TwoStage Stochastic Variational Inequalities: Theory, Algorithms and Applications
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Abstract
The stochastic variational inequality (SVI) provides a unified form of optimality conditions of stochastic optimization and stochastic games which have wide applications in science, engineering, economics and finance. In the recent two decades, onestage SVI has been studied extensively and widely used in modeling equilibrium problems under uncertainty. Moreover, the recently proposed twostage SVI and multistage SVI can be applied to the case when the decision makers want to make decisions at different stages in a stochastic environment. The twostage SVI is a foundation of multistage SVI, which is to find a pair of “hereandnow” solution and “waitandsee” solution. This paper provides a survey of recent developments in analysis, algorithms and applications of the twostage SVI.
Keywords
Twostage stochastic variational inequality Twostage stochastic complementary problem Twostage stochastic gamesMathematics Subject Classification
90C15 90C331 Introduction
The variational inequality (VI) represents the firstorder optimality conditions of optimization problems and models equilibrium problems, which plays a key role in optimization and operations research. The definition of VI is as follows.
Definition 1
However, in many real applications in finance, management, engineering and science, the decision makers have to make sequential decisions in an uncertain environment. In such situations, the deterministic VI may not be suitable. Motivated by those applications, onestage, twostage and multistage SVIs arouse the attention of scholars. Onestage SVI has been investigated for many years [3, 4]. But the investigation of multistage SVI and even twostage SVI has just begun [5]. In this paper, starting from onestage SVI, we will introduce the model, the motivation and recent progress of twostage SVI (and extend to multistage SVI), including theoretical results, algorithms and applications.
The organization of the paper is as follows. Starting from onestage SVI, the motivations, the models and the properties of the twostage SVI (the multistage SVI) are introduced in Sect. 2. In Sect. 3, we introduce several approximation methods and algorithms for different models of twostage SVI (multistage SVI). Two applications of twostage SVI are shown in Sect. 4. Section 5 gives several final remarks about the challenge of this research area.
Through out this paper, we use \(\xi :\varOmega \rightarrow \mathbb {R}^d\) to denote a random vector in probability space \((\varOmega , \mathcal{F}, {P})\) with support set \(\varXi \subset \mathbb {R}^d\). For function \(f(\cdot , \cdot ):\mathbb {R}^n\times \mathbb {R}^m\rightarrow \mathbb {R}\), \(\partial _x f(x, y)\) denotes the Clarke subdifferential of f w.r.t. x.
2 TwoStage Stochastic Variational Inequalities: Modeling and Analysis
In this section, we will give the motivation and model of twostage stochastic variational inequalities and introduce the analysis in the literature. We will start the model of onestage SVI firstly and then turn to twostage SVI.
2.1 From OneStage SVI to TwoStage SVI
As a stochastic generalization of the deterministic VI under uncertainties, onestage SVI has been investigated deeply and applied widely. However, a good formulation of a VI in a stochastic environment is not straightforward. We use the following onestage stochastic game to introduce the motivation of different onestage SVI formulations firstly.
Example 1
(Motivation of onestage SVI) We consider a duopoly market where two firms compete to supply a homogeneous product (or service) noncooperatively in future. They need to make a decision on the quantity of production based on the other firm’s decision in an uncertain environment.
