiable properties of $g(x,t)$. From [37], $g(·,t)$ is continuously differentiable over S, satisfying

$${\nabla }_{x}g(x,t)=F(x,t)-({\nabla }_{x}F(x,t)-\alpha G)(H(x,t)-x),\forall t\in T$$

(5)

Definition 5

([38]). Let $g\left(t\right)$ and $\mathcal{P}$ be real-valued bounded functions on a closed interval $[a,b]$. Taking a partition of the interval $a=x_0<x_1<x_2,\dots <x_{n-1}<x_n=b,$ and setting $\delta =\underset{1\le i\le n}{max}(t_i-t_{i-1})$, we consider the Riemann sum $\underset{\begin{array}{c}\delta \to 0\\ n\to \infty \end{array}}{lim}{\displaystyle \sum _{i=1}^n}g\left(z_i\right)[\mathcal{P}\left(t_i\right)-\mathcal{P}\left(t_{i-1}\right)]$, denoted by $\underset{\begin{array}{c}\delta \to 0\\ n\to \infty \end{array}}{lim}{\displaystyle \sum _{i=1}^n}g\left(z_i\right)\Delta \mathcal{P}\left(t_i\right)$ (where $t_{i-1}\le z_i\le t_i,i=1,2,\cdots ,n)$. If the sum tends to a fixed number I as $\delta \to 0$, then I is called the Stieltjes integral, or sometimes the Riemann–Stieltjes integral. The Stieltjes integral of g with respect to t is denoted ${\int }_a^{b}g\left(t\right)d\mathcal{P}\left(t\right)$ or sometimes simply by $\int gd\mathcal{P}$.

Lemma 1

([38]). If the function $g\left(t\right)$ is continuous on $[a,b]$ and $\mathcal{P}\left(t\right)$ is monotonic nonincreasing or nondecreasing function, then ${\int }_0^{1}g\left(t\right)d\mathcal{P}\left(t\right)$ exists.

Let ${\lambda }_{max}$ and ${\lambda }_{min}$ represent the largest and smallest eigenvalues of the positive definite matrix G, respectively. For any $x\in R^n$, we have

$$\begin{array}{c}\hfill \sqrt{{\lambda }_{min}}\parallel x\parallel \le {\parallel x\parallel }_G\le \sqrt{{\lambda }_{max}}\parallel x\parallel ,\end{array}$$

(6)

Let $\parallel A\parallel$ denote the spectral norm and ${\parallel A\parallel }_{\mathcal{F}}$ denote the Frobenius matrix norm of matrix A. We know that

$$\parallel A\parallel \le {\parallel A\parallel }_{\mathcal{F}},$$

(7)

and

$${\parallel A\parallel }_{\mathcal{F}}\le \sum _{j=1}^n\parallel A_j\parallel ,$$

(8)

where $A_j$ is the jth column vector of A.

For convenience, we give the relationship between $\parallel A\parallel$ and ${\parallel A\parallel }_{\mathcal{F}}$ and the relationship between $\parallel x\parallel$ and ${\parallel x\parallel }_G$ in advance. These definitions and properties will be used in the later theorems.

3. The ${\mathit{SC}}^{\mathbf{1}}$ Property of $\mathit{\theta }\left(\mathit{x}\right)$

Definition 6

([39]). A function $g:R^n\to R$ is called an $S{C}^1$ function if its gradient is semi-smooth and g is continuously differentiable.

Lemma 2

([39]). Let $x_0\in S$. Suppose that

(1)

there exists an integrable function $\kappa :R\to R_{+}$ and a neighborhood $N\left(x_0\right)$ of $x_0$ satisfying

$$|g(x_0,t)-g(x,t)|\le k\left(t\right)\parallel x_0-x\parallel ,\forall x\in N\left(x_0\right),a.e.t\in T;$$

(2)

$g(x,t)$ is semi-smooth at $x_0$ for $a.e.t\in T$;

(3)

there exist an integrable function $\zeta :R\to R_{+}$ and a neighborhood $N\left(0\right)$ of $0\in R^n$ satisfying

$$\frac{\parallel g^{\prime }(x_0+h,t)-g^{\prime }(x_0,t)\parallel }{\parallel h\parallel }\le \zeta \left(t\right),\forall h\in N\left(0\right),a.e.t\in T.$$

Then $\theta \left(x\right)$ is semi-smooth at $x_0$.

