# Convergence Analysis of the Approximation Problems for Solving Stochastic Vector Variational Inequality Problems

Authors: Meiju Luo; Kun Zhang
Journal: Complexity
 (2020)

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## Abstract

In this paper, we consider stochastic vector variational inequality problems (SVVIPs). Because of the existence of stochastic variable, the SVVIP may have no solutions generally. For solving this problem, we employ the regularized gap function of SVVIP to the loss function and then give a low-risk conditional value-at-risk (CVaR) model. However, this low-risk CVaR model is difficult to solve by the general constraint optimization algorithm. This is because the objective function is nonsmoothing function, and the objective function contains expectation, which is not easy to be computed. By using the sample average approximation technique and smoothing function, we present the corresponding approximation problems of the low-risk CVaR model to deal with these two difficulties related to the low-risk CVaR model. In addition, for the given approximation problems, we prove the convergence results of global optimal solutions and the convergence results of stationary points, respectively. Finally, a numerical experiment is given.


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## Body

## 1. Introduction

Variational inequality problems (VIPs) are a class of equilibrium optimization problems. They are powerful tools to solve large-scale optimization problems and equilibrium problems, and they have widely used in economic equilibrium, optimal control, countermeasure theory, and transportation planning. Recalling the variational inequality problems, denoted by VIX,f is to find a vector x∗∈X such that(1)x−x∗Tfx∗≥0,∀x∈X,where X⊆Rn is a nonempty closed convex set and f:X⟶Rn is a given function.However, some elements may involve uncertain data in practice. For example, the demands are generally difficult to be determined in supply chain network because they vary with the change in income level. In addition, in traffic equilibrium problems, the users attempt to minimize travel cost leads the equilibrium flows to uncertain.Since the existence of the stochastic elements, the solutions of stochastic variational inequality problems would change with stochastic elements, respectively. In order to meet the needs in practice, many authors began to consider the following stochastic variational inequality problems, denoted by SVIX,f, which requires an x∗∈X such that(2)x−x∗Tfx∗,ξω≥0,∀x∈X,ξω∈Ω,a.s.,where ξω:Ω⟶ℚ is a stochastic vector defined on probability space Ω,ℱ,P. ℚ is a support of the probability space. f:X×ℚ⟶Rn is a given function, and a.s. is the abbreviation for “almost surely” under the given probability measure. For simplicity, we use ξ instead of ξω in this paper.At present, scholars have obtained many theoretical results in the study of stochastic variational inequality problems. Because of the existence of stochastic variables, it is difficult to obtain a solution of stochastic variational inequality problems satisfying all stochastic variables. In order to solve these problems, scholars gave reasonable deterministic model of stochastic variational inequality problems and regarded the solutions of this deterministic model as the solutions of stochastic variational inequality problems. For example, the expected value (EV) method [1, 2, 3] and the expected residual minimization (ERM) method [4, 5, 6] focused on minimizing the average or expected residual. In addition, conditional value-at-risk model was presented by Chen and Lin [7]. This model can better avoid risks.In real life, many problems have multiple evaluation indicators. In order to make these evaluation indicators comprehensively optimal, scholars began to study the vector variational inequality problems. The vector variation inequality problems are powerful tools for studying optimization problems, quantitative economics, and equilibrium problems. As a result, in-depth research about vector variational inequality problems has been obtained (see [8, 9, 10, 11, 12, 13] for details).Similar to the practical applications of variational inequality problems, vector variational inequality problems will also encounter the situations that the practical problems contain stochastic variables. Hence, this paper considers stochastic vector variational inequality problems (SVVIPs). The definition of stochastic vector variational inequality problems is as follows: findingx∗∈X such that(3)x−x∗TF1x∗,ξω,…,x−x∗TFmx∗,ξω∉−intR+m,∀x∈X,a.s.ξω∈ℚ,where X⊆Rn is a nonempty closed convex set, Fj:Rn×ℚ⟶Rn is a vector valued function for each j=1,…,m, ξω:Ω⟶ℚ is a stochastic vector defined on probability space Ω,ℱ,P with support set ℚ⊂Rr, and a.s. is the abbreviation for “almost surely” under the given probability measure. For simplicity, in this paper, we use ξ to denote the ξω.Because of the existence of the stochastic variableξ, generally, there is no vector x∗ satisfying (3) for all ξ∈ℚ. In order to give reasonable solutions of (3), we will present a low-risk deterministic model. And the solutions of this model are regarded as the solutions of SVVIP.In this paper, we assumeℚ⊂Rr is a nonempty and compact set. Fx,ξ is a continuously differentiable with respect to x, and Fx,ξ is continuously differentiable with respect to ξ∈ℚ. ρ⋅ is continuity probability density function. For convenience, we use ∇xgx to denote gradient of g:Rn⟶R with respect to x. ⋅G denotes G norm, that is, xG=xTGx.The remainder of this paper is organized as follows. In Section2, we give a definition, a lemma, and the equivalent formulation of SVVIP along with regularized gap function of SVVIP. In Section 3, we present a CVaR deterministic model for solving SVVIP and then regard the solutions of this CVaR model as the solutions of SVVIP. Furthermore, we employ sample average approximation and smoothing method to solve this CVaR model. In Section 4, the convergence proof of global optimal solutions sequence of the CVaR approximation problems is given. In Section 5, the convergence result of stationary points sequence of corresponding CVaR approximation problems is considered. In Section 6, a numerical experiment is considered. In Section 7, conclusions are given.


