Abstract

In this paper, a Stieltjes integral approximation method for uncertain variational inequality problem (UVIP) is studied. Firstly, uncertain variables are introduced on the basis of variational inequality. Since the uncertain variables are based on nonadditive measures, there is usually no density function. Secondly, the expected value model of UVIP is established after the expected value is discretized by the Stieltjes integral. Furthermore, a gap function is constructed to transform UVIP into an uncertain constraint optimization problem, and the optimal value of the constraint problem is proved to be the solution of UVIP. Finally, the convergence of solutions of the Stieltjes integral discretization approximation problem is proved.

1. Introduction

VIP is a significant branch of inequality and a classical problem in mathematics, which has attracted many scholars. Through the unremitting efforts of many mathematicians, VIP has developed into an important subject with rich content and broad prospects in mathematical programming. These achievements involve rich mathematical theories, optimization theory, economics and engineering (see [1–7]), and so on. For the classical VIP, , there is a point such that where is closed convex and is a vector-valued function. Chen and Fukushima [8] presented the regularized gap function as follows: where matrix is symmetric and positive definite square and parameter . indicates the G-norm, which is given by , . It means , and iff is a solution of VIP . On the basis of these theories, we convert the VIP (1) into an optimization problem as follows:

Generally, the minimization problem (3) does not involve uncertainties. However, it is just an ideal situation. All of these characteristics may lead to the uncertainty. Therefore, many researchers have systematically studied variational inequalities with random variables. That is, where is a stochastic sample space and the mapping . Due to the randomness of the function , there is generally no solution to problem (4). By calculating expected value over , problem (4) is transformed into as follows:

This problem is widely used in economics, management, and operations research. It was investigated in references such as [9–11]. Based on probability theory, the SVIP in literature [8] is studied. It is well known that probability is based on repeated tests, so it must have a large number of historical sample data to estimate probability. But in most conditions, it is hard to model a probability distribution due to the nonrepeatability of events, such as unprecedented sudden natural disasters, crisis management and emergency of acute infectious diseases, and so on. Liu [12] created uncertainty theory, which is based on nonadditive measure, to deal with these uncertain phenomena.

In the past few years, uncertainty theory has become a very fruitful subject. At the same time, many successful applications have been made at home and abroad (see [12–22]). Chen and Zhu [23] introduced the uncertain variable into the VIP and established the uncertain variational inequality problem (UVIP). They constructed the expected value model to solve the UVIP as follows: where is the set of uncertain variables and the mapping .

Based on uncertainty theory, an approximation problem on UVIP is studied in this paper. It is clear that SVIP and UVIP are both natural generalizations of deterministic variational inequalities. Other contents of this paper are as follows. The second section reviews the basic concepts and properties of some uncertainty theories, including uncertain variables and uncertain expectations. In Section 3, research on the convergence of the approximation problem generated by the Stieltjes integral discrete approximation method (SDA for short) will be finished. Finally, a conclusion summarizes and prospects the future research work.

2. Preliminaries

In this section, we will give some definitions and lemmas. Firstly, we collect the concepts and properties in uncertainty space. Supposed that is a nonempty set and is a -algebra over . Then, is called a measurable space; each element in is called an event. So presents the belief degree that occurs. So is an uncertainty space, which is defined by . To deal with belief degrees rightly, Liu [12] presented three axioms as follows: (1)(2)(3), where are sequence of events

Definition 1 (see [12]). Let . If the following exists, then is the expected value of uncertain variable .

Theorem 2 (see [12]). Let and be the uncertainty distribution of . If exists, then

Theorem 3 (see [12]). Let and be the uncertainty distribution of . If exists,

3. SDA Method and Its Convergence

In this section, we will provide the convergence of SDA method and regularized gap functions on the set on the basis of uncertainty theory. It turns out that for , there is an optimal solution for problem (1). Therefore, we can find a fixed point such that where and is an uncertain space. Furthermore, we present the regularized gap function where parameter and matrix is positive definite and symmetric. Now, we can convert (10) into an optimization problem as follows:

In this section, in order to solve problem (12), we will propose a Stieltjes integral discrete approximation method (abbreviated as SDA), and the convergence of the method is studied. In most cases, there is no density function in the uncertain distribution. Then, it is difficult to calculate the uncertain expectation directly, so we use the Stieltjes integral to calculate. The distribution function is discretized before that, and we introduced the following definitions.

Definition 4 (division of interval by the Stieltjes integral [24]). Let be a bounded function on the interval and be a bounded variation function on , and make a division of interval and a group of “intermediate points,” , and make a sum:

Set . When , the sum tends to a certain finite limit; then, is said to be integrable about on the interval . This limit is recorded as .

From the division of interval by the Stieltjes integral (10), we have ; the expectation of is

According to the arbitrariness of , let and ; then,

Therefore, we have the discrete approximation of (12) as follows:

Definition 5 (see [1]). Let be a symmetric positive definitive matrix and be a convex subset of . is a solution set of the following optimization model: where the operator is a skewed projection mapping for fixed .

Definition 6. In addition, we made the following assumptions in this section: (1) is a nonempty and compact set of (2)There exists a function which is integrable and Suppose that (1) and (2) hold, we call as -bounded function.

The following theorem will provide the uniform convergence of the approximate problem (12).

Theorem 7. Suppose that is -bounded function on , , it is continuous with respect to . Then, we have (a) is finite and continuous(b) uniformly converges to and (c) uniformly converges to and

Proof. (a)Since is continuous on , , and , when , it holds Then, we have is an integrable function, so it is monotonous, and the range of the function is between zero and one. Therefore, it is bounded; it means that is continuous. From Definition 6, is -bounded function; we have Since is integrable, we have which is finite. Therefore, (a) is hold. (b)From equation (15), it can be seen that and it means that , , when ; it holds From the fact that is arbitrary, so From the fact that is arbitrary, uniformly converges to , that is, (c)It follows from Li et al. [25] that the problem is essentially equal to the problem . So it is easy to have that , , and where Let be a function defined by (11). , , and iff is a solution of FVIP . Therefore, is a solution of (16) iff it solves (10), so Since , we have Denote the smallest eigenvalue of by . Note that Further, we can conclude that On account that is nonempty and compact, so , it holds Furthermore, we can conclude Moreover, from the nonexpansive property of the projection operator, it holds Then, we can get From (a) and (b), uniformly converges to . So , when , such that From and , , , and , we can get Then, That is, uniformly converges to .