The market demand in future is characterized by a random inverse demand function \(p(q,\xi (\omega ))\), where \(p(q,\xi (\omega ))\) is the market price and q is the total supply to the market. Specifically, for each realization of the random vector \(\xi \), we obtain a different inverse demand function \(p(\cdot ,\xi )\). The uncertainty in the inverse demand function is then characterized by the distribution of the random variable \(\xi \). Firm i’s cost function for producing (supplying) a quantity of \(y_i\) in future is \(H_i(y_i, \xi )\), \(i=1,2\) with limit capacity \(c_i\). We assume \(H_i(y_i, \xi )\) is twice continuously differentiable, \(H_i'(y_i, \xi )\geqslant 0\) and \(H_i''(y_i, \xi )\geqslant 0\) for \(y_i\geqslant 0\), \(p(q,\xi )\) is twice continuously differentiable in q and \(p_q'(q,\xi )<0\) and \(p'_q(q,\xi )+qp''_{qq}(q,\xi )\leqslant 0\) for \(q\geqslant 0\) and \(\xi \in \varXi .\)
 For situation 1, to maximize each firm’s profit, they need to find \((y_1^*(\cdot ), y_2^*(\cdot ))\) such that it solves the following problem:for almost every (a.e.) \(\xi \in \varXi \), where \(y_{i}\) is for decision variable of the firm(s) other than i. Moreover, we can write down the Karush–Kuhn–Tucker (KKT) conditions for (2.1) as follows: for a.e. \(\xi \in \varXi \)$$\begin{aligned} \begin{array}{ll} \displaystyle \max _{y_i(\xi )} &{}\quad p(y_i(\xi )+y^*_{i}(\xi ),\xi )y_i(\xi ) H_i(y_i(\xi ), \xi ) \\ \text{ s.t. } &{}\quad 0\leqslant y_i(\xi ) \leqslant c_i, \end{array} \end{aligned}$$(2.1)and (2.2) is a “waitandsee” model.$$\begin{aligned}&0\leqslant \begin{pmatrix} y_i(\xi )\\ \mu _i(\xi ) \end{pmatrix} \nonumber \\&\perp \begin{pmatrix} p(y_i(\xi )+y_{i}(\xi ),\xi ) y_i(\xi )p_q'(y_i(\xi )+y_{i}(\xi ),\xi )+H_i'(y_i(\xi ), \xi ) + \mu _i(\xi )\\ c_i  y_i(\xi ) \end{pmatrix}\geqslant 0, \end{aligned}$$(2.2)
 For situation 2, we consider the expected residual minimization (ERM) formulation firstly. To maximize each firm’s profit in ERM formulation, they need to find \((y_1^*, y_2^*)\) such that it solves the following problem:where \(y=(y_1, y_2), \mu =(\mu _1,\mu _2)\)\(\phi \) is a residual function of onestage SVI in “waitandsee” model (2.2), e.g.,$$\begin{aligned} \min _{0\leqslant y_1\leqslant c_1, y_2\leqslant c_2} \;\; {{E}} [\phi (y, \mu , \xi )], \end{aligned}$$(2.3)and (2.3) is a onestage SVI in ERM formulation.$$\begin{aligned}&\phi (y, \mu , \xi )\\&\quad : = \left\ \min \left\{ \begin{pmatrix}y_1\\ \mu _1\\ y_2\\ \mu _2\end{pmatrix} , \begin{pmatrix} p(y_1+y_{2},\xi ) y_1p_q'(y_1+y_{2},\xi )+H_1'(y_1, \xi ) + \mu _1\\ c_1  y_1\\ p(y_2+y_1,\xi ) y_2p_q'(y_2+y_1,\xi )+H_2'(y_2, \xi ) + \mu _2\\ c_2  y_2 \end{pmatrix}\right\} \right\ _2^2 \end{aligned}$$
 We then consider the expectedvalue (EV) formulation. To maximize each firm’s profit in EV formulation, they need to find \((y_1^*, y_2^*)\) such that it solves the following problem:for almost every \(\xi \in \varXi \). Moreover, we can write down the KKT conditions for (2.4) as follows:$$\begin{aligned} \begin{array}{ll} \displaystyle \max _{y_i} &{}\quad {{E}}[p(y_i+y^*_{i},\xi )y_i H_i(y_i, \xi )] \\ \text{ s.t. } &{}\quad 0\leqslant y_i \leqslant c_i, \end{array} \end{aligned}$$(2.4)and (2.5) is a onestage SVI in EV formulation.$$\begin{aligned} 0\leqslant & {} \begin{pmatrix} y_i\\ \mu _i \end{pmatrix} \perp \begin{pmatrix} {E}[p(y_i+y_{i},\xi ) y_ip_q'(y_i+y_{i},\xi )+H_i'(y_i, \xi ) + \mu _i]\\ c_i  y_i \end{pmatrix}\nonumber \\\geqslant & {} 0, \text{ for } \text{ a.e. } \xi \in \varXi , \end{aligned}$$(2.5)
 1.“waitandsee” model: find \(x: \varXi \rightarrow \mathbb {R}^n\) such thatIn this model, for every scenario \(\xi \) in the further, we have \(x(\xi )\) as the solution and \(x(\cdot )\) is a measurable function on \(\varXi \). Since the solution \(x(\xi )\) depends on the further uncertain scenario, we call it “waitandsee” solution and the model “waitandsee” model. Although the model can give perfect solution for every scenario in the further, in many realworld applications when the decision needs to be made before we observe the uncertain scenario, it is unworkable.$$\begin{aligned} 0\in f(x(\xi ), \xi ) + {\mathcal {N}}_X(x(\xi ), \xi ), \;\; \text{ for } \text{ a.e. } \xi \in \varXi . \end{aligned}$$