## 2. Preliminaries

Definition 1 (see[14]).
The Clarke generalized gradient offx with respect to x is defined as(4)∂xfx≔convlimy⟶x,y∈Df⋅∇xfy,where Df⋅ denotes the set of points near x where function f:Rn⟶R near x is Frechét differentiable, ∇xfy denotes fy general gradient with respect to x, and conv denotes convex hull of a set.Lemma 1 (see [15]).
The SVVIP (3) is equivalent to the following stochastic variational inequalities with simple constraints: find x∗,λ∗∈X×Λ such that(5)x−x∗T∑j=1mλj∗Fjx∗,ξ≥0,∀x∈X,a.s.ξ∈ℚ.Here,Λ≔λ∈Rm:λj≥0,∑j=1mλj=1.Based on the regularized gap function proposed by Fukushima in [16], in this paper, we present the corresponding regularized gap function of problem (5) as follows:(6)gx,λ,ξ=maxy∈Xx−yT∑j=1mλjFjx,ξ−C2x−yG2,where C>0 is a given parameter and G is an n×n symmetric positive definite matrix. It follows from [16] that, for any x,λ∈X×Λ,(7)gx,λ,ξ=x−Hx,λ,ξT∑j=1mλjFjx,ξ−C2x−Hx,λ,ξG2,and gx,λ,ξ is continuously differentiable with respect to x and λ if Fjx,ξ,j=1,…,m is continuously differentiable with respect to x. Here,(8)Hx,λ,ξ=ProjX,Gx−C−1G−1∑j=1mλjFjx,ξ.Also based on the work of Fukushima in [16], it is easy to obtain that gx,λ,ξ≥0 and its value is equal to 0 if and only if x∗,λ∗ solves (5). Therefore, (5) can be transformed into the following constraint optimization problem:(9)mingx,λ,ξs.t.x∈X,λ∈Λ,ξ∈ℚ,a.s..Due to the existence of stochastic variables, problem (3) and problem (5) usually have no solutions. However, based on the needs of such problems in practical applications, we urgently need to give a reasonable solution which can be regarded as the solution of problem (3) or (5). Therefore, in the next section, we will present the conditional value-at-risk (CVaR) model for solving stochastic vector variational inequality problems.


