© Wuhan University 2024

0 Introduction

The deterministic variational inequality (VI) represents the first-order optimality conditions of optimization problems and models equilibrium problems, which plays a crucial role in optimization and operations research. For more details, see Refs.[1-3] and the references therein. However, in many real-world applications, such as economics, management, science, and engineering, the decision-makers have to make a decision or sequential decisions under uncertainties environment. In these situations, deterministic VI may not be suitable. Motivated by these applications, stochastic versions of various problems have arisen in the past several decades. In recent years, interest in stochastic variational inequalities (SVIs) has been revived among the optimization community^[4-13]. In particular, very recently, two-stage SVIs, where the decision-makers are confronted with two consecutive stages of uncertainties, are gaining ever-increasing popularity^[14-19]. Due to their strong modeling capabilities, two-stage SVIs and multistage SVIPs successfully capture a wide range of practical applications. In 2017, Chen et al^[15] proposed the model of two-stage nonlinear SVIs. For the recent development in two-stage and multistage SVIs, we refer to Refs. [15-20].

In practice, due to the inaccurate distribution of random variables, people may need to consider the stability of the solutions caused by distribution perturbations. In 2020, Jiang et al^[21] discussed the quantitative stability for a class of two-stage linear SVI-SCP problems with box-constrained. Afterward, Liu et al^[22] concentrated on the quantitative stability analysis of two-stage stochastic linear variational inequality problems with fixed recourse. Based on above works, in this paper, we consider the quantitative stability analysis of the following two-stage stochastic linear variational inequality problem:

$\{\begin{array}{l}\mathrm{0}\in \mathit{A}x+{\mathbb{E}}_p\left[\mathit{B}\left(\xi \right)y\left(\xi \right)\right]+{\mathit{q}}_{\mathrm{1}}+{\mathcal{N}}_S\left(x\right),\\ \mathrm{0}\le y\left(\xi \right)\perp \mathit{M}\left(\xi \right)y\left(\xi \right)+\mathit{N}\left(\xi \right)x+{\mathit{q}}_{\mathrm{2}}\left(\xi \right)\ge \mathrm{0},\mathrm{f}\mathrm{o}\mathrm{r}P-\mathrm{a}.\mathrm{e}.\xi \in \mathrm{\Xi },\end{array}$(1)

where $\mathit{A}\in {\mathbb{R}}^{n\times n}$, ${\mathit{q}}_{\mathrm{1}}\in {\mathbb{R}}^n$, $\mathit{M}(\cdot )\in {\mathbb{R}}^{m\times m}$, $\mathit{B}(\cdot ):{\mathbb{R}}^l\to {\mathbb{R}}^{n\times m}$, $\mathit{N}(\cdot ):{\mathbb{R}}^l\to {\mathbb{R}}^{m\times n}$, $\mathit{q}(\cdot ):{\mathbb{R}}^l\to {\mathbb{R}}^m$ are all matrix/vector-valued mappings, $\xi :\mathrm{\Omega }\to {\mathbb{R}}^l$ is defined in the space $(\mathrm{\Omega },\mathrm{ℱ},P)$, ${\mathbb{E}}_P$ is the mathematical expectation, $S\in {\mathbb{R}}^n$ is closed and convex, ${{N}}_S(x)$ denotes the normal cone to $S$ at $x$, and abbreviation a.e. stands for "almost every".

We can see that problem (1) is a generalized model of that in Ref. [21] and Ref. [22], where the authors assume that $\mathit{M}\left(\xi \right)$ is fixed and the constraint set $S$ is a bounded box set. It should be mentioned that the results in this paper are not a trivial extension from Ref. [21] and Ref. [22]. The main reason is that we need to consider the case that $\mathit{M}\left(\xi \right)$ is unfixed and $S$ is unbounded simultaneously.

As a same discussion in Ref. [22], we know that under the assumption that the second stage problem of (1) has a unique solution $y^{\mathrm{*}}\left(x,\xi \right)$, problem (1) is equivalent to the following optimization problem

$\underset{x\in S}{\mathrm{m}\mathrm{i}\mathrm{n}}{f}_P(x)$(2)

where $f_P:{\mathbb{R}}^n\to {\mathbb{R}}_{+}$ is the residual function:

$f_P\left(x\right):={\Vert x-P_S\left(x-\mathit{A}x-{\mathbb{E}}_P\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]-{\mathit{q}}_{\mathrm{1}}\right)\Vert }^{\mathrm{2}}.$(3)

The following notation is adopted. We let $\mathit{I}$ denote the identity matrix, ${\text{B}}$ denote the closed unit ball centered on zero, $\Vert \cdot \Vert$ denote the Euclidean norm, $S_{\mathrm{\infty }}$ denote the recession cone of a set $S$, and $K^{\mathrm{*}}$ denote the dual cone of a cone $K$. Let $P_k(\mathrm{\Xi }):=\{P\in P(\mathrm{\Xi }):{\mathbb{E}}_p\left[||\xi |{|}^k\right]<+\mathrm{\infty }\},d(a,\mathit{ℬ}):=\mathrm{i}\mathrm{n}{{\mathrm{f}}_b}_{\in \mathit{ℬ}}||a-b||$ and $d\left({A},\mathit{ℬ}\right):=\mathrm{s}\mathrm{u}{\mathrm{p}}_{a\in {A}}\mathrm{i}\mathrm{n}{\mathrm{f}}_{b\in \mathit{ℬ}}||a-b||$. For a given matrix $\mathit{A}$, ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(\mathit{A}\right)$ represents the minimal eigenvalue of $\mathit{A}$.

