Stochastic tensor complementarity problem: CVaR-ERM model and the convergence analysis of stationary points for its approximation problem
- Open Access
- 24-11-2025
- Research
Published in:
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by (Link opens in a new window)
Abstract
1 Introduction
In 2019, Che, Qi, and Wei proposed the stochastic tensor complementarity problem (STCP) in [2], which is to find an \(x\in R^{n}\) that satisfies where ω is an n-dimensional stochastic variable defined on the probability space \((\Omega , \textbf{F}, P)\), \(\mathcal{B}(\omega )=\big(b_{i_{1} i_{2} \cdots i_{m}}(\omega )\big) \in S_{m, n}\) (symbols will be explained in the next section) is a stochastic tensor, \({q}(\omega )\in R^{n}\) is a stochastic vector, and \(\mathcal{F}(x,\omega )=\big(\mathcal{F}_{1}(x,\omega ),\ldots , \mathcal{F}_{n}(x,\omega )\big)^{T}\). “a.s.” is the abbreviation of “almost surely” under a given probability measure. In fact, it is impossible to find an \(x\in R^{n}\) that satisfies (1) for all \(\omega \in \Omega \). Therefore, researchers have opted to provide deterministic models for solving STCP (1), and regarded the solutions of these deterministic models as the solution of STCP (1).
$$\begin{aligned} {x} \geqslant 0, \quad \mathcal{F}({x,\omega}):=\mathcal{B}(\omega ) {x}^{m-1}+{q}( \omega ) \geqslant{0},\quad{x}^{T}\mathcal{F}({x,\omega})=0, \quad \omega \in \Omega , \text{ a.s. }, \end{aligned}$$
(1)
Based on the above considerations, scholars have studied the expected residual minimization (ERM) model for solving STCP and studied the properties of its solution set in [2, 5, 12, 14]. The ERM model for solving STCP is to find an \(x\in R^{n}\) that satisfies Here, \(\mathbb{E}\) represents the mathematical expectation with respect to the stochastic variable ω, and \(\Phi : R^{n} \times \Omega \rightarrow R^{n}\) denotes where \(\phi : R^{2} \rightarrow R\) is an NCP function defined as follows: Additionally, researchers have explored the algorithms for solving the ERM model of STCP in [6, 13]. It is worth noting that there are currently limited researches on STCP, with the majority focusing on the ERM model and a few studies yet on other models.
$$\begin{aligned} \min _{x \in R_{+}^{n}} \mathbb{E}\left [ \left \| \Phi \big( x, \mathcal{F}({x,\omega})\big)\right \| ^{2} \right ] . \end{aligned}$$
(2)
(3)
$$ \phi (a, b)=0 \Leftrightarrow a \geq 0, b \geq 0, a b=0. $$
Advertisement
In 2014, Xu and Yu [18], while investigating stochastic nonlinear complementarity problem, recognized the limitations of the ERM model. Specifically, they observed that the solutions obtained through solving the ERM model might not satisfy the feasibility condition in stochastic nonlinear complementarity problem for many stochastic variables \(\omega \in \Omega \), which implies that, when a real-world problem can be formulated as a stochastic nonlinear complementarity problem and demands a high level of feasibility, using the solutions obtained from solving the ERM model as the solutions of the stochastic nonlinear complementarity problem may result in significant errors. As a result, they proposed a new deterministic model for solving the stochastic nonlinear complementarity problem, termed CVaR-constrained stochastic programming reformulation, herein referred to as the CVaR-ERM model.
In our previous work [19], we studied the CVaR-ERM model for solving the STCP, which addresses the shortcomings of the ERM model for solving the STCP discussed in [6, 13]. Additionally, due to the difficulties associated with using existing algorithms to solve this model, we presented the approximate problem of this model by the smoothing method, the penalty function method, and the sample average approximation method and proved the convergence of the global optimal solution for the approximate problem under certain conditions.