 2.The expected residual minimization (ERM) formulation [6, 7, 8, 9, 10]: Find \(x\in \mathbb {R}^n\) such that it is a solution ofwhere \(\phi (\cdot , \xi ): X\rightarrow \mathbb {R}\) is a residual function of the VI\((X, f(\cdot , \xi ))\) for a.e. \(\xi \in \varXi \), that is,$$\begin{aligned} \min _{x\in X} \;\; {E}[\phi (x, \xi )], \end{aligned}$$Different with the “waitandsee model,” the solutions of the ERM formulation do not depend on the further uncertain scenario and the decision makers can make decisions before they observe further uncertainty. We call the solutions “hereandnow” solutions. Moreover, the ERM formulation can quantify the quality of the “hereandnow” solution. The value of \(\phi (x, \xi )\) can be considered as the “loss” due to failure of the equilibrium and hence can measure the quality of the “hereandnow” solution at scenario \(\xi \). Then, the expect value of \(\phi (x, \xi )\) measures the quality of the “hereandnow” solution in the sense of expectation. In [11], Chen et al. further investigated the ERM in the case when X depends on random vector \(\xi \).$$\begin{aligned} \phi (\cdot , \xi ) \geqslant 0, \;\;\;\;\; \text{ and } \;\;\;\;\; \phi (x, \xi )=0 \;\; \Leftrightarrow \;\; 0\in f(x, \xi ) + {\mathcal {N}}_X(x), \; \text{ for } \text{ a.e. } \xi \in \varXi . \end{aligned}$$
 3.The expectedvalue (EV) formulation [12, 13, 14, 15]: Find \(x\in \mathbb {R}^n\) such thatThe solutions of EV formulation are also “hereandnow” solutions. Similar as deterministic VI, the EV formulation can be used to represent the firstorder optimality conditions of onestage stochastic optimization and describe onestage stochastic Nash equilibrium. But the “hereandnow” solution of the EV formulation is made in the sense of expectation, for different scenarios in the further, the “hereandnow” solution may not be a good solution. Moreover, when we set \(G(x):={E}[f(x, \xi )]\), then the EV formulation is the same as the deterministic VI: \(0\in G(x)+{\mathcal {N}}_X(x)\). Similar as the ERM formulation, we can also reformulate the EV formulation as a minimization problem$$\begin{aligned} 0\in {E}[f(x, \xi )] + {\mathcal {N}}_X(x). \end{aligned}$$(2.6)where \(\theta : X\rightarrow \mathbb {R}_+\) is a residual function of the deterministic VI, that is,$$\begin{aligned} \min _{x\in X} \;\; \theta (x), \end{aligned}$$One popular residual function is the regularized gap function [16] as follows:$$\begin{aligned} \theta (\cdot ) \geqslant 0, \;\;\;\;\; \text{ and } \;\;\;\;\; \theta (x)=0 \;\; \Leftrightarrow \;\; 0\in G(x) + {\mathcal {N}}_X(x). \end{aligned}$$$$\begin{aligned} \theta (x) = \max _{v\in X} \;\;\langle xv, G(x) \rangle  \frac{\alpha }{2}\Vert xv\Vert ^2. \end{aligned}$$(2.7)
Since the solutions of the EV formulation and ERM formulation are all “hereandnow” solutions, we can call the two formulations “hereandnow” model.
In the case when we consider twostage decisions under uncertain environment, the mathematical tools of onestage SVI are not enough. To elicit twostage SVI, we consider an extension of Example 1:
Example 2
(Motivation of twostage SVI) Similar as Example 1, we consider a duopoly market where two firms compete to supply a homogeneous product (or service) noncooperatively in future. The only difference with Example 1 is that neither of the firms has an existing capacity, and thus, they must make a decision at the present time on their capacity for future supply of quantities in order to have enough time to build the necessary facilities.
 (i)
\(p(q,\xi )\) is twice continuously differentiable in q and \(p_q'(q,\xi )<0\) for \(q\geqslant 0\) and \(\xi \in \varXi ;\)
 (ii)
\(p'_q(q,\xi )+qp''_{qq}(q,\xi )\leqslant 0\), for \(q\geqslant 0\) and \(\xi \in \varXi \).
In what follows, we give the definition of twostage SVI.