1 Preliminaries

In this section, we introduce a class of pseudo metrics known as the $\zeta$-structure metric^[23,24].

Definition 1   (Ref. [25], $\zeta$-structure metrics) Let $\mathrm{\mathscr{H}}$ be a set of real-value measurable functions on $\mathrm{\Xi }$. For any two probability measures $P,Q\in {P}\left(\mathrm{\Xi }\right)$, the function

${\mathbb{D}}_{\mathrm{\mathscr{H}}}(P,Q):=\underset{h\in \mathrm{\mathscr{H}}}{\mathrm{s}\mathrm{u}\mathrm{p}}\left|{\mathbb{E}}_P[h(\xi )]-{\mathbb{E}}_Q[h(\xi )]\right|$

is called the $\zeta$-structure metric between $P$ and $Q$ induced by $\mathrm{\mathscr{H}}$. Also, $\mathrm{\mathscr{H}}$ is called the generator of ${\mathbb{D}}_{\mathrm{\mathscr{H}}}(\cdot ,\cdot )$.

For $p\ge \mathrm{1}$, if we take

${\mathit{H}}_{\mathrm{F}{\mathrm{M}}_p}:=\left\{h:\mathrm{\Xi }\to \mathbb{R}:\left|h({\xi }_{\mathrm{1}})-h\left({\xi }_{\mathrm{2}}\right)\right|\le \mathrm{m}\mathrm{a}\mathrm{x}\left\{\mathrm{1},{\Vert {\xi }_{\mathrm{1}}\Vert }^{\mathrm{p}-\mathrm{1}},{\Vert {\xi }_{\mathrm{2}}\Vert }^{\mathrm{p}-\mathrm{1}}\right\}\Vert {\xi }_{\mathrm{1}}-{\xi }_{\mathrm{2}}\Vert \right\},$

then we obtain the following $p$-th order Fortet-Mourier metric

${\zeta }_p(P,Q):=\underset{h\in {\mathrm{\mathscr{H}}}_{\mathrm{F}{\mathrm{M}}_p}}{\mathrm{s}\mathrm{u}\mathrm{p}}\left|{\mathbb{E}}_P[h(\xi )]-{\mathbb{E}}_Q[h(\xi )]\right|,$

which is widely used in the stochastic programming problems^[22-24].

Let $S^{\mathrm{*}}\left(P\right)$ and $v\left(p\right)$ be the solution set and optimal value of (2), respectively. The growth function of problem (2) is defined by

${\psi }_P(\pi ):=\mathrm{m}\mathrm{i}\mathrm{n}\{f_P(x):d(x,S^{\mathrm{*}}(P))\ge \pi ,x\in S\}.$

Its inverse function is

${\psi }_P^{-\mathrm{1}}(t):=\mathrm{s}\mathrm{u}\mathrm{p}\{\pi \in {\mathbb{R}}_{+}:{\psi }_P(\pi )\le t\}.$(4)

The following proposition plays a key role in this paper.

Proposition 1   (Ref. [26]) Let $\mathit{M}\left(\xi \right)$ be a P-matrix (all principal minors are positive) for every $\xi \in \mathrm{\Xi }.$ Then,

(i) The second stage problem of (1) has a unique solution $y^{\mathrm{*}}\left(x,\xi \right)$, and

$y^{\mathrm{*}}\left(x,\xi \right)=-W\left(x,\xi \right)\left(\mathit{N}\left(\xi \right)x+{\mathit{q}}_{\mathrm{2}}\left(\xi \right)\right),$(5)

where $W(x,\xi ):=[\mathit{I}-D(x,\xi )(\mathit{I}-\mathit{M}{(\xi ))]}^{-\mathrm{1}}D(x,\xi )$ and

$D_{jj}(x,\xi )=\{\begin{array}{cc}\mathrm{1},& \mathrm{i}\mathrm{f}(\mathit{M}(\xi )y^{\mathrm{*}}(x,\xi )+\mathit{N}(\xi )x+{\mathit{q}}_{\mathrm{2}}{(\xi ))}_j\le y_j^{\mathrm{*}}(x,\xi ),\\ \mathrm{0},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\end{array}$

$\mathrm{f}\mathrm{o}\mathrm{r}j=\mathrm{1},\cdots ,m;$

(ii)

$\Vert y^{\mathrm{*}}(x_{\mathrm{1}},\xi )-y^{\mathrm{*}}(x_{\mathrm{2}},\xi )\Vert \le \underset{{}_{J\in {J}}}{\mathrm{m}\mathrm{a}\mathrm{x}}\Vert {\mathit{M}}_{J\times J}^{-\mathrm{1}}(\xi )\Vert \Vert \mathit{N}(\xi )\Vert \Vert x_{\mathrm{1}}-x_{\mathrm{2}}\Vert ,$(6)

where ${\mathit{M}}_{J\times J}(\xi )$ is the sub-matrix of $\mathit{M}\left(\xi \right)$, and $J$ denotes the power set of $\{\mathrm{1,2},\cdots ,m\}.$

2 Quantitative Stability Analysis

2.1 Existence of Solutions

We need the following assumption in this section.

