An Approximation Method for Variational Inequality with Uncertain Variables

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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][2][3][4][5][6][7]), and so on.For the classical VIP, ∀v ∈ S, there is a point u ∈ S ∈ R n such that where S ≠ ∅ is closed convex and f : S ⟶ R n is a vectorvalued function.Chen and Fukushima [8] presented the regularized gap function as follows: where matrix G is symmetric and positive definite square and parameter γ > 0. k•k G indicates the G-norm, which is given by kuk G = ffiffiffiffiffiffiffiffiffiffiffi u T Gu p , u ∈ R n .It means RðuÞ ≥ 0, ∀u ∈ S, and RðuÞ = 0 iff u is a solution of VIP ð f , SÞ.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 Ω is a stochastic sample space and the mapping f : R n × Ω ⟶ R n .Due to the randomness of the function f , there is generally no solution to problem (4).By calculating expected value E½ f ðu † , ωÞ over ω, 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][10][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][13][14][15][16][17][18][19][20][21][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 Ξ is the set of uncertain variables and the mapping f : R n × Ξ ⟶ R n .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.

Preliminaries
In this section, we will give some definitions and lemmas.Firstly, we collect the concepts and properties in uncertainty space.Supposed that Γ is a nonempty set and L is a σalgebra over Γ.Then, ðΓ, LÞ is called a measurable space; each element Λ in Γ is called an event.So MfΛg ∈ R presents the belief degree that Λ occurs.So ðΓ, L, MÞ is an uncertainty space, which is defined by Ξ.To deal with belief degrees rightly, Liu [12] presented three axioms as follows: (1) MfΓg = 1 i=1 MfΛ i g, where Λ 1 , Λ 2 , ⋯ are sequence of events Definition 1 (see [12]).Let ζ ∈ Ξ.If the following exists, then E½ζ is the expected value of uncertain variable ζ.