## 3. Low-Risk CVaR Model and Its Approximation Problems

We introduce the definition of value-at-risk (VaR) firstly before giving the low-risk CVaR model of (3). For a variable x and a parameter u∈R+, the risk value less than level u and its probability is defined as(10)Pqx,ξ≤u=∫qx,ξ≤uρξdξ,where ρξ denotes probability density function of ξ. For a given confidence level α, the definition of VaRαx about x is given as follows:(11)VaRαx≔minu∈RPqx,ξ≤u≥α,where qx,ξ:Rn×ℚ⟶R is the loss function. Even though VaR is a popular risk measure, it has many disadvantages such as a lack of subadditivity, convexity, and fails to be coherent in the sense. Luckily, CVaR can cover the shortage of VaR; that is, CVaR satisfies coherence and convexity. Moreover, Rockafellar and Uryasev have indicated that when minimizing CVaR, VaR can be obtained as the result of CVaR’s auxiliary in [17, 18].CVaR is a low-risk model. Many scholars have used the CVaR model to solve stochastic complementarity problems [19], stochastic variational inequalities problems [9], and stochastic second-order cone complementarity problems [20]. There are also scholars who use CVaR to deal with stochastic constraints such as [21, 22].For a given confidence levelα, CVaR is defined as the conditional expectation of the loss associated with x relative to that loss being VaRα or greater, that is,(12)CVaRαx≔1−α−1∫qx,ξ≥VaRαxqx,ξρξdξ.Hence, the optimization model of minimizing the CVaR can be denoted by(13)minCVaRαx.We then introduce the function Qαx,u:Rn×R+⟶R as follows:(14)Qαx,u=u+11−α∫qx,ξ−u+ρξdξ=u+1−α−1Eqx,ξ−u+,where E denotes mathematical expectation and t+=maxt,0. Following from the conclusion has given in [17, 18], we have that minimizing CVaRαx is equivalent to minimize the function Qαx,u. So CVaR model (3) is equivalent to the following constraint optimization problem:(15)minx,u∈Rn×R+Qαx,u.In this paper, we apply regularized gap function (6) and combine the optimal problem (9) to give the loss function in (15). Then, the low-risk CVaR model for solving SVVIP is given as follows:(16)minx,λ,u∈X×Λ×R+Θx,λ,u≔u+1−α−1Egx,λ,ξ−u+.Here, α∈0,1 denotes a given confidence level.In fact, problem (16) is difficult to compute. This is because the model contains an expectation, which is difficult to calculation in general, and ⋅+ is not differentiable everywhere. Hence, we will employ the sample average approximation method and use the smoothing function presented by Li in [23] to obtain the corresponding approximation problem.Sample average approximation (SAA) method uses the sample average value to approximate the expected [24]. Let ℚk≔ξ1,…,ξNk denote the set of independently and identically distributed stochastic samples from ℚ, and suppose that Nk tends to infinity as k increases. For an integrable function ψ:ℚ⟶R, the strong law of large numbers guarantees that this procedure converges with probability one (abbreviated by “w.p.1.”), that is,(17)limk⟶∞1Nk∑i=1Nkψξi=Eψξ,w.p.1..We then present the SAA problem of the low-risk CVaR model by applying the sample average approximate method as follows:(18)minx,λ,u∈X×Λ×R+Θkx,λ,u≔u+1−α−11Nk∑i=1Nkgx,λ,ξi−u+.The smoothing form of functiont+, which is presented by Li et al. in [23], is given as follows:(19)hμt=μlnexpt/μ+1.In addition, they have proved that the following properties hold:(20)limμ⟶0hμt=t+,0≤hμt−t+≤μln2.In order to express easily, let(21)hx,λ,ξ,u≔gx,λ,ξ−u+.Then, for anyμ>0, we use(22)hμx,λ,ξ,u≔μlnexpgx,λ,ξ−u/μ+1to denote the smoothing function of hx,λ,ξ,u. According to the properties (20) of the smoothing function, we have(23)limμ⟶0hμx,λ,ξ,u=hx,λ,ξ,u,(24)0≤hμx,λ,ξ,u−hx,λ,ξ,u≤μln2.Finally, we combine the sample average approximation method and the smoothing function together to give smoothing sample average approximation (SSAA) problem of the low-risk CVaR model as follows:(25)minx,λ,u∈X×Λ×R+Θμkx,λ,u≔u+1−α−11Nk∑i=1Nkμlnexpgx,λ,ξi−u/μ+1.