Assumption 1 Let $\mathit{M}\left(\xi \right)$ be a P-matrix for every $\xi \in \mathrm{\Xi }$. Moreover, there exists a continuous function ${\kappa }_{\mathit{M}}\left(\xi \right):\mathrm{\Xi }\to {\mathbb{R}}_{++}$, such that

$\underset{J\in {J}}{\mathrm{m}\mathrm{a}\mathrm{x}}\Vert \begin{array}{c}{\mathit{M}}_{J\times J}^{-\mathrm{1}}\left(\xi \right)\end{array}\Vert \le \frac{\mathrm{1}}{{\kappa }_{\mathit{M}}\left(\xi \right)}$

for any $\xi \in \mathrm{\Xi }.$

In the sequel, we will study the existence of solutions to problem (1) and its distribution perturbed problem under $Q$, i.e.,

$\{\begin{array}{l}\mathrm{0}\in \mathit{A}x+{\mathbb{E}}_Q\left[\mathit{B}\left(\xi \right)y\left(\xi \right)\right]+{\mathit{q}}_{\mathrm{1}}+{{N}}_S\left(x\right),\\ \mathrm{0}\le y\left(\xi \right)\perp \mathit{M}\left(\xi \right)y\left(\xi \right)+\mathit{N}\left(\xi \right)x+{\mathit{q}}_{\mathrm{2}}\left(\xi \right)\ge \mathrm{0},\mathrm{f}\mathrm{o}\mathrm{r}Q-\mathrm{a}.\mathrm{e}.\xi \in \mathrm{\Xi }.\end{array}$(7)

In the following, we employ the concept of pseudo monotonicity to establish the existence assertion. For this purpose, we need the definition of pseudo monotonicity. For $O=P,Q$, define the mapping ${\mathit{\Phi }}_O:{\mathbb{R}}^n\to {\mathbb{R}}^n$ as

${\mathit{\Phi }}_O\left(x\right)=\mathit{A}x+{\mathbb{E}}_O\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]+{\mathit{q}}_{\mathrm{1}}.$

Recall that ${\mathit{\Phi }}_O$ is pseudo monotone (Ref. [2], Definition 2.3.1) if

$〈x_{\mathrm{1}}-x_{\mathrm{2}},{\mathit{\Phi }}_O\left(x_{\mathrm{2}}\right)〉\ge \mathrm{0}\Rightarrow 〈x_{\mathrm{1}}-x_{\mathrm{2}},{\mathit{\Phi }}_O\left(x_{\mathrm{1}}\right)〉\ge \mathrm{0}.$

The following proposition presents the existence of solutions to problem (1).

Proposition 2   Suppose that Assumption 1 hold. Then the following statement hold.

(i) If $S\in {\mathbb{R}}^n$ is bounded and

${\int }_{\Xi }\Vert \mathit{B}\left(\xi \right)\Vert \underset{J\in {J}}{\mathrm{m}\mathrm{a}\mathrm{x}}\Vert \begin{array}{c}{\mathit{M}}_{J\times J}^{-\mathrm{1}}\left(\xi \right)\end{array}\Vert \Vert \mathit{N}\left(\xi \right)\Vert P\left(\mathrm{d}\xi \right)<\mathrm{\infty }.$

Then, problem (1) has a nonempty solution set.

(ii) If ${\mathit{\Phi }}_P$ is pseudo monotone on $S$ and there exists a vector ${\mathit{x}}^{\mathrm{r}\mathrm{e}\mathrm{f}}\in S$ satisfying ${\mathit{\Phi }}_P\left({\mathit{x}}^{\mathrm{r}\mathrm{e}\mathrm{f}}\right)\in \mathrm{i}\mathrm{n}\mathrm{t}(S_{\mathrm{\infty }}{)}^{\mathrm{*}}$. Then, problem (1) has a nonempty, convex and compact solution set.

Proof   (i) The proof is similar to that in Ref. [22], we omit it. (ii) It directly follows from Theorem 2.3.5 in Ref. [2].

Assumption 2 Suppose that the random coefficients in problem (1) depend affine linearly on $\xi =\left({\xi }^{\mathrm{1}},{\xi }^{\mathrm{2}},\cdots ,{\xi }^l\right)$$\in \mathrm{\Xi }$, i.e.

$\mathit{\Lambda }(\xi )={\mathit{\Lambda }}_{\mathrm{0}}+{\xi }^{\mathrm{1}}{\mathit{\Lambda }}_{\mathrm{1}}+{\xi }^{\mathrm{2}}{\mathit{\Lambda }}_{\mathrm{2}}+\cdots +{\xi }^l{\mathit{\Lambda }}_l,$

where $\mathit{\Lambda }\left(\xi \right)=\mathit{B}\left(\xi \right),\mathit{M}\left(\xi \right),\mathit{N}\left(\xi \right)$ or ${\mathit{q}}_{\mathrm{2}}\left(\xi \right)$.

We first have the following result.