Theorem 4.

Let $x_0\in S$, $N\left(x_0\right)$ be a neighborhood of $x_0$. Suppose that

(1)

there exist an integrable function $\kappa :R\to R_{+}$, such that ${\int }_T\kappa \left(t\right)d\mathcal{P}\left(t\right)<+\infty$, satisfying

$$|g(x_1,t)-g(x_2,t)|\le \kappa \left(t\right)\parallel x_1-x_2\parallel ,\forall x_1,x_2\in N\left(x_0\right),a.e.t\in T.$$

(2)

For $a.e.t\in T$, $g(x,t)$ is $S{C}^1$ function on $N\left(x_0\right)$. There is nonnegative function such that ${\int }_{T}L\left(t\right)d\mathcal{P}\left(t\right)<+\infty$, satisfying $\nabla g(x,t)$ is Lipschitz continuous,

$$|\nabla g(x_1,t)-\nabla g(x_2,t)|\le L\left(t\right)\parallel x_1-x_2\parallel .$$

(3)

There exists an integrable function $\zeta :R\to R_{+}$ and $N\left(0\right)$ of $0\in R^n$ satisfying

$$\frac{\parallel s^{\prime }(x_0+h,t)-s^{\prime }(x_0,t)\parallel }{\parallel h\parallel }\le \zeta \left(t\right),\forall h\in N\left(0\right),a.e.t\in T,$$

where $s(x,t)=\nabla g(x,t)$, and ∇ represents gradient.

Then $\theta \left(x\right)$ is $S{C}^1$ function on $N\left(x_0\right)$.

Proof. 

By condition (2), the gradient of $g(·,t)$ is semi-smooth, and $g(·,t)$ is continuous and differentiable. It is easy to see that $s(x,t)$ is directional differentiable, Lipschitz continuous, and

$$s^{\prime }(x_0+h,t)-s^{\prime }(x_0,t)=o{(\parallel h\parallel )}^{\prime }.$$

Next, we show $\theta \left(x\right)$ is $S{C}^1$ function. First, we show $\theta \left(x\right)$ is differentiable at $x_0$. By definition of $\theta \left(x\right)$ and condition (1), we obtain

$$\begin{array}{cc}\hfill |\theta \left(x_1\right)-\theta \left(x_2\right)|& =|{\int }_{T}g(x_1,t)d\mathcal{P}\left(t\right)-{\int }_{T}g(x_2,t)d\mathcal{P}\left(t\right)|\hfill \\ & \le {\int }_T|g(x_1,t)-g(x_2,t)|d\mathcal{P}\left(t\right)\hfill \\ & \le \parallel x_1-x_2\parallel {\int }_T\kappa \left(t\right)d\mathcal{P}\left(t\right),\hfill \end{array}$$

for $\forall x_1,x_2\in N\left(x_0\right)$ and $t\in T$. So $\theta \left(x\right)$ is differentiable at $x_0$. By condition (1), for $h\in R^n$

$$l^{-1}|g(x_0+lh,t)-g(x_0,t)|\le \kappa \left(t\right)\parallel h\parallel .$$

Note that

$$\begin{array}{cc}\hfill \underset{l\to 0}{lim}\frac{\theta (x_0+lh,t)-\theta (x_0,t)}{l}& =\underset{l\to 0}{lim}\frac{{\int }_{T}g(x_0+lh,t)d\mathcal{P}\left(t\right)-{\int }_{T}g(x_0,t)d\mathcal{P}\left(t\right)}{l}\hfill \\ & ={\int }_T\underset{l\to 0}{lim}\frac{g(x_0+lh,t)-g(x_0,t)}{l}d\mathcal{P}\left(t\right),\hfill \end{array}$$

so $\nabla \theta \left(x\right)$ is directional differentiable, denoted

$${\theta }^{\prime }(x_0;h)={\int }_T{g}^{\prime }(x_0;h,t)d\mathcal{P}\left(t\right).$$

Since $g^{\prime }(x_0;h,t)=\nabla g^{\prime }{(x_0,t)}^{H}h$, ${\theta }^{\prime }(x_0;h)$ is linear with respect to h, from Lipschitz continuity and Gateaux differentiablity of $\theta \left(x\right)$, $\theta \left(x\right)$ is Frechet differentiable.