In fact, we are more interested in the convergence of the stationary points of the approximation problem, which stems from practical considerations, as the non-convex nature of the CVaR-ERM model and its approximation problem can lead to the prevalence of local rather than global optimal solutions during the solving process in certain scenarios. However, we encountered obstacles when proving the convergence of stationary points. The complexity of the gradient of \(\left \| \Phi \big( x,\mathcal{F}(x,\omega ) \big) \right \|^{2}\) made it impossible to apply the traditional proof methods of stationary points convergence to our research. Fortunately, we overcame the previous obstacles by leveraging the property that \(\nabla _{x}\left \| \Phi \big( x,\mathcal{F}(x,\omega ) \big) \right \|^{2}\) is a singleton set, thus successfully proving the convergence of stationary points of the approximation problem. This is also a major contribution of this paper.
Throughout this paper, e denotes the n-dimensional vector \((1,1,\ldots ,1)^{T}\). \(e_{i}\) represents the n-dimensional unit vector with the i-th element being 1. \(B (x)\) represents a neighborhood of point x. For a sequence, \({\mathop {\overline{\lim}}}\) is used to represent the outer limit of the sequence, which is the set of all accumulation points of the sequence. “conv” represents a convex hull of a set. \(\nabla _{x} f(x)\in R^{q\times n}\) represents the Jacobi matrix of the vector valued function \(f: R^{n}\rightarrow R^{q}\) at x. When \(a\in R\) is a real number, \({(a) _{+}} \) represents \(\max \left \lbrace a,0\right \rbrace \) and \(\max \left \lbrace a,0\right \rbrace =\frac{\left | a\right |+a }{2}\). For a vector x, \({(x)_{+}}\) represents the n-dimensional vector with the i-th component is \(\max \left \lbrace x_{i},0\right \rbrace \). \(\| x\|\) represents the 2 norm of x, i.e. \(\|x \|=\sqrt{\sum \limits _{i=1}^{n}{{{\left | {{x}_{i}} \right |}^{2}}}}\). \({{\left \| M \right \|_{F}}}\) represents the Frobenius norm of the matrix \(M \in {{R}^{n}\times {R}^{m}}\), i.e. \({{\left \| M \right \|_{F}}}=\sqrt{\sum \limits _{j=1}^{m}{\sum \limits _{i=1}^{n}{{{\left | {{m}_{ij}} \right |}^{2}}}}}\). \(\|\mathcal{B}\|_{\mathcal{F}}\) denotes the Frobenius norm of tensor \(\mathcal{B} \in S_{m, n}\), which is \(\|\mathcal{B}\|_{\mathcal{F}}=\sqrt {\sum _{i_{1}, i_{2}, \ldots , i_{m}=1}^{n} b_{i_{1} i_{2} \cdots i_{m}}^{2}}\). When \(\mathscr{Y}\) is a compact set composed of vectors, \(\left \| \mathscr{Y}\right \|_{S} \) represents the Frobenius norm of \(\mathscr{Y}\), i.e. \(\left \| \mathscr{Y}\right \| _{S} = \underset{y\in \mathscr{Y}}{\mathop{sup}}\,\left \|y \right \| \). \(d(x, \mathscr{Z}): =\underset{z\in \mathscr{Z}}{\mathop{\inf }}\, \left \|x-z \right \| \) represents the distance from point x to set \(\mathscr{Z} \). For two compact sets \(\mathscr{X}\) and \(\mathscr{Z}\), there holds \(\mathbb{D}(\mathscr{X},\mathscr{Z}): = \underset{x\in \mathscr{X}}{\mathop{\sup }}\, d(x, \mathscr{Z})\).
Advertisement
The organization of this paper is as follows. In Sect. 2, we introduce relevant concepts related to tensors and some definitions and lemmas that will be used later. In Sect. 3, We introduced the CVaR-ERM model for solving STCP, presented its approximation problem, and analyzed the convergence of the stationary points of the approximation problem by leveraging the property that \(\nabla _{x}\left \| \Phi \big( x,\mathcal{F}(x,\omega ) \big) \right \|^{2}\) is a singleton set. Finally, we summarize the entire paper in Sect. 4.