Lemma 1   Let Assumption 1 and Assumption 2 be satisfied, and ${\kappa }_{\mathit{M}}\left(\xi \right)\ge \kappa >\mathrm{0}$. Then for all ${\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}}\in \mathrm{\Xi }$$\mathrm{a}\mathrm{n}\mathrm{d}x\in S\bigcap r{\text{B}}$, there exists a constant $L>\mathrm{0}$ such that

$\Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{1}}\right)y^{\mathrm{*}}\left(x,{\xi }_{\mathrm{1}}\right)-\mathit{B}\left({\xi }_{\mathrm{2}}\right)y^{\mathrm{*}}\left(x,{\xi }_{\mathrm{2}}\right)\end{array}\Vert \le L\mathrm{m}\mathrm{a}\mathrm{x}{\left\{\mathrm{1},\Vert \begin{array}{c}{\xi }_{\mathrm{1}}\end{array}\Vert ,\Vert \begin{array}{c}{\xi }_{\mathrm{2}}\end{array}\Vert \right\}}^{\mathrm{2}}\Vert \begin{array}{c}{\xi }_{\mathrm{1}}-{\xi }_{\mathrm{2}}\end{array}\Vert .$

Proof   By Assumption 1 and ${\kappa }_{\mathit{M}}\left(\xi \right)\ge \kappa >\mathrm{0}$, for any ${\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}}\in \mathrm{\Xi }$, one has

$\begin{array}{l}\Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{1}}\right)y^{\mathrm{*}}\left(x,{\xi }_{\mathrm{1}}\right)-\mathit{B}\left({\xi }_{\mathrm{2}}\right)y^{\mathrm{*}}\left(x,{\xi }_{\mathrm{2}}\right)\end{array}\Vert \\ \le \Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \Vert \begin{array}{c}W\left(x,{\xi }_{\mathrm{1}}\right)\end{array}\Vert \Vert \begin{array}{c}\left(\mathit{N}\left({\xi }_{\mathrm{2}}\right)-\mathit{N}\left({\xi }_{\mathrm{1}}\right)\right)x+{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{2}}\right)-{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \\ +\Vert \mathit{B}\left({\xi }_{\mathrm{2}}\right)W\left(x,{\xi }_{\mathrm{2}}\right)-\mathit{B}\left({\xi }_{\mathrm{1}}\right)W\left(x,{\xi }_{\mathrm{1}}\right)\Vert \Vert (\mathit{N}\left({\xi }_{\mathrm{2}}\right)x+{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{2}}\right)\Vert \\ \le \Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \Vert \begin{array}{c}W\left(x,{\xi }_{\mathrm{1}}\right)\end{array}\Vert \left(\Vert \begin{array}{c}\mathit{N}\left({\xi }_{\mathrm{2}}\right)-\mathit{N}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \Vert x\Vert +\Vert \begin{array}{c}{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{2}}\right)-{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \right)\\ +\left(\Vert \mathit{B}\left({\xi }_{\mathrm{2}}\right)\Vert \Vert W\left(x,{\xi }_{\mathrm{2}}\right)-W\left(x,{\xi }_{\mathrm{1}}\right)\Vert +\Vert \mathit{B}\left({\xi }_{\mathrm{2}}\right)-\mathit{B}\left({\xi }_{\mathrm{1}}\right)\Vert \Vert W\left(x,{\xi }_{\mathrm{1}}\right)\Vert \right)(\Vert \mathit{N}\left({\xi }_{\mathrm{2}}\right)\Vert ||x||+\Vert {\mathit{q}}_{\mathrm{2}}({\xi }_{\mathrm{2}})\Vert )\\ \le \Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \underset{J\in {J}}{\mathrm{m}\mathrm{a}\mathrm{x}}\Vert \begin{array}{c}{\mathit{M}}_{J\times J}^{-\mathrm{1}}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \left(\Vert \begin{array}{c}\mathit{N}\left({\xi }_{\mathrm{2}}\right)-\mathit{N}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \Vert x\Vert +\Vert \begin{array}{c}{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{2}}\right)-{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \right)\\ +(\Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{2}}\right)\end{array}\Vert \left[\underset{J\in {J}}{\mathrm{m}\mathrm{a}\mathrm{x}}\Vert \begin{array}{c}{\mathit{M}}_{J\times J}^{-\mathrm{1}}\left({\xi }_{\mathrm{2}}\right)\end{array}\Vert +\underset{J\in {J}}{\mathrm{m}\mathrm{a}\mathrm{x}}\Vert \begin{array}{c}{\mathit{M}}_{J\times J}^{-\mathrm{1}}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \right]+\Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{2}}\right)-\mathit{B}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \underset{J\in {J}}{\mathrm{m}\mathrm{a}\mathrm{x}}\Vert \begin{array}{c}{\mathit{M}}_{J\times J}^{-\mathrm{1}}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert )\cdot \left(\Vert \begin{array}{c}(\mathit{N}\left({\xi }_{\mathrm{2}}\right)\end{array}\Vert \Vert x\Vert +\Vert \begin{array}{c}{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{2}}\right)\end{array}\Vert \right)\\ \le \frac{\mathrm{1}}{\kappa }\Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \left(\Vert \begin{array}{c}\mathit{N}\left({\xi }_{\mathrm{2}}\right)-\mathit{N}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \Vert x\Vert +\Vert \begin{array}{c}{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{2}}\right)-{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \right)+\left(\frac{\mathrm{2}}{\kappa }\Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{2}}\right)\end{array}\Vert +\frac{\mathrm{1}}{\kappa }\Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{2}}\right)-\mathit{B}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \right)\left(||\mathit{N}({\xi }_{\mathrm{2}})||||x||+{\mathit{q}}_{\mathrm{2}}({\xi }_{\mathrm{2}})||\right),\end{array}$

where the first inequality follows from (5), and the third inequality follows from the fact that $W\left(x,\xi \right)\le \underset{J\in {J}}{\mathrm{m}\mathrm{a}\mathrm{x}}\Vert \begin{array}{c}{\mathit{M}}_{J\times J}^{-\mathrm{1}}\left(\xi \right)\end{array}\Vert$. On the other hand, by Assumption 2 and $\Vert x\Vert \le r$, we have that there exists positive constant $L^{\text{'}}$ such that