Note that $\theta \left(x\right)$ is differentiable at $x_0$, $\nabla \theta \left(x_0\right)={\int }_T\nabla g(x_0,t)d\mathcal{P}\left(t\right)$, and $\nabla g(x,t)$ is semi-smooth at $x_0$ and is Lipschitz continuous. We get

$$\begin{array}{cc}\hfill |\nabla \theta \left(x_1\right)-\nabla \theta \left(x_2\right)|& =|{\int }_T\nabla g(x_1,t)d\mathcal{P}\left(t\right)-{\int }_T\nabla g(x_2,t)d\mathcal{P}\left(t\right)|\hfill \\ & \le {\int }_T|\nabla g(x_1,t)-\nabla g(x_2,t)|d\mathcal{P}\left(t\right)\hfill \\ & \le \parallel x_1-x_2\parallel {\int }_{T}L\left(t\right)d\mathcal{P}\left(t\right),\hfill \end{array}$$

where ${\int }_{T}L\left(t\right)d\mathcal{P}\left(t\right)<+\infty$, so $\forall x_1,x_2\in N\left(x_0\right),a.e.t\in T$, $\nabla \theta \left(x\right)$ is Lipschitz continuous.

Finally, we will show $\nabla \theta \left(x\right)$ is semi-smooth. Let $\nu \left(x\right):=\nabla \theta \left(x\right)={\int }_T\nabla g\left(t\right)d\mathcal{P}\left(t\right),$ we need to show $\nabla \theta \left(x\right)$ is directional differentiable and ${\nu }^{\prime }(x+h)-{\nu }^{\prime }\left(x\right)=o(\parallel h\parallel )$.

By condition (3), we have

$$\begin{array}{cc}\hfill \frac{\parallel {\nu }^{\prime }(x+h)-{\nu }^{\prime }\left(x\right)\parallel }{\parallel h\parallel }& ={\int }_T\frac{\parallel \nabla g^{\prime }(x+h,t)-\nabla g^{\prime }(x,t)\parallel }{\parallel h\parallel }d\mathcal{P}\left(t\right)\hfill \\ & ={\int }_T\frac{\parallel s^{\prime }(x+h,t)-s^{\prime }(x,t)\parallel }{\parallel h\parallel }d\mathcal{P}\left(t\right)\hfill \\ & \le {\int }_T\zeta \left(t\right)d\mathcal{P}\left(t\right)\hfill \\ & <+\infty .\hfill \end{array}$$

Since $s^{\prime }(x,t)$ is semi-smooth, $\underset{h\to 0}{lim}\frac{\parallel {\nu }^{\prime }(x+h)-{\nu }^{\prime }\left(x\right)\parallel }{\parallel h\parallel }={\int }_T\underset{h\to 0}{lim}\frac{\parallel {\nu }^{\prime }(x+h)-{\nu }^{\prime }\left(x\right)\parallel }{\parallel h\parallel }d\mathcal{P}\left(t\right)=0.$ By Lemma 2, $\nabla \theta \left(x\right)$ is semi-smooth. So $\theta \left(x\right)$ is $S{C}^1$ on $N\left(x_0\right)$. □

4. Convergence of the FERM Model

Definition 7.