Definition 2
2.2 Multistage SVI
In [22], Rockafellar and Wets first proposed the formwork of the multistage SVI. The multistage SVI is an extension of the twostage SVI and can deal with multistage problems of optimization and equilibrium in a stochastic setting which involves actions that respond to increasing levels of information. Moreover, Rockafellar and Sun [23, 24] extended the progressive hedging method to solve multistage SVIs.
Under (2.27), suppose \((x^*(\xi ), y^*(\xi ), w_1^*(\xi ))\) (\(w_2(\xi )\equiv 0\)) is a solution of (2.26) with \(x^*(\xi )=x^*\) for all \(\xi \in \varXi \) and \({E}[w_1^*(\xi )]=0\). Then, taking expectation at first line of (2.26), we have (2.16)–(2.17). Conversely, suppose \((x^*, y^*(\xi ))\) is a solution of (2.16)–(2.17). By (2.16), \({E}[\varPhi (x^*, y^*(\xi ), \xi )]\in N_{D}(x^*)\), there exists \(w_1^*(\xi )\) such that \(\varPhi (x^*, y^*(\xi ), \xi )w_1^*(\xi )\in N_{D}(x^*)\) and \({E}[w_1^*(\xi )]=0\), which implies \((x^*(\xi ), y^*(\xi ), w_1^*(\xi ))\) with \(x^*(\xi )=x^*\) and \({E}[w_1^*(\xi )]=0\) is a solution of (2.26). Then, (2.16)–(2.17) and (2.26) are equivalence.
Definition 3
3 Algorithms and Approximation Methods
In this section, we consider the algorithms and approximation methods for twostage and multistage SVI. We first introduced the ERM formulation of (2.20)–(2.21) as follows.
3.1 ERM Solution Procedure
Definition 4
 1.
\(r(u,\xi )\geqslant 0\) for all \(u\in C(\xi )\), \( \text{ a.e. } \xi \in \varXi \);
 2.For any \(u: C\times \varXi \rightarrow \mathbb {R}^n\), it holds that$$\begin{aligned}&0\in F( {\bar{u}}(x, \xi ), \xi )+{\mathcal {N}}_{C(\xi )}({\bar{u}}(x, \xi )) \\&\quad \Leftrightarrow r({\bar{u}}(x,\xi ),\xi ) = 0 \text{ and } {\bar{u}}(x,\xi )\in C(\xi ), \text{ for } \text{ a.e. } \xi \in \varXi . \end{aligned}$$
Assumption 1
Assume (i) W has full row rank and (ii) \(C(\xi )\subseteq K\), a compact convex set for all \(\xi \).
Theorem 1
Theorem 1 means that problem (3.2) is a twostage stochastic program with complete recourse. However, the objective function of problem (3.2) involves minimizers of constrained quadratic programs for \(\xi \in \varXi \) and is not necessarily differentiable even when the sample is finite.
Assumption 2
The functions \(F(\cdot , \xi )\) and \(G(\cdot )\) are continuously differentiable for all \(\xi \in \varXi \). Moreover, for any compact set \(Y\subset \mathbb {R}^m\), there are functions \(d, \rho : \varXi \rightarrow \mathbb {R}_+\) such that \(\Vert F(y, \xi )\Vert \leqslant d_\xi \) and \(\Vert \nabla F(y, \xi )\Vert \leqslant \rho _\xi \) for all \(y\in Y\), where \(d\in L_1^\infty \) and \(\rho \in L_1^1\).
Lemma 1
Theorem 2
Note that since problem (3.4) is equivalent to (3.5), we can replace (3.4) by (3.5) in Theorem 2.
Under Assumptions 12, similar as Lemma 1, Chen et. al. [20] proved the smoothness of the objective function in the secondstage problem of (3.5) ([20, Proposition 3.8]). Moreover, they also considered the convexity of (3.5) ([20, Proposition 3.9 and Corollary 3.11]).
Theorem 3
(Convergence theorem) Suppose Assumptions 1–2 hold. Then, \(\phi _N\) converges to \(\phi \) a.s.uniformly on the compact set \({\bar{D}}\) such that \(S, S^*\subseteq {\bar{D}}\). Let \(\{x_N\}\) be a sequence of minimizers of problem (3.6) generated by iid samples. Then, \(\{x_N\}\) is \({\mathcal {P}}\)a.e. bounded and any accumulation point \(x^*\) of \(\{x_N\}\) as \(v\rightarrow \infty \) is \({\mathcal {P}}\)a.e. a solution of (3.5).