2 Preliminary
In this section, we give some basic definitions and properties that will be used later in the article.
An m-order \(\left (v_{1}\times {{v}_{2}}\times \cdots \times {{v}_{m}} \right ) \)-dimensional tensor \(\mathcal{B}\) over the real number field is a multi-dimensional array of the following form where \([n]\) denotes \(\left \lbrace 1,2,\ldots ,n\right \rbrace \) and n, \(v_{1},{v}_{2},\ldots , {v}_{m}\) and \(m\ge 2\) are positive integers. If \(v_{1} = {v}_{2} =\cdots = {v}_{m}=n\), then \(\mathcal{B}\) is called an m-order n-dimensional real tensor. We use \(S_{m, n}\) to represent the set of all m-order n-dimensional real tensors. A symmetric tensor refers to a tensor \(\mathcal{B}=\left (b_{i_{1} i_{2} \cdots i_{m}}\right ) \in S_{m, n}\) that remains unchanged under any permutation of \(i_{1}, i_{2}, \ldots , i_{m}\). A semi-symmetric tensor refers to a tensor \(\mathcal{B}=\left (b_{i_{1} i_{2} \cdots i_{m}}\right ) \in S_{m, n}\) that remains constant under any permutation of \(i_{2}, \ldots , i_{m}\).
$$ \mathcal{B}=\left (b_{i_{1} i_{2} \cdots i_{m}}\right ),\quad b_{i_{1} i_{2} \cdots i_{m}}\in R^{v_{1}\times {{v}_{2}}\times \cdots \times {v}_{m}}, \quad i_{j}\in [v_{j}],\quad j\in [m], $$
For a vector \(x\in R^{n} \) and a tensor \(\mathcal{B}=\left (b_{i_{1} i_{2} \cdots i_{m}}\right ) \in S_{m, n}\), \(\mathcal{B}{x}^{m-1}\) is a vector whose components is \(\mathcal{B} x^{m-2}\) is a matrix with the following elements For \(r\in [m-1]\), we have If \(r=m-1\), then \(\left \|\cdot \right \|_{O } \) represents \(\left \|\cdot \right \| \). If \(r=m-2\), then \(\left \|\cdot \right \|_{O } \) represents \(\left \|\cdot \right \|_{F } \). If \(r\leq m-3\), then \(\left \|\cdot \right \|_{O } \) represents \(\left \| \cdot \right \|_{\mathcal{F}}\). The proof of (5) can be seen in Lemma 1 of [17]. If the tensor \(\mathcal{B}\) is a semi-symmetric tensor, then \(\nabla \mathcal{B}{x}^{m-1}=(m- 1)\mathcal{B}{x}^{m-2}\) and \(\nabla \left ( \mathcal{B}{x}^{m-1}\right ) _{i}=(m-1)\left ( \mathcal{B}{x}^{m-2}\right )_{i}\) for \(i\in [n]\), the proof of which is shown in Lemma 4 of [17].
$$\begin{aligned} \left (\mathcal{B}{x}^{m-1}\right )_{i}=\sum _{i_{2}, \ldots , i_{m}=1}^{n} b_{i i_{2} \cdots i_{m}} x_{i_{2}} \cdots x_{i_{m}}, \quad \forall i \in [n]. \end{aligned}$$
(4)
$$\begin{aligned} \left (\mathcal{B} x^{m-2}\right )_{i j}=\sum _{i_{3}, \ldots , i_{m}=1}^{n} b_{i j i_{3} \cdots i_{m}} x_{i_{3}} \cdots x_{i_{m}}, \quad \forall i, j \in [n]. \end{aligned}$$
$$\begin{aligned} \left \| \mathcal{B}{x}^{r}\right \|_{O } \le \left \| \mathcal{B} \right \|_{\mathcal{F}} \left \| {x}\right \|^{r}. \end{aligned}$$
(5)
Definition 1
[8] A set-valued map \(\Upsilon : \mathcal{X}\to 2^{R^{n}} \) is uniformly compact near x, if there is a neighborhood \(B(\bar{x})\) of x̄ such that the closure of \(\bigcup \limits _{x\in B(\bar{x})}{\Upsilon (x)}\) is compact.