SDA Method and Its Convergence
In this section, we will provide the convergence of SDA method and regularized gap functions Rðu, ζÞ on the set S on the basis of uncertainty theory.It turns out that for γ > 0, there is an optimal solution for problem (1).Therefore, we can find a fixed point where f : R n × Ξ ⟶ R n and Ξ is an uncertain space.Furthermore, we present the regularized gap function where parameter γ > 0 and matrix G 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 f ðxÞ be a bounded function on the interval ½a, b and κðxÞ be a bounded variation function on ½a, b, and make a division of interval T : Advances in Mathematical Physics a group of "intermediate points," x i−1 ≤ ξ ≤ x i ði = 1, 2,⋯,nÞ, and make a sum: Set δðTÞ = max 0≤i≤n jx i − x i−1 j.When δðTÞ ⟶ 0, the sum tends to a certain finite limit; then, f ðxÞ is said to be R − S integrable about κðxÞ on the interval ½a, b.This limit is recorded as Ð b a f ðxÞdx.From the division of interval by the Stieltjes integral (10), we have According to the arbitrariness of ΔΦ, let Therefore, we have the discrete approximation of (12) as follows: Definition 5 (see [1]).Let G ∈ R n×n be a symmetric positive definitive matrix and S be a convex subset of R n .Θ S,G ðuÞ is a solution set of the following optimization model: where the operator Θ S,G : R n ⟶ S is a skewed projection mapping for fixed u ∈ R n .Definition 6.In addition, we made the following assumptions in this section: (1) S is a nonempty and compact set of R n (2) There exists a function φðζÞ which is integrable and Suppose that (1) and ( 2) hold, we call f ðu, ζÞ as ϕbounded function.
The following theorem will provide the uniform convergence of the approximate problem (12).Theorem 7. Suppose that f ðu, ζÞ is ϕ-bounded function on S, ∀ζ ∈ Ξ, it is continuous with respect to u.Then, we have (a) E½ f ðu, ζÞ is finite and continuous (b) E½ f N ðu, ζÞ uniformly converges to E½ f ðu, ζÞ and Proof.
(a) Since f ðu, ζÞ is continuous on S, ∀ε > 0, and ∃δ > 0, Then, we have ϕðζÞ 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 E½ f ðu, ζÞ is continuous.From Definition 6, f ðu, ζÞ is ϕ-bounded function; we have Since ϕðζÞ is integrable, we have Ð ϕðtÞdΦðtÞ which is finite.Therefore, (a) is hold.
(b) From equation ( 15), it can be seen that 3 Advances in Mathematical Physics and it means that ∀ε > 0, ∃N 0 > 0, when N > N 0 ; it holds From the fact that u is arbitrary, so From the fact that ε is arbitrary, E½ f N ðu, ζÞ uniformly converges to E½ f ðu, ζÞ, that is, where Let Rðu, ζÞ: R n × Ξ ⟶ R n > 0 be a function defined by (11).∀u ∈ S, ζ ∈ Ξ, and Rðu, ζÞ = 0 iff u is a solution of FVIP ð f , SÞ.Therefore, u is a solution of ( 16) iff it solves (10), so Since Rðu, ζÞ ≥ 0, we have Denote the smallest eigenvalue of G by λ min .Note that ffiffiffiffiffiffiffiffi ffi Further, we can conclude that On account that S is nonempty and compact, so ∃K > 0, it holds Furthermore, we can conclude Moreover, from the nonexpansive property of the projection operator, it holds Advances in Mathematical Physics Then, we can get From (a) and (b), E½ f N ðu, ζÞ uniformly converges to E½ f ðu, ζÞ.So ∀δ > 0, when N > N 0 , ∃N 0 such that From ku − P ðu, ζÞk < ð2/γ ffiffiffiffiffiffiffiffi ffi λ min p ÞK and ku − P N ðu, ζÞk < ð2/γ ffiffiffiffiffiffiffiffi ffi Then, That is, R N ðu, ζÞ uniformly converges to Rðu, ζÞ.
Since the condition of uniform convergence is strong, there will be inevitable mistakes in the calculation process.Here, we weaken the condition of the function and then prove it.
Definition 8 (see [26]).Let f f n g ∞ n=1 be a sequence and the function f be lower semicontinuous.f f n g epiconverges to f : Proof.To prove P N ðu N , ζÞ approaches to P ðu, ζÞ, we will prove the following: (a) ∀fu N g s:t: lim Firstly, we prove (a).Recall that

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By [27] and (29), we can get that P N ðu N , ζÞ is the unique optimal solution of min u∈S R N ðu N , ζÞ; and P ðu, ζÞ is the only optimal solution of min u∈S Rðu, ζÞ.So we have Because E½ f N ðu N , ζÞ epiconverges to the function E½ f ðu, ζÞ if for any u, ∀fu N g s:t: lim Then, From ∀fu N g s:t: lim n⟶∞ u N = u, so for any ε > 0, ∃N 1 > 0, Then, for any ε > 0, ∃N 2 ∈ max fN 0 , N 1 g, N > N 2 , we have By u N ∈ S, v ∈ S, γ > 0, so ðu N − vÞ T , ðγ/2ÞkGj are finite, and E½ f ðu, ζÞ is finite; then, R N ðu N , ζÞ approaches Rðu, ζÞ.And P N ðu N , ζÞ is the only optimal solution to min u∈S R N ðu N , ζÞ; and P ðu, ζÞ is the only optimal solution to min u∈S Rðu, ζÞ.
So P N ðu N , ζÞ approaches to P ðu, ζÞ; that is, for any δ > 0, ∃N 3 ∈ max fN 0 , N 1 g, N > N 3 , we have Next, we prove (b).Because E½ f N ðu N , ζÞ epiconverges to the function E½ f ðu, ζÞ if ∃fv N g s:t: lim That means that there exists a sequence fv N g converging to v, and it holds From (a) and (b), we can get that P N ðu N , ζÞ approaches to P ðu, ζÞ, so the proof is completed.Proof.To prove fR N ðu N , ζÞg epiconverge to fRðu, ζÞg, we will prove the following: (I) If for any u, ∀fu N g s:t: lim First of all, we prove (I).Because for any u, E½ f N ðu N , ζÞ epiconverges to the function E½ f ðu, ζÞ, ∀fu N g s:t: lim We then obtain From fu N g converging to u, so for any By (51) in Lemma 9 and (56), that is, for any ε 3 > 0, ∃N 3 ∈ max fN 0 , N 1 g, N > N 3 , we have P N ðu N , ζÞ − P ðu, ζÞ ≥ −ε 3 .We can get Obviously, u N ∈ S, P N ðu N , ζÞ is the the unique optimal solution of problem min u∈S R N ðu N , ζÞ; and P ðu, ζÞ is the the unique optimal solution of problem min u∈S Rðu, ζÞ, so Furthermore, we prove (II).From Because E½ f N ðu N , ζÞ epiconverges to the function E½ f ðu, ζÞ, ∃fv N g s:t: lim