3.2 Progressive Hedging Algorithm for SVI
The convergence of Algorithm 1 is given in [23] as follows:
Theorem 4
[23, Theorem 2] As long as the (monotone) variational inequality (2.30) has at least one solution, the sequence of pairs \((x^{\nu }(\cdot ), w^{\nu }(\cdot ))\) generated by Algorithm 1 will converge to pair \(({\bar{x}}(\cdot ), {\bar{w}}(\cdot ))\) satisfying (2.31) and thus furnish \({\bar{x}}(\cdot )\) as a solution to (2.30). The decrease will surely be at a linear rate if, in particular, the sets \(C(\xi )\) are polyhedral and functions \(F(\cdot , \xi )\) are affine.
This version of progressive hedging inherits from the one for general stochastic variational inequalities in the preceding section the property that, as long as a solution exists, the sequence of iterates \((x^{\nu }(\cdot ), y^{\nu }(\cdot ), w^{\nu }(\cdot ), z^{\nu }(\cdot ))\) will converge to a particular solution \(({\bar{x}}(\cdot ),{\bar{y}}(\cdot ),{\bar{w}}(\cdot ),{\bar{z}}(\cdot ))\). They also consider the variant version of the algorithm when the parameters r are different in the x part and y part and apply Algorithm 2 to multistage stochastic optimization problem.
3.3 Discrete Approximation Methods
When random vectors follow a continuous distribution, the PHA cannot be applied to the twostage SVI. In this case, Chen et al. [30] proposed a discrete approximation method for twostage SLCP and Chen et al. [31] investigated the sample average approximation (SAA) method for twostage stochastic generalized equation (SGE).
Discrete approximation for twostage SLCP For the twostage SLCP (2.18)–(2.19), Chen et al. [30] firstly investigated the existence and uniqueness of a solution under the assumptions as follows.
Assumption 3
Under Assumption 3, some properties of the twostage SLCP are given in [30] as follows:
Proposition 1
 (i)
The twostage SLCP (2.18)–(2.19) has a unique solution \((x^*, y^*(\cdot ))\in \mathbb {R}^n\times \mathcal{Y}\).
 (ii)
 (iii)The first equation of SLCP (2.18)–(2.19) can be reformulated aswhere$$\begin{aligned} 0\leqslant & {} x \; \perp \; (A  {E}[B(\xi ) W(x,\xi )N(\xi )])x \nonumber \\&{E}[B(\xi ) W(x,\xi )q_2(\xi )] + q_1\geqslant 0, \end{aligned}$$(3.16)$$\begin{aligned} \Vert (A  {E}[B(\xi ) W(x,\xi )N(\xi )])^{1}\Vert \leqslant \frac{1}{{E}[\kappa (\xi )]}<+\infty . \end{aligned}$$
 (iv)LetThen, F is Lipschitz continuous and every matrix \(V_x\) in the Clarke generalized Jacobian \(\partial F(x)\) (see definition in [32, Section 2.6]) is nonsingular with \(\Vert V_x^{1}\Vert \leqslant {\bar{d}}\) for some constant \({\bar{d}}>0\) which is independent of x.$$\begin{aligned} F(x):= & {} \min \big (x, (A  {E}[B(\xi ) W(x,\xi )N(\xi )])x\nonumber \\&{E}[B(\xi ) W(x,\xi )q_2(\xi )] + q_1\big ). \end{aligned}$$(3.17)
Besides the existence and uniqueness of the twostage SVI, the globally Lipschitz continuity and formulation (3.15) of \({\bar{y}}(\cdot , \xi )\) allow us to substitute \({\bar{y}}(\cdot , \xi )\) into the firststage stochastic function \(A x + {E}[B(\xi )y(\xi )] + q_1\) and rewrite the twostage SVI as onestage SVI (3.16). Then, they proposed their discrete approximation method in [30] as follows.
Theorem 5
 (i)
 (ii)
If, in addition, \(\max _{i\in \{1, \ldots , K\}} \varDelta (\varXi _i^K) \rightarrow 0\), then \(\{(x^K, y^K( \cdot ))\}\) is bounded on \(\mathbb {R}^n\times \mathcal{Y}\), where the boundedness of \(y^K( \cdot )\) is in the sense of the norm topology of \({\mathcal {L}}_1(\mathcal{Y})\).