$\begin{array}{l}\frac{\mathrm{1}}{\kappa }\Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \left(\Vert \begin{array}{c}\mathit{N}\left({\xi }_{\mathrm{2}}\right)-\mathit{N}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \Vert x\Vert +\Vert \begin{array}{c}{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{2}}\right)-{\mathit{q}}_{\mathrm{2}}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \right)\\ \le \frac{L\text{'}}{\kappa }(r+\mathrm{1})(\mathrm{1}+\Vert \begin{array}{c}{\xi }_{\mathrm{1}}\end{array}\Vert )\Vert \begin{array}{c}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}\end{array}\Vert \le \frac{\mathrm{2}L\text{'}}{\kappa }(r+\mathrm{1})\mathrm{m}\mathrm{a}\mathrm{x}\{\mathrm{1},\Vert \begin{array}{c}{\xi }_{\mathrm{1}}\end{array}\Vert \}\Vert \begin{array}{c}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}\end{array}\Vert \\ \le \frac{\mathrm{2}L\text{'}}{\kappa }(r+\mathrm{1})\mathrm{m}\mathrm{a}\mathrm{x}\{\mathrm{1},{\Vert \begin{array}{c}{\xi }_{\mathrm{1}}\end{array}\Vert ,\Vert \begin{array}{c}{\xi }_{\mathrm{2}}\end{array}\Vert \}}^{\mathrm{2}}\Vert \begin{array}{c}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}\end{array}\Vert .\end{array}$

Moreover, there exist positive constants $L^{″}$ and $L^{‴}$ such that

$\frac{\mathrm{2}}{k}\Vert \mathit{B}({\xi }_{\mathrm{2}})\Vert +\frac{\mathrm{1}}{K}\Vert \mathit{B}({\xi }_{\mathrm{2}})-\mathit{B}({\xi }_{\mathrm{1}})\Vert \le \frac{L\text{'}\text{'}}{k}(\mathrm{2}(\mathrm{1}+\Vert {\xi }_{\mathrm{2}}\Vert )+\Vert {\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}\Vert )\le \frac{\mathrm{2}L\text{'}\text{'}}{k}(\mathrm{1}+\Vert {\xi }_{\mathrm{2}}\Vert )\Vert {\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}\Vert$

and

$\Vert \mathit{N}({\xi }_{\mathrm{2}})\Vert \Vert x\Vert +\Vert {\mathit{q}}_{\mathrm{2}}({\xi }_{\mathrm{2}})\Vert \le L^{‴}(r+\mathrm{1})(\mathrm{1}+\Vert {\xi }_{\mathrm{2}}\Vert ).$

Therefore, we obtain that

$\begin{array}{l}\left(\frac{\mathrm{2}}{\kappa }\Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{2}}\right)\end{array}\Vert +\frac{\mathrm{1}}{\kappa }\Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{2}}\right)-\mathit{B}\left({\xi }_{\mathrm{1}}\right)\end{array}\Vert \right)\left(\Vert \mathit{N}({\xi }_{\mathrm{2}})\Vert \Vert x\Vert +\Vert {\mathit{q}}_{\mathrm{2}}({\xi }_{\mathrm{2}})\Vert \right)\\ \le \frac{\mathrm{2}{L}^{″}{L}^{‴}}{\kappa }\left(r+\mathrm{1}\right){\left(\mathrm{1}+\Vert \begin{array}{c}{\xi }_{\mathrm{2}}\end{array}\Vert \right)}^{\mathrm{2}}\Vert \begin{array}{c}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}\end{array}\Vert \le \frac{\mathrm{8}{L}^{″}{L}^{‴}}{\kappa }\left(r+\mathrm{1}\right)\mathrm{m}\mathrm{a}\mathrm{x}{\left\{\mathrm{1},\Vert \begin{array}{c}{\xi }_{\mathrm{2}}\end{array}\Vert \right\}}^{\mathrm{2}}\Vert \begin{array}{c}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}\end{array}\Vert \\ \le \frac{\mathrm{8}{L}^{″}{L}^{‴}}{\kappa }\left(r+\mathrm{1}\right)\mathrm{m}\mathrm{a}\mathrm{x}{\left\{\mathrm{1},\Vert \begin{array}{c}{\xi }_{\mathrm{1}}\end{array}\Vert ,\Vert \begin{array}{c}{\xi }_{\mathrm{2}}\end{array}\Vert \right\}}^{\mathrm{2}}\Vert \begin{array}{c}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}\end{array}\Vert .\end{array}$

To summarize the above estimation, we have that

$\Vert \begin{array}{c}\mathit{B}\left({\xi }_{\mathrm{1}}\right)y^{\mathrm{*}}\left(x,{\xi }_{\mathrm{1}}\right)-\mathit{B}\left({\xi }_{\mathrm{2}}\right)y^{\mathrm{*}}\left(x,{\xi }_{\mathrm{2}}\right)\end{array}\Vert \le \frac{\mathrm{2}\left(L^{\text{'}}+\mathrm{4}{L}^{″}{L}^{‴}\right)}{\kappa }\left(r+\mathrm{1}\right)\mathrm{m}\mathrm{a}\mathrm{x}{\{\mathrm{1},\Vert \begin{array}{c}{\xi }_{\mathrm{1}}\end{array}\Vert ,\Vert \begin{array}{c}{\xi }_{\mathrm{2}}\end{array}\Vert \}}^{\mathrm{2}}\Vert \begin{array}{c}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}\end{array}\Vert .$

We then complete the proof by letting $L:=\frac{\mathrm{2}\left(L^{\text{'}}+\mathrm{4}{L}^{″}{L}^{‴}\right)}{\kappa }\left(r+\mathrm{1}\right)$.