Lemma 1
[8] A set-valued map \(\Upsilon : \mathcal{X}\to 2^{R^{n}} \) is uniformly compact near x. Then ϒ is upper semi-continuous if and only if ϒ is closed.
Lemma 2
[7] If \(\varphi : \Omega \to R\) is an integrable function on Ω, then we have where \(\rho :\Omega \to \left [ 0,\infty \right )\) represents the probability density function.
$$\begin{aligned} \lim _{k\rightarrow +\infty}\frac{1}{N_{k}}\sum _{\omega ^{j}\in \Omega _{k}}\varphi (\omega ^{j})\rho (\omega ^{j})=\mathbb{E}[ \varphi (\omega )], \end{aligned}$$
(6)
Definition 2
[4] \(f:R^{n}\to R\) is a local Lipschitz continuous function. The Clarke generalized gradient of f at x is where \(D_{f}\) denotes the set of points that make \(f(x)\) differentiable.
$$\begin{aligned} {\partial}_{x} f ( x ) = conv\Big\{ \underset{{k}\to +\infty}{\mathop{{\lim }}}\,{{\nabla }_{x}}{{f }}({{x}^{k}}) | \underset{k\to +\infty }{\mathop{\lim }}\,{{x}^{k}}=x,\{{{x}^{k}}\} \subseteq {D_{f} } \Big\} , \end{aligned}$$
Definition 3
[4] \(H: {R}^{n} \rightarrow {R}^{m}\) is locally Lipschitz continuous near \(x \in R^{n}\). The Clarke generalized Jacobian matrix of H at x is where \(D_{H}\) represents the set of points that make H differentiable.
$$ \partial _{x} H(x):=\mathrm{conv}\left \{\lim _{k \rightarrow +\infty} \nabla _{x} H(x^{k})^{T} \in {R}^{m \times n}: \exists \{x^{k}\} \subseteq D_{H}\right . s.t. \left .\lim _{k \rightarrow +\infty} x^{k}=x \right \}, $$
Specifically, if \(m=1\), \(\partial _{x} H(x) \subseteq {R}^{n}\) degenerates into the Clarke generalized gradient of H at x. Furthermore, according to Proposition 2.6.2 in [4], we have where the right-hand side represents the matrix set in \({R}^{n \times m}\), and the i-th column is given by the generalized gradient of \(H_{i}\).
$$ \partial _{x} H(x)^{T} \subseteq \partial _{x} H_{1}(x) \times \cdots \times \partial _{x} H_{m}(x), $$
The definition of Fischer-Burmeister (FB) function is We choose the FB function to define (3) because the square of FB function is continuous and differentiable everywhere. However, the gradient of \(\Phi \big (x, \mathcal {F} (x, \omega ) \big)\) with respect to x is complex. To help readers understand the main content better, we will provide a detailed analysis of the Clarke generalized Jacobian matrix of \(\Phi \big (x, \mathcal {F} (x, \omega ) \big)\).