Define the minimum of ${\theta }^n\left(x\right)$ as follows:

$$min{\theta }^n\left(x\right)=\sum _{t^i\in T_{\delta }}g(x,z_i)\Delta \mathcal{P}\left(t_i\right)$$

(9)

where $T_{\delta }=\{t_i|i=1,2,\cdots ,n_{\delta }\}$ is a set satisfying $n_{\delta }\to \infty$ as $\delta \to 0.$

We study the behavior of the approximations to the FERM problem (3) as follow

$$\begin{array}{c}min{\theta }^n\left(x\right)=\sum _{t_i\in T_{\delta }}g(x,z_i)\Delta \mathcal{P}\left(t_i\right),\hfill \\ s.t.x\in S.\hfill \end{array}$$

(10)

We consider the limiting behavior of problems (10) below.

4.1. Convergence of Global Optimal Solutions

Let $S^{*}$ and $S_n^{*}$ be the sets of optimal solutions of problems (3) and (10).

Lemma 3.

$\theta \left(x\right)=\underset{\begin{array}{c}\delta \to 0\\ n\to \infty \end{array}}{lim}{\theta }^n\left(x\right)$ holds for any given $x\in S$.

Proof. 

Since

$$\begin{array}{c}\hfill \theta \left(x\right)=E\left[g(x,\xi )\right]={\int }_T(x,t)d\mathcal{P}\left(t\right),\end{array}$$

while

$$\begin{array}{c}\hfill {\theta }^n\left(x\right)=\sum _{t^i\in T_{\delta }}g(x,z_i)\Delta \mathcal{P}\left(t_i\right).\end{array}$$

therefore $\underset{\begin{array}{c}\delta \to 0\\ n\to \infty \end{array}}{lim}{\displaystyle \sum _{t_i\in T_{\delta }}}g(x,z_i)\Delta \mathcal{P}\left(t_i\right)={\int }_{T}g(x,t)d\mathcal{P}\left(t\right)$, so $\theta \left(x\right)=\underset{\begin{array}{c}\delta \to 0\\ n\to \infty \end{array}}{lim}{\theta }^n\left(x\right).$ □

Theorem 5.

Let $\overline{x}$ be an accumulation point of $\left\{x^n\right\}$, $x^n\in S_n^{*}$, then $\overline{x}\in S^{*}$.

Proof. 

Let $\underset{n\to \infty }{lim}{x}_n=\overline{x}$. It holds that $\overline{x}\in S$. Let $B\subseteq S$ be a compact convex set, and $x^n\in B$. By the continuity of ${\nabla }_{x}g$ on the compact set $B\times T$, $\exists C\in R^{+}$ and we have

$$\parallel {\nabla }_{x}g(x,z)\parallel \le C,\forall (x,z)\in B\times T.$$

(11)

For each $x^n$ and each $z_i$, the next equation holds from the mean-value theorem:

$$g(x^n,z_i)-g(\overline{x},z_i)=g_x^{\prime }{(y^{ni},z_i)}^T(x^n-\overline{x}),$$

where $y^{ni}={\alpha }_{ni}{x}^n+(1-{\alpha }_{ni})\overline{x}\in B$ with ${\alpha }_{ni}\in [0,1]$. It then follows from Lemma 3 that

$$\begin{array}{cc}\hfill |{\theta }^n\left(x^n\right)-{\theta }^n\left(\overline{x}\right)|& =|\sum _{t^i\in T_{\delta }}(g(x^n,z_i)-g(\overline{x},z_i))\Delta \mathcal{P}\left(t_i\right)|\hfill \\ & \le \sum _{t_i\in T_{\delta }}|(g(x^n,z_i)-g(\overline{x},z_i))|\Delta \mathcal{P}\left(t_i\right)\hfill \\ & =\sum _{t_i\in T_{\delta }}|{\nabla }_{x}g{(y^{ni},z_i)}^T(x^n-\overline{x})|\Delta \mathcal{P}\left(t_i\right)\hfill \\ & \le \sum _{t_i\in T_{\delta }}\parallel {\nabla }_{x}g(y^{ni},z_i)\parallel ·\parallel (x^n-\overline{x})\parallel \Delta \mathcal{P}\left(t_i\right)\hfill \\ & \le C\parallel x^n-\overline{x}\parallel \sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\hfill \\ & \stackrel{n\to \infty }{\to }0,\hfill \end{array}$$

noting that

$$|{\theta }^n\left(x^n\right)-\theta \left(\overline{x}\right)|\le |{\theta }^n\left(x^n\right)-{\theta }^n\left(\overline{x}\right)|+|{\theta }^n\left(\overline{x}\right)-\theta \left(\overline{x}\right)|.$$