We continue to give some lemmas.

Lemma 2   Under the same conditions of Lemma 1 and let $P,Q\in {{P}}_{\mathrm{2}}\left(\mathrm{\Xi }\right)$. Then $\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)$ is integrable on Ξ under distributions $P$ and $Q$.

Proof   By Proposition 2 and the fact that $W\left(x,\xi \right)\le \underset{{}_{J\in {J}}}{\mathrm{m}\mathrm{a}\mathrm{x}}\Vert {\mathit{M}}_{J\times J}^{-\mathrm{1}}\left(\xi \right)\Vert$, we have

$\Vert \begin{array}{c}\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\end{array}\Vert =\Vert \begin{array}{c}\mathit{B}\left(\xi \right)\left(-W\left(x,\xi \right)\left(\mathit{N}\left(\xi \right)x+{\mathit{q}}_{\mathrm{2}}\left(\xi \right)\right)\right)\end{array}\Vert$

$\begin{array}{l}\le \Vert \mathit{B}\left(\xi \right)\Vert \Vert W\left(x,\xi \right)\Vert \Vert \mathit{N}\left(\xi \right)x+{\mathit{q}}_{\mathrm{2}}\left(\xi \right)\Vert \\ \le \underset{J\in {J}}{\mathrm{m}\mathrm{a}\mathrm{x}}\Vert {\mathit{M}}_{J\times J}^{-\mathrm{1}}\left(\xi \right)\Vert \left(\Vert \mathit{B}\left(\xi \right)\Vert \Vert \mathit{N}\left(\xi \right)\Vert \Vert x\Vert +\Vert \mathit{B}\left(\xi \right)\Vert \Vert {\mathit{q}}_{\mathrm{2}}\left(\xi \right)\Vert \right)\\ \le \frac{\mathrm{1}}{{\kappa }_{\mathit{M}}\left(\xi \right)}\left(\Vert \mathit{B}\left(\xi \right)\Vert \Vert \mathit{N}\left(\xi \right)\Vert \Vert x\Vert +\Vert \mathit{B}\left(\xi \right)\Vert \Vert {\mathit{q}}_{\mathrm{2}}\left(\xi \right)\Vert \right)\le \frac{\mathrm{1}}{\kappa }\left(\Vert \mathit{B}\left(\xi \right)\Vert \Vert \mathit{N}\left(\xi \right)\Vert \Vert x\Vert +\Vert \mathit{B}\left(\xi \right)\Vert \Vert {\mathit{q}}_{\mathrm{2}}\left(\xi \right)\Vert \right)\end{array}$(8)

Then we can easily verify that under Assumption 2 and $P,Q\in {{P}}_{\mathrm{2}}\left(\mathrm{\Xi }\right)$, the right side of (8) is integrable. This completes the proof.

Lemma 3   Under the same assumptions in Lemma 2, it has

$\Vert \begin{array}{c}{\mathbb{E}}_P\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]-{\mathbb{E}}_Q\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]\end{array}\Vert \le \sqrt[]{n}L{\zeta }_{\mathrm{3}}\left(P,Q\right)$

for all $x\in S\bigcap {\text{B}},$where $L$ comes from Lemma 1.

Proof   By Lemma 1, we get

$\begin{array}{l}\left|{\left[\mathit{B}\left({\xi }_{\mathrm{1}}\right)y^{\mathrm{*}}\left(x,{\xi }_{\mathrm{1}}\right)\right]}_i-{\left[\mathit{B}\left({\xi }_{\mathrm{2}}\right)y^{\mathrm{*}}\left(x,{\xi }_{\mathrm{2}}\right)\right]}_i\right|\le \Vert \left[\mathit{B}\left({\xi }_{\mathrm{1}}\right)y^{\mathrm{*}}\left(x,\xi {\mathrm{1}}_{\mathrm{1}}\right)\right]-\left[\mathit{B}\left({\xi }_{\mathrm{1}}\right)y^{\mathrm{*}}\left(x,{\xi }_{\mathrm{1}}\right)\right]\Vert \\ \le L\mathrm{m}\mathrm{a}\mathrm{x}{\left\{\mathrm{1},\Vert {\xi }_{\mathrm{1}}\Vert ,\Vert {\xi }_{\mathrm{2}}\Vert \right\}}^{\mathrm{2}}\Vert {\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}\Vert ,\end{array}$

i.e., $\frac{{\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]}_i}{L}\in {{G}}_{\mathrm{F}\mathrm{M}\mathrm{3}},i=\mathrm{1,2},\cdots ,n.$ Thus,