$$\begin{aligned} {{\phi}}(a,b)& =\sqrt{{{a}^{2}}+{{b}^{2}}}-(a+b). \end{aligned}$$
The Clarke generalized Jacobian matrix of \(\Phi \big( x,\mathcal{F}( x,\omega )\big)\) is \({\partial}_{x}\Phi \big( x,\mathcal{F}( x,\omega )\big)\) [4] with When \(\big( x_{i},{{\mathcal{F}}_{i}}(x,\omega )\big)\ne \left (0,0 \right )\), we have\ where When \(\big( x_{i},{{\mathcal{F}}_{i}}(x,\omega )\big)= \left (0,0 \right )\), we have
$$\begin{aligned} {\partial}_{x}\Phi \big( {x},\mathcal{F}({x},\omega )\big)\subseteq { \partial}_{x}\phi _{1} \big({x}_{1},\mathcal{F}_{1}( {x},\omega ) \big)\times \cdots \times{\partial}_{x}\phi _{n} \big( {x}_{n}, \mathcal{F}_{n}( {x},\omega )\big). \end{aligned}$$
(7)
$$\begin{aligned} {\partial}_{x}\phi \big( x_{i},\mathcal{F}_{i}( x,\omega )\big)= \left \lbrace {\nabla}_{x}\phi \big( x_{i},\mathcal{F}_{i}( x,\omega ) \big)\right \rbrace , \end{aligned}$$
(8)
$$\begin{aligned} {\nabla}_{x}\phi \big( x_{i},\mathcal{F}(x,\omega )\big) = \Big( \frac{{{x}_{i}}}{\sqrt{{{x}_{i}^{2}} +{{\mathcal{F}}_{i}}(x,\omega )^{2}}}-1 \Big){{e}_{i}}+\Big( \frac{{{\mathcal{F}}_{i}}(x,\omega )}{\sqrt{{{x}_{i}^{2}}+{{\mathcal{F}}_{i}}(x,\omega )^{2}}}-1 \Big){{\nabla }_{x}}{{\mathcal{F}}_{i}}(x,\omega ). \end{aligned}$$
(9)
$$\begin{aligned} {\partial}_{x}\phi \big( x_{i},\mathcal{F}_{i}( x,\omega )\big) \subseteq \left \{ \left ({{\eta }_{i}} -1\right ){{e}_{i}}+\left ( {{ \gamma }_{i}} -1\right )\nabla _{x} {{\mathcal{F}}_{i}}\left ( x, \omega \right )|\eta _{i}^{2}+\gamma _{i}^{2}\le 1 \right \}. \end{aligned}$$
(10)
Next, we will explain the gradient of \(\left \| \Phi \big( x,\mathcal{F}(x,\omega ) \big) \right \|^{2}\) at x̄. By (7), we have
$$\begin{aligned} &{\nabla}_{x}\left \| \Phi \big( \bar{x},\mathcal{F}(\bar{x},\omega ) \big) \right \|^{2} \\ =&\;2{\partial}_{x}\Phi \big( \bar{x},\mathcal{F}( \bar{x},\omega ) \big)\cdot \Phi \big( \bar{x},\mathcal{F}( \bar{x},\omega )\big) \\ \subseteq &\;2{\partial}_{x}\phi _{1}\big( \bar{x}_{1},\mathcal{F}_{1}( \bar{x},\omega )\big)\cdot \phi _{1} \big( \bar{x}_{1},\mathcal{F}_{1}( \bar{x},\omega ) \big)+\cdots +2{\partial}_{x}\phi _{n} \big( \bar{x}_{n}, \mathcal{F}_{n}( \bar{x},\omega )\big)\cdot \phi _{n} \big( \bar{x}_{n}, \mathcal{F}_{n}( \bar{x},\omega ) \big). \end{aligned}$$
(11)
For simplicity, we analyze \({\partial}_{x}\phi _{i} \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x}, \omega )\big)\cdot \phi _{i} \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ) \big)\), \(i\in [n]\). For any \(i\in [n]\), let \({\Gamma }_{(i,\bar{x})}=\{ \omega \in \Omega | \big( {{\bar{x}}_{i}}, \mathcal{F}_{i}(\bar{x},\omega ) \big)= (0,0)\}\).