## 4. Convergence of Global Optimal Solutions

We next consider the convergence of the smoothing sample average approximation problem (25). In this section and the next section, we only consider the most complex convergence, and let μk⟶0 as k⟶∞. Here, μ varies as k increases.Theorem 1.
Suppose that, for eachk, xk,λk,uk is a global optimal solution of problem (25), and x¯,λ¯,u¯ is an accumulation point of sequence xk,λk,uk. Then, x¯,λ¯,u¯ is a global optimal solution of the low-risk CVaR model (16), which holds with probability one.Proof.
Without loss of generality, we assume thatlimk⟶∞xk,λk,uk=x¯,λ¯,u¯. Let B is a compact set including the whole sequence xk,λk,uk. Z⊆R+ is a compact set including μk. Because hμx,λ,ξ,u is continuously differentiable on compact set B×ℚ×Z, we can have that there exists a constant D>0 such that, for any x,λ,ξ,u∈B×ℚ×Z, the following formulation holds:(26)∇hμx,λ,ξ,u≤D.
In addition, by mean value theorem, for eachxk,λk,uk,ξi and μk, there exists αki∈0,1 such that xki,λki,ξi,uki=αkixk,λk,ξi,uk+1−αkix¯,λ¯,ξi,u¯∈B, and we then obtain that(27)hμkxk,λk,ξi,uk−hμkx¯,λ¯,ξi,u¯=∇hμkxki,λki,ξi,ukiTxk,λk,uk−x¯,λ¯,u¯.
Hence, by (26) and (27), we obtain that(28)Θμkkxk,λk,uk−Θμkkx¯,λ¯,u¯≤uk−u¯+1−α−11Nk∑i=1Nkhμkxk,λk,ξi,uk,−hμkx¯,λ¯,ξi,u¯≤uk−u¯+1−α−11Nk∑i=1Nk∇hμkxki,λki,ξi,uki⋅xk,λk,uk−x¯,λ¯,u¯≤uk−u¯+D1−α⋅1Nk∑i=1Nkxk,λk,uk−x¯,λ¯,u¯⟶k⟶∞0,w.p.1.
On the other hand, following from (17) and (24), we have(29)Θμkkx¯,λ¯,u¯−Θx¯,λ¯,u¯≤1−α−11Nk∑i=1Nkhμkx¯,λ¯,ξi,u¯−Ehx¯,λ¯,ξ,u¯≤1−α−11Nk∑i=1Nkhμkx¯,λ¯,ξi,u¯−hx¯,λ¯,ξi,u¯+1Nk∑i=1Nkhx¯,λ¯,ξi,u¯−Ehx¯,λ¯,ξ,u¯≤1−α−1μkln2+1Nk∑i=1Nkhx¯,λ¯,ξi,u¯−Ehx¯,λ¯,ξ,u¯⟶k⟶∞0,w.p.1.
Similar to (29), we have(30)limk⟶∞Θμkkx,λ,u=Θx,λ,u,w.p.1.
Because(31)Θμkkxk,λk,uk−Θx¯,λ¯,u¯≤Θμkkxk,λk,uk−Θμkkx¯,λ¯,u¯|+|Θμkkx¯,λ¯,u¯−Θx¯,λ¯,u¯,it then follows from (28) and (29) that(32)limk⟶∞Θμkkxk,λk,uk=Θx¯,λ¯,u¯,w.p.1.
Moreover, for everyk, since xk,λk,uk is a global optimal solution of problem (25), we have(33)Θμkkxk,λk,uk≤Θμkkx,λ,u,∀x,λ,u∈X×Λ×R+.
Taking limits to (33), according to (30) and (32), we have(34)Θx¯,λ¯,u¯≤Θx,λ,u,w.p.1.∀x,λ,u∈X×Λ×R+.