(12)

From Lemma 3 and (12), it holds that

$$\begin{array}{c}\hfill \underset{\begin{array}{c}\delta \to 0\\ n\to \infty \end{array}}{lim}{\theta }^n\left(x^n\right)=\theta \left(\overline{x}\right).\end{array}$$

(13)

Since, for each sufficiently large n, $x^n\in S_n^{*}$, which means that there exists $ϵ>0$ such that

$$\begin{array}{c}\hfill {\theta }^n\left(x^n\right)\le {\theta }^n\left(x\right)+ϵ.\end{array}$$

(14)

This holds for any $x\in S$. Letting $n\to \infty$ in (14) and taking into account (13) and Lemma 2, $\theta \left(\overline{x}\right)\le \theta \left(x\right)+ϵ$ holds implies $\overline{x}\in S^{*}$. □

4.2. Convergence of Stationary Points

Theorem 6.

Let $\underset{n\to \infty }{lim}{x}^n=\overline{x}$. Then,

$$\underset{\begin{array}{c}\delta \to 0\\ n\to \infty \end{array}}{lim}\nabla {\theta }^n\left(x^n\right)=\nabla \theta \left(\overline{x}\right).$$

Proof. 

Let $B\subseteq S$ be a compact convex set and $x^n\in B$. Because $F_x^{\prime },g_x^{\prime },H$, and ${\left(F_j\right)}_x^{″}$ are continuous on the compact set $B\times T$, for any $(x,z)\in B\times T$, $\exists C\ge sup\{\parallel x^n\parallel ,n=1,2,\dots \}$ satisfying

$$\parallel F_x^{\prime }(x,z)\parallel \le C,$$

(15)

$$\parallel H(x,z)\parallel \le C,$$

(16)

$$\parallel {\left(F_j\right)}_x^{″}(x,z)\parallel \le C,j=1,\cdots ,n,$$

(17)

where ${\left(F_j\right)}_x^{″}(x,z)$ denotes the Hessian matrix of $F_j(x,z)$ with respect to x.

We first show that

$$\underset{\begin{array}{c}\delta \to 0\\ n\to \infty \end{array}}{lim}\sum _{t^i\in T_n}\Delta \mathcal{P}\left(t^i\right)\parallel {\nabla }_{x}F(x^n,z_i)-{\nabla }_{x}F(\overline{x},z_i)\parallel =0.$$

(18)

In fact, from (7) and (8), we have

$$\begin{array}{cc}\hfill \sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\parallel F_x^{\prime }(x^n,z_i)-F_x^{\prime }(\overline{x},z_i)\parallel & \le \sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right){\parallel F_x^{\prime }(x^n,z_i)-F_x^{\prime }(\overline{x},z_i)\parallel }_F\hfill \\ & \le \sum _{j=1}^n\sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\parallel {\left(F_j\right)}_x^{\prime }(x^n,z_i)-{\left(F_j\right)}_x^{\prime }(\overline{x},z_i)\parallel ,\hfill \end{array}$$

(19)

From the mean-value theorem, for any fixed j and for each $x^n$ and $z^i$, there exists $y^{nij}={\alpha }_{nij}{x}^n+(1-{\alpha }_{nij})\overline{x}\in B$ with ${\alpha }_{nij}\in [0,1]$ such that

$$\begin{array}{cc}\hfill \sum _{j=1}^n\sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\parallel {\left(F_j\right)}_x^{\prime }(x^n,z_i)-{\left(F_j\right)}_x^{\prime }(\overline{x},z_i)\parallel & \le \sum _{j=1}^n\sum _{t^i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\parallel {\left(F_j\right)}_x^{″}(y^{kij},z_i)\parallel \parallel x^n-\overline{x}\parallel \hfill \\ & \le C\parallel x^n-\overline{x}\parallel \sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\hfill \\ & \stackrel{n\to \infty }{\to }0,\hfill \end{array}$$