(i) When \(\omega \in {\Gamma }_{(i,\bar{x})}\), we have \(\phi _{i} \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ) \big)=0\). So there holds
$$ {\partial}_{x}\phi _{i} \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x}, \omega )\big)\cdot \phi _{i} \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ) \big)=\left \lbrace 0\right \rbrace . $$
(ii) When \(\omega \in {\Omega / {{\Gamma }_{(i,\bar{x})}}}\), we have The above content will be used in the proof of Lemma 3.
$$ {\partial}_{x}\phi _{i} \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x}, \omega )\big)\cdot \phi _{i} \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ) \big)=\left \lbrace {{\nabla }_{x}}{{\phi }_{i}} \big(\bar{x}_{i},\mathcal{F}_{i}({\bar{x}},\omega )\big)\cdot{{\phi }_{i}} \big({\bar{x}_{i}},\mathcal{F}_{i}(\bar{x},\omega )\big)\right \rbrace . $$
3 Approximation problem and the convergence analysis of its stationary points
As a natural continuation of our previous work [19], in this paper, we mainly focus on exploring the convergence properties of the stationary points of the approximate problem of the CVaR-ERM model for solving STCP. For the convenience of readers, we first elaborate on the formulation of the approximate problem of the CVaR-ERM model for solving STCP before discussing the convergence of the stationary points.
3.1 CVaR-ERM model and its approximation problem
The CVaR-ERM model for solving STCP is to find an \(x\in R^{n}\) that satisfies where \(\alpha _{i}\in (0,1)\) (\(i\in [n]=\left \lbrace 1,2,\ldots ,n\right \rbrace \)) are given very small numbers. By introducing auxiliary variables \(y_{1},y_{2},\ldots ,y_{n}\), we have from [16] that For simplicity, we set \({{H}_{i}}\left ( x,{{y}_{i}},\omega \right ):=y_{i}{{\alpha }_{i}}+{{ \left ( -{{\mathcal{F}}_{i}}({x},{\omega })-y_{i} \right )}_{+}}\), \(i \in [n]\), \(y:=\left ( y_{1},y_{2},\ldots , y_{n}\right ) ^{T}\), \(H(x,y,\omega ):=\big( {{H}_{1}}\left ( x,{{y}_{1},\omega} \right ), \ldots , {{H}_{n}}( x,{{y}_{n}},\omega ) \big)^{T}\), \(h(x,y):=\mathbb{E}[{H}(x, y, \omega )]\). Then, we get the following equivalent form of problem (12):
$$\begin{aligned} {\mathop{\min }}\ & g(x):=\mathbb{E}\big[ {{\left \| \Phi \big( x, \mathcal{F}(x,\omega ) \big) \right \|}^{2}} \big] \\ {\mathrm{s.t.}} & x \ge 0 , \\ &CVaR_{{{\alpha }_{i}}}\big(- \mathcal{F}_{i}({x,\omega}) \big)\le 0, \quad i\in [n], \end{aligned}$$
(12)
$$ CVaR_{{{\alpha }_{i}}}\big(- \mathcal{F}_{i}({x,\omega}) \big)= \underset{{y}_{i}}{\mathop{\min }} \,\,\,\mathbb{E}\big[ y_{i}{{ \alpha }_{i}}+{{\left ( -{{\mathcal{F}}_{i}}({x},{\omega })-y_{i} \right )}_{+}}\big],\quad i\in [n]. $$
$$\begin{aligned} {\mathop{\min _{(x, y) }}}\ & g(x):=\mathbb{E}\big[ {{\left \| \Phi \big( x,\mathcal{F}(x,\omega ) \big) \right \|}^{2}} \big] \\ {\mathrm{s.t.}} & x \ge 0 , \\ &{{h}}(x, y) \leq 0. \end{aligned}$$
(13)
Note that the function \(h(x,y)\) in the problem (13) is a max function and is non-smoothing, which makes it impossible to solve by using general smooth optimization algorithms. To resolve this issue, the smoothing technique is applied to approximate the non-smoothing constraints of problem (13).