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We then obtain that there exists a sequence fv N g converging to v; it holds By fv N g converging to v, so for any ε > 0, ∃N 1 > 0, N > N 1 , we have And since Rðv, ζÞ ≥ 0, we have where λ min indicate the smallest eigenvalue of G. Further, we can conclude that Because S is a compact and nonempty set on R n , then ∃M > 0; it holds Furthermore, it is not difficult to show that From (63), v N − v ≤ ε 2 , by (53) in Lemma 9; that is, for any ε 3 > 0, ∃N 3 ∈ max fN 0 , N 1 g, N > N 3 , we have P N ðu N , ζÞ − P ðu, ζÞ ≤ ε 3 .kE½ f ðv, ζÞk < M, and kv − P ðv, ζÞk < ð2/γ ffiffiffiffiffiffiffiffi ffi λ min p ÞM and kv N − P N ðv N , ζÞk < ð2/γ ffiffiffiffiffiffiffiffi ffi λ min p ÞM; we have It means that there exists a sequence fv N g that converges to v, so that From (60) and (72), we can get that fR N ðu N , ζÞg epi-converge to fRðu, ζÞg.
Therefore, we have that Since, by Theorem 10, the sequence fR N ðu N , ζÞg epiconverges to Rðu, ζÞ, that is, for every sequence fu N g converging to u, we have Then, Because ε is arbitrary, we have The conclusion follows from (76) and (82) immediately.
Theorem 12. Suppose that E½ f N ðu N , ζÞ epiconverge to E½ f ðu, ζÞ.Suppose that function f ðu, ζÞ is uniformly monotone with respect to u, there exists a function ΨðuÞ which is nonnegative integrable, ∀u, v ∈ R n , ∀N > 0, Here, u N is an optimal solution of ( 16), and E½ΨðζÞ > 0.Then, the sequence fu N g converges to the unique solution of (10).
arginf R and arginf R N are the optimal solution sets of (11) and (16) E½ΨðζÞ > 0, so ðu − vÞ T fE½ f ðu, ζÞ − E½ f ðv, ζÞg ≥ 0. By the arbitrariness of u, v, we have So, u = v means the uniqueness of the solution to problem (10), denoted by u † .It is not difficult to show that u † is also the unique solution of (12).Therefore, u † is a unique cluster point of the bounded sequence fu N g.

Conclusions
In this paper, we studied the SDA method for solving the UVIP.By constructing the gap function (11), the uncertain variational inequality problem is transformed into an optimization problem (12).Then, we propose the SDA method to solve it.Also, we research the convergence of the optimization problem.Finally, the correctness of the SDA method is proved; that is, the solution of the approximation problem (16) obtained by the SDA method converges to the solution of the original uncertain variational inequality (10).
In this paper, we have done some work on the Stieltjes integral discrete approximation of uncertain variational inequalities and obtained the related theoretical results, which have good theoretical and practical significance.Future studies are as follows: we can consider the displacement gap function to establish the correlation model; and we can consider to apply this method to the solution of uncertain complementary functions.
Let u N k ∈ arginf R N k , u ∈ arginf R.By Theorem 11, we have lim k⟶∞ u N k = u ∈ S.And from the 9 Advances in Mathematical Physics assumptions, it shows that E½ f ð•, is uniformly monotone and E½ΨðζÞ > 0. So we have