(20)

where the second inequality follows from (1). We have (18) immediately from (19) and (20). In a similar way, we can show that

$$\underset{\begin{array}{c}\delta \to 0\\ n\to \infty \end{array}}{lim}\sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\parallel F(x^n,z_i)-F(\overline{x},z_i)\parallel =0.$$

(21)

From the nonexpansive property of $Pro{j}_{S,G}$, (6) and (21), it holds that

$$\begin{array}{c}\sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\parallel H(\overline{x},z_i)-H(x^n,z_i)\parallel \hfill \\ \le {\lambda }_{min}^{-\frac{1}{2}}\sum _{t^i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right){\parallel H(\overline{x},z_i)-H(x^n,z_i)\parallel }_G\hfill \\ \le {\lambda }_{min}^{-\frac{1}{2}}\sum _{t^i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\parallel (\overline{x}-{\alpha }^{-1}{G}^{-1}F(\overline{x},z_i)-{(x^n-{\alpha }^{-1}{G}^{-1}F(x^n,z_i)\parallel }_G\hfill \\ \le {\lambda }_{max}^{\frac{1}{2}}{\lambda }_{min}^{-\frac{1}{2}}\sum _{t^i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)(\parallel x^k-\overline{x}\parallel +{\alpha }^{-1}\parallel G^{-1}\parallel \parallel F(x^n,z_i)-F(\overline{x},z_i)\parallel )\hfill \\ \stackrel{n\to \infty }{\to }0.\hfill \end{array}$$

(22)

On the other hand, by (15), (16), (18) and (22), we have

$$\begin{array}{c}\sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\parallel F_x^{\prime }(x^n,z_i)H(x^n,z_i)-F_x^{\prime }(\overline{x},z_i)H(\overline{x},z_i)\parallel \hfill \\ =\sum _{t^i\in T_{\delta }}\Delta \mathcal{P}\left(t^i\right)\parallel F_x^{\prime }(x^n,z_i)(H(x^n,z_i)-H(\overline{x},z_i))\hfill \\ +(F_x^{\prime }(x^n,z_i)-F_x^{\prime }(\overline{x},z_i))\parallel H(\overline{x},z_i)\parallel \hfill \\ \le \sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\parallel F_x^{\prime }(x^n,z_i)\parallel H(x^n,z_i)-H(\overline{x},z_i)\parallel \hfill \\ +\parallel F_x^{\prime }(x^n,z_i)-F_x^{\prime }(\overline{x},z_i)\parallel \parallel H(\overline{x},z_i)\parallel \hfill \\ \le C·\sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)(\parallel (H(x^n,z_i)-H(\overline{x},z_i)\parallel +\parallel F_x^{\prime }(x^n,z_i)-F_x^{\prime }(\overline{x},z_i)\parallel )\hfill \\ \stackrel{n\to \infty }{\to }0.\hfill \end{array}$$

(23)

Noting that $C\ge sup\{\parallel x^n\parallel ,k=1,2,\dots \}$, we have from (15) and (18) that

$$\begin{array}{c}\sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)\parallel F_x^{\prime }(x^n,z_i)x^n-F_x^{\prime }(\overline{x},z_i)\overline{x}\parallel \hfill \\ \le \sum _{t^i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)(\parallel F_x^{\prime }(x^n,z_i)x^n-F_x^{\prime }(\overline{x},z_i)\overline{x}\parallel \parallel x^n\parallel )+\parallel F_x^{\prime }(\overline{x},z_i)\parallel \parallel x^n-\overline{x}\parallel \hfill \\ \le C·\sum _{t^i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)(\parallel F_x^{\prime }(x^n,z_i)-F_x^{\prime }(\overline{x},z_i)\parallel +\parallel x^n-\overline{x}\parallel )\hfill \\ \stackrel{n\to \infty }{\to }0.\hfill \end{array}$$

(24)