$\left|{\int }_{\mathrm{\Xi }}\frac{{\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]}_i}{L}\left(P-Q\right)\left(\mathrm{d}\xi \right)\right|\le {\zeta }_{\mathrm{3}}\left(P,Q\right),i=\mathrm{1,2},\cdots ,n.$

Then we get that

$\Vert \begin{array}{c}{\int }_{\mathrm{\Xi }}\frac{\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]}{L}\left(P-Q\right)\left(\mathrm{d}\xi \right)\end{array}\Vert ={\left({\displaystyle \sum _{i=\mathrm{1}}^n}{\left|{\int }_{\mathrm{\Xi }}\frac{{\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]}_i}{L}\left(P-Q\right)\left(\mathrm{d}\xi \right)\right|}^{\mathrm{2}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}\le \sqrt[]{n}{\zeta }_{\mathrm{3}}\left(P,Q\right)$

Hence $\Vert \begin{array}{c}{\int }_{\mathrm{\Xi }}\mathit{B}(\xi )y^{\mathrm{*}}(x,\xi )\left(P-Q\right)\left(\mathrm{d}\xi \right)\end{array}\Vert \le \sqrt[]{n}L{\zeta }_{\mathrm{3}}(P,Q).$ This completes the proof.

Lemma 4   Under the same assumptions of Lemma 3, there holds that ${\mathit{\Phi }}_P\left(x\right)\to {\mathit{\Phi }}_Q\left(x\right)$ as ${\zeta }_{\mathrm{3}}(P,Q)\to \mathrm{0}$.

Proof   For any fixed $x$, by Lemma 3, we have

$\begin{array}{l}\Vert \begin{array}{c}{\mathit{\Phi }}_P\left(x\right)-{\mathit{\Phi }}_Q\left(x\right)\end{array}\Vert =\Vert \begin{array}{c}\left(\mathit{A}x+{\mathbb{E}}_P\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]+q_{\mathrm{1}}\right)-\left(\mathit{A}x+{\mathbb{E}}_Q\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]+q_{\mathrm{1}}\right)\end{array}\Vert \\ =\Vert \begin{array}{c}{\mathbb{E}}_P\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]-{\mathbb{E}}_Q\left[\mathit{B}\left(\xi \right)y^{\mathrm{*}}\left(x,\xi \right)\right]\end{array}\Vert \le \sqrt[]{n}L{\zeta }_{\mathrm{3}}\left(P,Q\right).\end{array}$

This completes the proof.

We next give the first main result of this paper.

Proposition 3   (i) Let Assumption 1 and Assumption 2 hold, ${\kappa }_{\mathit{M}}\left(\xi \right)\ge \kappa >\mathrm{0}$ and $Q\in {{P}}_{\mathrm{2}}(\mathrm{\Xi })$. Then the perturbed problem (7) is solved when $S\in {\mathbb{R}}^n$ is bounded.

(ii) Let assumptions of Proposition 2 (ii) and Lemma 4 hold. Then, there exists $\tau >\mathrm{0}$ such that the solution set of the problem (7) is nonempty convex and compact when ${\zeta }_{\mathrm{3}}\left(P,Q\right)<\tau .$

Proof   (i) Firstly, by Assumption 2, we know that there exists $L_{\mathrm{1}}>\mathrm{0}$, such that

$\Vert \mathit{B}\left(\xi \right)\Vert \le \sqrt[]{L_{\mathrm{1}}}\left(\mathrm{1}+\Vert \xi \Vert \right),\Vert \mathit{N}\left(\xi \right)\Vert \le \sqrt[]{L_{\mathrm{1}}}\left(\mathrm{1}+\Vert \xi \Vert \right)$

Then, by Assumption 1 and ${\kappa }_{\mathit{M}}\left(\xi \right)\ge \kappa >\mathrm{0}$, we have

$\Vert \mathit{B}\left(\xi \right)\Vert \underset{J\in {J}}{\mathrm{m}\mathrm{a}\mathrm{x}}\Vert \begin{array}{c}{\mathit{M}}_{J\times J}^{-\mathrm{1}}\left(\xi \right)\end{array}\Vert \le \frac{L_{\mathrm{1}}{\left(\mathrm{1}+\Vert \xi \Vert \right)}^{\mathrm{2}}}{{\kappa }_{\mathit{M}}\left(\xi \right)}<\frac{L_{\mathrm{1}}\left(\mathrm{1}+\mathrm{2}\Vert \xi \Vert +{\Vert \xi \Vert }^{\mathrm{2}}\right)}{\kappa }.$

As $Q\in P_{\mathrm{2}}\left(\mathrm{\Xi }\right)$, it has

$\mathrm{0}<{\mathbb{E}}_Q\left[\frac{L_{\mathrm{1}}\left(\mathrm{1}+\mathrm{2}\Vert \xi \Vert +{\Vert \xi \Vert }^{\mathrm{2}}\right)}{\kappa }\right]<+\mathrm{\infty }.$

The following proof is similar to that of Proposition 2 (i), and we omit it.