The Chen-Harker-Kanzow-Smale (CHKS) smoothing function [3] of \({{H}_{i}}\left ( x,{{y}_{i}},\omega \right )=y_{i}{{\alpha }_{i}}+{{ \left ( -{{\mathcal{F}}_{i}}({x},{\omega })-y_{i} \right )}_{+}} \left ( i\in [n]\right ) \) is where \(\mu \ge 0 \) is the smoothing parameter.
$$ {H}_{i}\left (x, y_{i},\omega , \mu \right ):=y_{i} \alpha _{i}+ \frac{\sqrt{4 \mu ^{2}+\left ( -y_{i}-\mathcal{F}_{i}(x,\omega )\right ) ^{2}}-y_{i}-\mathcal{F}_{i}(x,\omega )}{2}, \quad i\in [n], $$
We define \({H}(x, y, \omega , \mu ):=\left ({H}_{1}\left (x, y_{1},\omega , \mu \right ), \ldots , {H}_{n}\left (x, y_{n},\omega , \mu \right ) \right )^{T}\) and \({h}(x, y,\mu ):= \mathbb{E}[{H}(x, y, \omega ,\mu )]\). Then, we obtain the smooth approximation problem of (13) as follows:
(14)
The objective function and constraints in problem (14) both involve mathematical expectations. Generally, estimating the expected value is challenging. Therefore, we use the sample average approximation (SAA) method to estimate the expected value and construct an optimization problem with only non-negative constraint using the penalty function method. By generating independent and identically distributed samples \(\Omega _{k}=\left \lbrace \omega ^{j}|j=1,2,\ldots ,N_{k}\right \rbrace \) (when \(k\to +\infty \), we have \(N_{k}\to +\infty \)) using the quasi-Monte Carlo method, we obtain the smoothing penalty approximation problem based on SAA of (13) as follows: where
$$\begin{aligned} {\mathop{\min _{(x, y)}}} & {{\Theta }^{k}}(x,y):={{g}^{k}}(x)+{{ \sigma }^{k}}{{\| {({h^{k}(x,y,{\mu }^{k})})_{+}} \|}^{2}}\text{ } \\ {\mathrm{s.t.}} &x\ge 0, \end{aligned}$$
(15)
$$\begin{aligned}& {{g}^{k}}(x):=\frac{1}{{{N}_{k}}}\!\!\sum \limits _{{{\omega }^{j}} \in {{\Omega }_{k}}}{{{\left \| \Phi ( x,\mathcal{F}(x,{{\omega }^{j}}) ) \right \|}^{2}}}\rho (\omega ^{j}),\\& h^{k}(x,y,{\mu }^{k}):=\frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}} \in {{\Omega }_{k}}}{H}( x,y,{{\omega }^{j},{\mu }^{k}} ) \rho ( \omega ^{j}). \end{aligned}$$
3.2 Convergence analysis of stationary points
In general, (15) is a non-convex optimization problem. Therefore, it is necessary to study the convergence analysis of the stationary points of (15). To this end, we first give some assumption conditions.
(A1)
We assume that Ω is a non-empty compact set on \(R^{n}\).
(A2)
\(\mathcal{B}(\omega )\) is a semi-symmetric tensor almost surely.
(A3)
(i) When \(k\to +\infty \), we assume that \({{\mu }^{k}}\to 0\), \({{\sigma }^{k}}\to +\infty \) and \({{\sigma }^{k}}{{\mu }^{k}}\to 0\).
(ii) When \(\underset{k\to +\infty }{\mathop{\lim }}\,(x^{k},y^{k}) =(\bar{x}, \bar{y})\), it holds that \(\underset{k\to +\infty }{\mathop{\lim }}\,{{\sigma }^{k}}\left \|(x^{k},y^{k})-( \bar{x},\bar{y}) \right \|=0\).