Thus, it follows from (5), (21)–(24) that

$$\begin{array}{cc}\hfill \parallel {\left({\theta }_x^{\prime }\right)}^n\left(x^n\right)-{\left({\theta }_x^{\prime }\right)}^n\left(\overline{x}\right)\parallel & \le \sum _{t_i\in T_{\delta }}\Delta \mathcal{P}\left(t_i\right)(\parallel F(x^n,z_i)-F(\overline{x},z_i)\parallel \hfill \\ & +\parallel F_x^{\prime }(x^n,z_i)H(x^n,z_i)-F_x^{\prime }(\overline{x},z_i)H(\overline{x},z_i)\parallel \hfill \\ & +\parallel F_x^{\prime }(x^n,z_i)x^n-F_x^{\prime }(\overline{x},z_i)\overline{x}\parallel \hfill \\ & +\alpha \parallel G\parallel \parallel H(\overline{x},z_i)-H(x^n,z_i)\parallel +\alpha \parallel G\parallel \parallel x^n-\overline{x}\parallel )\hfill \\ & \stackrel{n\to \infty }{\to }0.\hfill \end{array}$$

(25)

In a similar way with Lemma 3, we have $\underset{\begin{array}{c}\delta \to 0\\ n\to \infty \end{array}}{lim}{\left({\theta }_x^{\prime }\right)}^n\left(\overline{x}\right)={\theta }_x^{\prime }\left(\overline{x}\right)$, so we get the conclusion of this lemma immediately from (25). □

Now, we consider the limiting behavior of the stationary points of problems (10).

Definition 8.

Assume that $S=\{x\in R^n|{\tau }_i\left(x\right)\le 0,{\tau }_i:R^n\to R,i=1,2,\cdots ,m\}$, where, ${\tau }_i\left(x\right)$ are all continuous and differentiable convex functions. If a Lagrange multiplier vector ${\mu }^n\in R^m$ satisfies

$${\left({\theta }_x^{\prime }\right)}^n\left(x^n\right)+\sum _{i=1}^m{\mu }_i^n\left({\left({\tau }_i\right)}_x^{\prime }\right)\left(x^n\right)=0,$$

(26)

$$0\le {\mu }^n,\tau \left(x^n\right)\le 0and{\left({\mu }^n\right)}^T\tau \left(x^n\right)=0,$$

(27)

then the point $x^n$ is a stationary point of (10).

If there is a Lagrange multiplier vector $\overline{\mu }\in R^m$ satisfing

$${\theta }_x^{\prime }\left(\overline{x}\right)+\sum _{i=1}^m\overline{{\mu }_i}\left({\left({\tau }_i\right)}_x^{\prime }\right)\left(\overline{x}\right)=0,$$

(28)

$$0\le \overline{\mu },\tau \left(\overline{x}\right)\le 0and{\left(\overline{\mu }\right)}^T\tau \left(\overline{x}\right)=0,$$

(29)

then $\overline{x}$ is a stationary point of problem (3).

Definition 9.

If $\exists y\in R^n$ such that ${\tau }_i\left(y\right)<0,i=1,2,\cdots ,m$, then the Slater’s constraint qualification holds.

Theorem 7.

Let $x^n$ be stationary point of (10) and $\overline{x}$ be an accumulation point of $\left\{x^n\right\}$, then $\overline{x}$ is stationary point of (3) under the Slater’s constraint qualification condition.

Proof. 

Let $\underset{n\to \infty }{lim}{x}^n=\overline{x}$ and ${\mu }^n$ be corresponding multiplier vectors of (26) and (27).