(ii) We first claim that there exists $\tau >\mathrm{0}$ such that ${\mathit{\Phi }}_Q$ is pseudo monotone on $S$ whenever ${\zeta }_{\mathrm{3}}\left(P,Q\right)<\tau$. Since ${\mathit{\Phi }}_P$ is pseudo monotone on $S,\mathrm{i}.\mathrm{e}.\forall x_{\mathrm{1}},x_{\mathrm{2}}\in S$,

$〈x_{\mathrm{1}}-x_{\mathrm{2}},{\mathit{\Phi }}_P\left(x_{\mathrm{2}}\right)〉\ge \mathrm{0}\Rightarrow 〈x_{\mathrm{1}}-x_{\mathrm{2}},{\mathit{\Phi }}_P(x_{\mathrm{1}})〉\ge \mathrm{0}.$(9)

Suppose that for any $\tau >\mathrm{0}$, there exists $Q$ satisfies ${\zeta }_{\mathrm{3}}\left(P,Q\right)<\tau$ such that ${\mathit{\Phi }}_Q$ is not pseudo monotone on $S$. This implies that

$〈x_{\mathrm{1}}(\tau )-x_{\mathrm{2}}(\tau ),{\mathit{\Phi }}_Q\left(x_{\mathrm{2}}(\tau )\right)〉\ge \mathrm{0}\Rightarrow 〈x_{\mathrm{1}}(\tau )-x_{\mathrm{2}}(\tau ),{\mathit{\Phi }}_Q\left(x_{\mathrm{1}}(\tau )\right)〉<\mathrm{0},$(10)

for some $x_{\mathrm{1}}\left(\tau \right),x_{\mathrm{2}}\left(\tau \right)\in S$ with $\Vert x_{\mathrm{1}}(\tau )\Vert =\Vert x_{\mathrm{2}}(\tau )\Vert =\mathrm{1}.$ Since $x_{\mathrm{1}}\left(\tau \right)$ and $x_{\mathrm{2}}\left(\tau \right)$ are bounded, without loss of generality, let $\underset{\tau \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}{x}_{\mathrm{1}}(\tau )=x_{\mathrm{1}}\in S$ and $\underset{\tau \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}{x}_{\mathrm{2}}(\tau )=x_{\mathrm{2}}\in S$. By Lemma 4, we know that ${\mathit{\Phi }}_Q\to {\mathit{\Phi }}_P$ as $\tau \to \mathrm{0}$. Therefore

$\begin{array}{l}〈x_{\mathrm{1}}\left(\tau \right)-x_{\mathrm{2}}\left(\tau \right),{\mathit{\Phi }}_Q\left(x_{\mathrm{1}}\left(\tau \right)\right)〉\\ =〈x_{\mathrm{1}}\left(\tau \right)-x_{\mathrm{2}}\left(\tau \right),{\mathit{\Phi }}_P\left(x_{\mathrm{1}}\left(\tau \right)\right)+{\mathit{\Phi }}_Q\left(x_{\mathrm{1}}\left(\tau \right)\right)-{\mathit{\Phi }}_P\left(x_{\mathrm{1}}\left(\tau \right)\right)〉\\ =〈x_{\mathrm{1}}\left(\tau \right)-x_{\mathrm{2}}\left(\tau \right),{\mathit{\Phi }}_P\left(x_{\mathrm{1}}\left(\tau \right)\right)〉+〈x_{\mathrm{1}}\left(\tau \right)-x_{\mathrm{2}}\left(\tau \right),{\mathit{\Phi }}_Q\left(x_{\mathrm{1}}\left(\tau \right)\right)-{\mathit{\Phi }}_P\left(x_{\mathrm{1}}\left(\tau \right)\right)〉\\ =〈x_{\mathrm{1}}\left(\tau \right)-x_{\mathrm{1}}+x_{\mathrm{1}}-x_{\mathrm{2}}+x_{\mathrm{2}}-x_{\mathrm{2}}\left(\tau \right),{\mathit{\Phi }}_P\left(x_{\mathrm{1}}\left(\tau \right)\right)〉+〈x_{\mathrm{1}}\left(\tau \right)-x_{\mathrm{2}}\left(\tau \right),{\mathit{\Phi }}_Q\left(x_{\mathrm{1}}\left(\tau \right)\right)-{\mathit{\Phi }}_P\left(x_{\mathrm{1}}\left(\tau \right)\right)〉\\ \begin{array}{l}=〈x_{\mathrm{1}}\left(\tau \right)-x_{\mathrm{1}},{\mathit{\Phi }}_P\left(x_{\mathrm{1}}\left(\tau \right)\right)〉+〈x_{\mathrm{1}}-x_{\mathrm{2}},{\mathit{\Phi }}_P\left(x_{\mathrm{1}}\left(\tau \right)\right)〉+〈x_{\mathrm{2}}-x_{\mathrm{2}}\left(\tau \right),{\mathit{\Phi }}_P\left(x_{\mathrm{1}}\left(\tau \right)\right)〉\\ +〈x_{\mathrm{1}}\left(\tau \right)-x_{\mathrm{2}}\left(\tau \right),{\mathit{\Phi }}_Q\left(x_{\mathrm{1}}\left(\tau \right)\right)-{\mathit{\Phi }}_P\left(x_{\mathrm{1}}\left(\tau \right)\right)〉.\end{array}\end{array}$

Then we have that

$\underset{\tau \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}〈x_{\mathrm{1}}\left(\tau \right)-x_{\mathrm{2}}\left(\tau \right),{\mathit{\Phi }}_Q\left(x_{\mathrm{1}}\left(\tau \right)\right)〉=〈x_{\mathrm{1}}-x_{\mathrm{2}},{\mathit{\Phi }}_P\left(x_{\mathrm{1}}\right)〉.$