## 5. Convergence of Stationary Points

In fact, SSAA problem (25) is a nonconvex optimal problem usually. Hence, when solving this problem, we usually obtain stationary points instead of global optimal solutions. Therefore, it is necessary to consider the convergence of stationary point of (25). In this section, suppose that(35)X≔x∈Rn:cix≤0,i=1,…,p;ljx=0,j=1,…,q,where cix,ljx,i=1,…,p,j=1,…,q, are all continuous differentiable functions. We firstly give the definitions of stationary point of (16) and (25), respectively.Definition 2.
Letdμkkixk,λk,ξi,uk≔expgxk,λk,ξi−uk/μk, for each fixed k, if there exist Lagrange multipliers δk=δ1k,…,δpkT∈Rp, βk=β1k,…,βqkT∈Rq, τk∈R, ηk=η1k,…,ηmkT∈Rm, and ςk∈R making xk,λk,uk satisfy(36)1−α−11Nk∑i=1Nkdμkkixk,λk,ξi,uk1+dμkkixk,λk,ξi,uk⋅∇xgxk,λk,ξi+∑i=1pδik∇cixk+∑j=1qβjk∇ljxk=0,(37)1−α−11Nk∑i=1Nkdμkkixk,λk,ξi,uk1+dμkkixk,λk,ξi,uk⋅∇λgxk,λk,ξi+τke−ηk=0,(38)1−1−α−11Nk∑i=1Nkdμkkixk,λk,ξi,uk1+dμkkixk,λk,ξi,uk−ςk=0,(39)δk≥0,cxk≤0,δkTcxk=0,(40)ηk≥0,λk≥0,ηkTλk=0,(41)eTλk−1=0,lxk=0,(42)uk≥0,ςk≥0,ukςk=0.
We callxk,λk,uk is a stationary point of (25), where e=1,…,1T∈Rm.Definition 3.
If there exist Lagrange multipliersδ^=δ^1,…,δ^pT∈Rp,β^=β^1,…,β^qT∈Rq,τ^∈R, η^=η^1,…,η^mT∈Rm, and ς^∈R making x^,λ^,u^ satisfy(43)0∈∂xΘx^,λ^,u^+∑i=1pδ^i∇cix^+∑j=1qβ^j∇ljx^,(44)0∈∂λΘx^,λ^,u^+τ^e−η^,(45)0∈∂uΘx^,λ^,u^,(46)δ^≥0,cx^≤0,δ^Tcx^=0,(47)η^≥0,λ^≥0,η^Tλ^=0,(48)eTλ^−1=0,lx^=0,(49)u^≥0,ς^≥0,u^ς^=0.
We callx^,λ^,u^ is a stationary point of (16), where e=1,…,1T∈Rm.Theorem 2.
Suppose that for eachk, xk,λk,uk is a stationary point of problem (25), x^,λ^,u^ is an accumulation point of the sequence xk,λk,uk and the Lagrange multipliers δk=δ1k,…,δpkT∈Rp, βk=β1k,…,βqkT∈Rq, τk∈R, ηk=η1k,…,ηmkT∈Rm, and ςk∈R are all bounded. Then, x^,λ^,u^ is a stationary point of the low-risk CVaR model (16), which holds with probability one.Proof.
Without loss of generality, we assume that(50)limk⟶∞xk,λk,uk=x^,λ^,u^,(51)limk⟶∞δk=δ^,limk⟶∞βk=β^,limk⟶∞τk=τ^,limk⟶∞ηk=η^,limk⟶∞ςk=ς^.
LetB≔xx−x^≤1, C≔uu−u^≤1, D≔δδ−δ^≤1, and T≔ββ−β^≤1. Since gx,λ,ξ is continuously differentiable on compact set B×Λ×ℚ, we have that there exists a constant M>0 such that, for any x,λ,ξ∈B×Λ×ℚ, the following formulation holds:(52)∇x,λgx,λ,ξ≤M.
By mean value theorem (Theorem 2.19 in [25]), we have that there exists β∈0,1 such that x˜,λ˜=βx′,λ′+1−βx′,λ′∈B×Λ and(53)gx′,λ′,ξ−gx″,λ″,ξ=∇x,λgx˜,λ˜,ξTx′,λ′−x″,λ″≤Mx′,λ′−x″,λ″.
Setting(54)ϑx,λ,ξ,u≔u+1−α−1gx,λ,ξ−u+,we have Θx,λ,u=Eϑx,λ,ξ,u. By (52), (53), and inequality x+−y+≤x−y, we have that(55)ϑx′,λ′,ξ,u′−ϑx″,λ″,ξ,u″≤u′−u″+11−αgx′,λ′,ξ−gx″,λ″,ξ+u′−u″≤u′−u″+11−αMx′,λ′−x″,λ″+u′−u″.
Hence, by Theorem 9 in Section2 [26], we obtain that(56)E∂xϑx^,λ^,ξ,u^⊆∂xΘx^,λ^,u^,E∂λϑx^,λ^,ξ,u^⊆∂λΘx^,λ^,u^,E∂uϑx^,λ^,ξ,u^⊆∂uΘx^,λ^,u^.
For simplicity, in order to prove0∈∂xΘx^,λ^,u^+∑i=1pδ^i∇cix^+∑j=1qβ^j∇ljx^, we only prove the following formulation holds:(57)0∈E∂xϑx^,λ^,ξ,u^+∑i=1pδ^i∇cix^+∑j=1qβ^j∇ljx^=E1−α−1⋅rx⋅∇xgx^,λ^,ξ+∑i=1pδ^i∇cix^+∑j=1qβ^j∇ljx^rx∈∂xgx^,λ^,ξ−u^+.
Here,(58)∂xgx^,λ^,ξ−u^+≔1,gx^,λ^,ξ−u^>0,0,1,gx^,λ^,ξ−u^=0,0,gx^,λ^,ξ−u^<0.
Setdμx,λ,ξ,u≔expgx,λ,ξ−u/μ and(59)ζμx,λ,ξ,u,δ,β=1−α−1dμx,λ,ξ,u1+dμx,λ,ξ,u∇xgx,λ,ξ+∑i=1pδi∇cix+∑j=1qβj∇ljx,μ≠0,1−α−1⋅r⋅∇xgx,λ,ξ+∑i=1pδ∇cix+∑j=1qβj∇ljx,μ=0,where r∈∂xgx,λ,ξ−u+.
We will prove thatζμ^x,λ,ξ,u,δ,β is upper semicontinuous on x^,λ^,u^ now. When μ≠0, by (5), we have that ζμ^x,λ,ξ,u,δ,β is a monotropic function near x^,λ^,u^. Therefore, since gx,λ,ξ is continuously differentiable about x, we obtain that ζμ^x,λ,ξ,u,δ,β is continuous on x^,λ^,u^. When μ=0, it is easy to obtain that ζ0x,λ,ξ,u,δ,β is closed on x^,λ^,u^. Combining (52) into account, we have ζ0x,λ,ξ,u,δ,β is uniform compact near x^,λ^,u^. Hence, by Lemma 3.5 of [7], we get that ζ0x,λ,ξ,u,δ,β is upper semicontinuous on x^,λ^,u^. Combining above argument together, ζμ^x,λ,ξ,u,δ,β is upper semicontinuous on x^,λ^,u^.
Note thatX is a compact set and gx,λ,ξ is continuously differentiable with respect to x, and we get that ζμx,λ,ξ,u,δ,β is bounded and compact set value function in B×Λ×ℚ×C×D×T. Therefore, for function ζμx,λ,ξ,u,δ,β, all the conditions in Lemma 3.4 [7] are hold. Because the formulation (36) of Definition 2 is equivalent to(60)0∈1Nk∑i=1Nkζμkxk,λk,ξi,uk,δk,βk,followed from Lemma 3.4 of [7], we obtain that the formulation (43) holds. Similar to the process of getting formulation (43), we obtain (44) and (45) immediately.
Moreover, by the continuity ofcix,ljx,=1,…,p,j=1,…,q and (50) and (51), taking limits to formulations (39)–(42), respectively, we know have (46)–(49) hold immediately, which mean that x^,λ^,u^ is a stationary point of (16).