 (iii)
\(\{x^K, y^K( \cdot )\}\) converges to the true solution \((x^*, y^*( \cdot ))\) of problem (2.18)–(2.19), where the convergence of \(\{y^K( \cdot )\} \rightarrow y^*( \cdot ) \) is in the sense of the norm topology of \({\mathcal {L}}_2(\mathcal{Y})\).
To establish the quantitative convergence analysis, we need more assumptions.
Assumption 4
\(M(\cdot )\), \(N(\cdot )\), \(q_2(\cdot )\) and \(B(\cdot )\) are Lipschitz continuous over a compact set containing \(\varXi \) with Lipschitz constant L.
Theorem 6
Assumption 5
For a.e. \(\xi \in \varXi \), problem (3.29) has a unique solution for any \(x\in \mathcal{X}\).
Assumption 6
For every \(\xi \) and \(x\in {\bar{\mathcal{X}}}(\xi )\), there is a neighborhood \(\mathcal{V}\) of x and a measurable function \(v(\xi )\) such that \(\Vert {\hat{y}}(x',\xi )\Vert \leqslant v(\xi )\) for all \(x'\in \mathcal{V}\cap {\bar{\mathcal{X}}}(\xi )\).
Lemma 2
Suppose that Assumptions 5 and 6 hold, and for a.e. \(\xi \in \varXi \) the multifunction \(\Gamma _2(\cdot ,\xi )\) is closed. Then for a.e. \(\xi \in \varXi \), the solution \({\hat{y}}(x,\xi )\) is a continuous function of \(x\in \mathcal{X}\).
Assumption 7
For any \(\delta \in (0,1)\), the multifunction \(\varDelta _\delta (\cdot )\) is outer semicontinuous.
The following lemma shows that this assumption holds under mild conditions.
Lemma 3
Suppose \(\varPsi (\cdot ,\cdot ,\cdot )\) is continuous, \(\Gamma _2(\cdot ,\cdot )\) is closed and Assumption 6 holds. Then, the multifunction \(\varDelta _\delta (\cdot )\) is outer semicontinuous.
Then, the almost sure convergence result is given as follows:
Theorem 7
 (a)
\({{\mathfrak {d}}}_N\rightarrow 0\) and \({\mathbb {D}}({\hat{{\mathcal {S}}}}_N,{\mathcal {S}}^*)\rightarrow 0\) w.p.1 as \(N\rightarrow \infty \).
 (b)In addition, assume that: (vi) for any \(\delta >0\), \(\tau >0\) and a.e. \(\omega \in \varOmega \), there exist \(\gamma >0\) and \(N_0\) such that for any \(x\in \mathcal{X}\cap X' + \gamma \,\mathcal{B}\) and \(N\geqslant N_0\), there exists \(z_x\in \mathcal{X}\cap X'\) such that^{1}Then, w.p.1 for N large enough it follows that$$\begin{aligned} \Vert z_x  x\Vert \leqslant \tau ,\;\; \Gamma _1(x) \subseteq \Gamma _1(z_x) + \delta \mathcal{B},\;\; \mathrm{and}\;\; \Vert {\hat{\phi }}_N(z_x)  {\hat{\phi }}_N(x)\Vert \leqslant \delta . \end{aligned}$$(3.34)where for \(\epsilon \geqslant 0\) and \(t\geqslant 0\),$$\begin{aligned} {\mathbb {D}}(\hat{{\mathcal {S}}}_N, {\mathcal {S}}^*) \leqslant \tau + R^{1}\left( \,\sup _{x\in \mathcal{X}\cap X'} \Vert \phi (x)  {\hat{\phi }}_N(x)\Vert \right) , \end{aligned}$$(3.35)$$\begin{aligned}&R(\epsilon ):= \inf _{x\in \mathcal{X}\cap X',\, d(x, {\mathcal {S}}^*)\geqslant \epsilon } d\big (0, \phi (x) + \Gamma _1(x)\big ), \\&R^{1}(t): = \inf \{ \epsilon \in \mathbb {R}_+: R(\epsilon ) \geqslant t \}. \end{aligned}$$
Note that in the case when \(\Gamma _1(\cdot ) := {\mathcal {N}}_D(\cdot )\) with a nonempty polyhedral convex set D, the first and second inequalities of (3.34) hold automatically.
To drive the exponential rate of convergence based on uniform large deviations theorem (cf., [33, 34, 35]), more assumptions are needed.