(i) Firstly, we demonstrate that $\left\{{\mu }^n\right\}$ is bounded. For convenience, denote

$${\upsilon }_n:=\sum _{i=1}^m{\mu }_i^n.$$

(30)

If $\left\{{\mu }^n\right\}$ is unbounded, then $\underset{k\to \infty }{lim}{\upsilon }_n=+\infty$. Further, taking a subsequence of ${\mu }^n$ such that $\overline{{\mu }_i}:=\underset{n\to \infty }{lim}\frac{{\mu }_i^n}{{\upsilon }_n}(i=1,2,\cdots ,m)$ exist. For every $i\notin \mathsf{{\rm Y}}\left(\overline{x}\right):=\{i|{\tau }_i\left(\overline{x}\right)=0,1\le i\le m\}$, we have ${\tau }_i\left(\overline{x}\right)\le 0$ by (27), while ${\left(\overline{{\mu }_i}\right)}^T{\tau }_i\left(\overline{x}\right)=0$, so it is easy to obtain that ${\mu }_i^{*}=0$. We then have from (30) that

$$\sum _{i\in \mathsf{{\rm Y}}\left(\overline{x}\right)}\overline{{\mu }_i}=\sum _{i=1}^m\overline{{\mu }_i}=1.$$

(31)

Note that $\left({\left({\tau }_i\right)}_x^{\prime }\right)$ is continuous for each i and, by Theorem 5, $\left\{{\left({\theta }_x^{\prime }\right)}^n\left(x^n\right)\right\}$ is convergent. Then, dividing (26) by ${\upsilon }_n$ and taking a limit, because of $\underset{n\to \infty }{lim}{\upsilon }_n=+\infty$, so $\underset{n\to \infty }{lim}\frac{{\left({\theta }_x^{\prime }\right)}^n\left(x^n\right)}{{\upsilon }_n}\to 0$ and we obtain

$$\sum _{i\in \mathsf{{\rm Y}}\left(\overline{x}\right)}\overline{{\mu }_i}\left({\left({\tau }_i\right)}_x^{\prime }\right)\left(\overline{x}\right)=\sum _{i=1}^m\overline{{\mu }_i}\left({\left({\tau }_i\right)}_x^{\prime }\right)\left(\overline{x}\right)=0.$$

(32)

From the Slater’s constraint qualification, $\exists y\in R^n$ satisfy ${\tau }_i\left(y\right)<0$. Moreover, ${\tau }_i,i=1,2,\cdots ,m$ is convex, It holds that

$$\begin{array}{c}\hfill {(y-\overline{x})}^T\left({\left({\tau }_i\right)}_x^{\prime }\right)\left(\overline{x}\right)\le {\tau }_i\left(y\right)-{\tau }_i\left(\overline{x}\right)={\tau }_i\left(y\right)<0,\forall i\in \mathsf{{\rm Y}}\left(\overline{x}\right).\end{array}$$

(33)

Since $\overline{{\mu }_i}\ge 0$ for each i by (27), following from (32) we get $\overline{{\mu }_i}\left({\left({\tau }_i\right)}_x^{\prime }\right)\left(\overline{x}\right)=0$, from (33) we get $\left({\left({\tau }_i\right)}_x^{\prime }\right)\left(\overline{x}\right)\ne 0$, so we naturally get that $\overline{{\mu }_i}=0$, $i\in \mathsf{{\rm Y}}\left(\overline{x}\right)$. This contradicts (31), so $\left\{{\mu }^n\right\}$ is bounded.

(ii) By (i), in ${\mu }^n$ exists a subsequence, we still denote it by ${\mu }^n$ such that $\overline{\mu }:=\underset{n\to \infty }{lim}{\mu }^n$ because ${\tau }_i$ and $\left({\left({\tau }_i\right)}_x^{\prime }\right)$ are all continuous for each i. From Theorem 5, we have

$$\underset{n\to \infty }{lim}{\left({\theta }_x^{\prime }\right)}^n\left(\overline{x}\right)={\theta }_x^{\prime }\left(\overline{x}\right).$$

Then (28) and (29) hold by taking a limit on both sides of (26) and (27). Thus, $\overline{x}$ is a stationary point of problem (3). □

5. Conclusions

We investigated the nonlinear variational inequality with fuzzy variable. The fuzzy nonlinear variational inequality model was established. $S{C}^1$ property of the FERM model The convergence of the algorithm of the fuzzy model was presented, based on the fuzzy event space being compact. The convergence of global optimal solutions and the convergence of stationary points were analysed. In the future, we will study the D gap function and its applications, fuzzy quasi-variational inequalities, and fuzzy weighted variational inequalities.