## 6. Numerical Experiment

In this section, we apply the approach introduced in the previous sections to solve the vector variational inequality problem. That is, we first formulate the vector variational inequality problem into the CVaR formulation (16) and then employ the SAA method and smoothing method to solve the CVaR model. In the following example, we use the function “fmincon” in Matlab2018b function to solve constraints optimization program (25). And we use the command “random” to obtain the independent and identically distributed samples.Example 1.
Setting(61)F1x,ξ=a1+ξx+b1,F2x,ξ=a2+ξx+b2,where ξ∈0,1 is a random variable with uniform distribution. Let X⊂R be the box constraint, that is, l≤x≤u and a1=50,a2=100,b1=70,b2=60,C=1,l=100,andu=500 along with G an identity matrix x0=10, λ1=0.5, λ2=0.5, and u=10. (Table 1)
From the numerical results, it can be seen that it is feasible to solve (25) by the methods we have proposed.Table 1
Computational results for the problem (25).
kxkλ1kλ2kuk1031000.98950.01050.00101041000.98940.01060.00101051000.98940.01060.00101061000.98940.01060.00101071000.98940.01060.0010


## 7. Conclusions

In this paper, we consider stochastic vector variational inequality problems (SVVIPs). Since for allξ∈ℚ, SVVIP may have no solutions generally. Therefore, we give a reasonable deterministic model, that is, conditional value-at-risk (CVaR) model and then regard the solutions of this model as the solutions of SVVIP. For solving this CVaR model, the sample average approximation method and smoothing method are employed to deal with the problems that the expectation is difficult to calculate and the objective function is nonsmoothing function, respectively. In addition, convergence analysis of global optimal solutions sequence and stationary points sequence of SSAA problems is considered. Finally, a numerical experiment indicates that the proposed approach is applicable.



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