Assumption 8
For a.e. \(\xi \in \varXi \), there exists a unique, parametrically CDregular [36] solution \({\bar{y}} = {\hat{y}}({\bar{x}}, \xi )\) of the secondstage generalized equation (2.17) for all \({\bar{x}} \in \mathcal{X}\).
Assumption 9
Assumption 10
For every \(x\in \mathcal{X}\) and \(i=1, \cdots , n\), the moment generating functions \(M^i_x(t)\) and \(M_\kappa (t)\) have finite values for all t in a neighborhood of zero.
Theorem 8
 (a)For sufficiently small \(\epsilon >0\), there exist positive constants \(\varrho =\varrho (\epsilon )\) and \(\varsigma =\varsigma (\epsilon )\), independent of N, such that$$\begin{aligned} P \left\{ \sup _{x\in \mathcal{X}}\big \Vert {\hat{\phi }}_N(x)\phi (x)\big \Vert \geqslant \epsilon \right\} \leqslant \varrho (\epsilon ) \mathrm{e}^{N\varsigma (\epsilon )}. \end{aligned}$$(3.36)
 (b)Assume in addition: (iv) the condition of part (b) in Theorem 7 holds and w.p.1 for N sufficiently large,(v) \(\phi (\cdot )\) has the following strong monotonicity property for every \(x^*\in {\mathcal {S}}^*\):$$\begin{aligned} {\mathcal {S}}^*\cap \mathrm{cl}\big (\mathrm{bd}(\mathcal{X}) \cap \mathrm{int}({\bar{\mathcal{X}}}_N)\big ) =\varnothing . \end{aligned}$$(3.37)where \(g:\mathbb {R}_+\rightarrow \mathbb {R}_+\) is such a function that function \({{\mathfrak {r}}}(\tau ):=g(\tau )/\tau \) is monotonically increasing for \(\tau >0\).$$\begin{aligned} (xx^*)^\top (\phi (x)\phi (x^*)) \geqslant g(\Vert xx^*\Vert ),\;\forall x\in \mathcal{X}, \end{aligned}$$(3.38)Then, \({\mathcal {S}}^*=\{x^*\}\) is a singleton, and for any sufficiently small \(\epsilon >0\), there exists N sufficiently large such thatwhere \(\varrho (\cdot )\) and \(\varsigma (\cdot )\) are defined in (3.36), and \({{\mathfrak {r}}}^{1}(\epsilon ):=\inf \{\tau >0:{{\mathfrak {r}}}(\tau )\geqslant \epsilon \}\) is the inverse of \({{\mathfrak {r}}}(\tau )\).$$\begin{aligned} P \left\{ {\mathbb {D}}(\hat{{\mathcal {S}}}_N, {\mathcal {S}}^*)\geqslant \epsilon \right\} \leqslant \varrho \left( {{\mathfrak {r}}}^{1}(\epsilon )\right) \exp \left( N\varsigma \big ({{\mathfrak {r}}}^{1}(\epsilon )\big )\right) , \end{aligned}$$(3.39)
Assumption 11
Theorem 9
 (a)
The secondstage SNCP (3.41) has a unique solution \({\hat{y}}(x, \xi )\) which is parametrically CDregular and the mapping \(x \mapsto {\hat{y}}(x, \xi )\) is Lipschitz continuous over \(\mathcal{X}\cap X'\), where \(X'\) is a compact subset of \(\mathbb {R}^n\).
 (b)The Clarke Jacobian of \({\hat{y}}(x, \xi )\) w.r.t. x is as followswhere \(M(x, y, \xi ) = \nabla _y \varPsi (x, y, \xi )\), \(L(x, {\hat{y}}(x, \xi ), \xi ) = \nabla _x \varPsi (x, {\hat{y}}(x, \xi ), \xi )\),$$\begin{aligned} \begin{array}{lll} &{}\partial {\hat{y}}(x,\xi )= \text{ conv }\left\{ \displaystyle {\lim _{z\rightarrow x}} \nabla _z {\hat{y}}(z,\xi ) : \nabla _z {\hat{y}}(z, \xi )\right. \\ &{}\left. ~~~\,~~\;\;\;\;\;\;\;= [I  J_{\alpha }(IM(z, {\hat{y}}(z,\xi ), \xi ))]^{1}D_{\alpha }L(z, {\hat{y}}(z,\xi ), \xi ) \right\} , \end{array} \end{aligned}$$\(J_{\alpha } \) is an mdimensional diagonal matrix and$$\begin{aligned} \alpha = \{i:({\hat{y}}(x, \xi ))_i>(\varPsi (x, {\hat{y}}(x, \xi ), \xi ))_i\}, \end{aligned}$$$$\begin{aligned} (J_{\alpha })_{jj} : = \left\{ \begin{array}{ll} 1, &{}\quad \mathrm{if } \; j\in \alpha ,\\ 0, &{}\quad \mathrm{otherwise }. \end{array} \right. \end{aligned}$$(3.44)
Assumption 12
For a.e. \(\xi \in \varXi \), \(\Theta (x, y(\xi ), \xi )\) is strongly monotone with parameter \(\kappa (\xi )\) at \((x, y(\cdot ))\in C\times \mathcal{Y}\), where \({E}[\kappa (\xi )] < +\infty \).
Theorem 10
 (a)
For any \((x, y(\cdot ))\in \mathrm{Sol}^*\), every matrix in \(\partial {\hat{\varPhi }}(x)\) is positive definite, and \({\hat{\varPhi }}\) and \(\phi \) are strongly monotone at x.
 (b)
Any solution \(x^*\in {\mathcal {S}}^*\cap X'\) of SVI (3.45) is CDregular and an isolate solution.
 (c)
Moreover, if replacing conditions (i) and (ii) by (iv) Assumption 12 holds over \(\mathbb {R}^n\times \mathcal{Y}\), then SVI (3.45) has a unique solution \(x^*\) and the solution is CDregular.
Here the definition of CDregular can be found in [36].
Theorem 11
Theorem 12
Theorem 13
4 Applications
Twostage SVI has wide applications in economics, traffic network, electricity markets, supply chain problems, finance, risk management under uncertain environment. One type of twostage SVI involves making a “hereandnow” decisions at the present time to meet the uncertainty that are revealed at a later time. This is one of a motivation of both twostage stochastic optimization and twostage stochastic Nash equilibrium problem (SNEP), such as Example 2. [37] discussed a scenariobased dynamic oligopolistic problem under uncertainty. In electricity market, [15, 38, 39, 40] considered capacity expansion problem under uncertain environment. [41] investigated the supplyside risk in uncertainty power market. [42, 43] discussed twosettlement markets consisting of a deterministic (firststage) spot market and a stochastic (secondstage) market known as the forward. [44] presented a stochastic complementarity model of equilibrium in an electric power market with uncertainty power demand. [45] presented Nash equilibrium models of perfectly competitive capacity expansion involving riskaverse participants in the presence of discrete state uncertainties and pricing mechanisms of different kinds. [46] modeled a production and supply competition of a homogenous product under uncertainty in an oligopolistic market by a twostage SVI, and used the model to describe the market share observation in the world market of crude oil. Here we introduce two important applications which form traffic equilibrium problems and noncooperative multiagent games under uncertain environment.
Noncooperative multiagent games In [48], Pang et al. formally introduced and studied a noncooperative multiagent game under uncertainty and focused mainly on a twostage setting of the game where each agent is riskaverse as follows.
When the risk measure \({\mathcal {R}}_i={E}\), the players are riskneutral. Moreover, the risk measure \({\mathcal {R}}_i\), \(i=1, \cdots , n\) can be considered as several deviation measures include the standard deviation, lower and upper semideviations, mean absolute (semi)deviations, absolute (semi)deviations, the median and deviations derived from the CVaR. They then investigated several properties of meandeviation composite games with quadratic recourse, such as the continuity, regularization and differentiability of the secondstage optimal value function, and the reformulation of meandeviation composite game.
Pang et al. [48] bypassed the SVI framework and dealt with the riskaverse SNEP based largely on smoothing, regularization, sampled solution approach and the bestresponse scheme. But now, by [23, 24] and the sample average approximation method [31], we can solve the twostage SVI by the progressive hedging method.
5 Final Remarks

Can we solve nonmonotone twostage SVI using PHM?

How could we achieve better convergence rate by using sampling technology?

How to extend the discrete approximation methods to multistage SVI when The random vector follows a continuous distribution?

How to extend the SIV to dynamic twostage SVI?

How could we solve the twostage or even multistage SVI more effectively?

In the case when we only have limited information about the distribution of random vectors, could we model the twostage variational inequalities in the sense of distributional robustness or even robustness?
Footnotes
 1.
Recall that \({\hat{\phi }}_N(x)={\hat{\phi }}_N(x,\omega )\) are random functions defined on the probability space \((\varOmega ,\mathcal{F},{\mathcal {P}})\).
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