(iii) For every fixed point \((x,y_{i})\in {R^{n}_{+}}\times R\), it holds almost surely that
$$ \underset{k\to +\infty }{\mathop{\lim }}\, \frac{{{\sigma }^{k}}}{{{N}_{k}}}\sum \limits _{\omega ^{j}\in{\Omega}} \rho (\omega ^{j})\Big( {{{H_{i}(x, y_{i},\omega ^{j})}}}-\mathbb{E} \left [ {H_{i}( x, y_{i},\omega )}\right ]\Big)=0 ,\quad i\in [n]. $$
Remark 1
According to [7, 10], for every \(i\in [n]\) and fixed point \((x,y_{i}) \in {R_{+}^{n}} \times R\), the convergence rate of \(\left \lbrace \frac{1}{N_{k}}\sum _{\omega ^{j}\in \Omega _{k}} {{H}_{i}}( x,{{y}_{i}},\omega ^{j})\rho (\omega ^{j})-\mathbb{E}\left [ {{H}_{i}} \left ( x,{{y}_{i}},\omega \right )\right ]\right \rbrace \), is of order \(O({{k}^{-\frac{1}{2}}})\). We choose \({{\mu }^{k}}={{k}^{-\varepsilon}}\) and \({{\sigma }^{k}}={{k}^{\nu }}\), where \(\nu \in (0,\frac{1}{2})\), \(\varepsilon >\nu \). Then, \(\mathrm{(i)}\) and \(\mathrm{(iii)}\) in \(\mathrm{(A3)}\) hold. Since \((\bar{x}, \bar{y})\) is an accumulation point of \((x^{k}, y^{k})\), at least one subsequence exists to satisfy \(\mathrm{(ii)}\) in \(\mathrm{(A3)}\).
Remark 2
According to Theorem 2.1 in [15], it is easy to see that for any stochastic tensor \(\widehat{\mathcal{ A}}(\omega )\in S_{m,n}\), there exists a unique semi-symmetric stochastic tensor \(\widehat{\mathcal{A}}(\omega )\in S_{m,n}\) such that As a result, assumption \(\mathrm{(A2)}\) holds.
$$ \mathcal{A}(\omega )x^{m-1}=\widehat{\mathcal{A}}(\omega )x^{m-1} \quad \text{a.s.}. $$
Definition 4
\((\bar{x},\bar{y})\) is called a stationary point of (13) if there exists Lagrange multipliers \(\bar{\upsilon}\in R^{n}\) and \(\bar{\iota}\in R^{n}\) such that
$$\begin{aligned} &0\in \begin{pmatrix} {\nabla}_{x} g(\bar{x}) \\ 0 \end{pmatrix} + \begin{pmatrix} \sum \limits _{i=1}^{n}{{\bar{\upsilon}}_{i}}{{\partial }_{x}}h_{i}( \bar{x},\bar{y}_{i}) \\ \sum \limits _{i=1}^{n}{{\bar{\upsilon}_{i}}}{{\partial }_{y}}h_{i}( \bar{x},\bar{y}_{i}) \end{pmatrix} + \begin{pmatrix} -\sum \limits _{i=1}^{n}{\bar{\iota }}_{i}{e_{i}} \\ 0 \end{pmatrix} , \end{aligned}$$
(16)
$$\begin{aligned} &\bar{\upsilon}\ge 0,\quad h(\bar{x},\bar{y})\leq 0, \quad \bar{\upsilon}^{T}h(\bar{x},\bar{y})=0, \end{aligned}$$
(17)
$$\begin{aligned} &\bar{\iota}\ge 0,\quad -\bar{x}\leq 0,\quad \bar{\iota}^{T} \bar{x}=0. \end{aligned}$$
(18)
Definition 5
For each k, \(( x^{k}, y^{k})\) is called a stationary point of (15) if there exists Lagrange multiplier \(\iota ^{k}\in R\) such that
$$\begin{aligned} &0= \begin{pmatrix} \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} { \nabla}_{x}\left \| \Phi \big( x^{k},\mathcal{F}(x^{k},\omega )\big) \right \|^{2} \rho (\omega ^{j}) \\ 0 \end{pmatrix} \\ &\qquad {}+ \begin{pmatrix} \sum \limits _{i=1}^{n}2{{\sigma }}^{k}\Big( \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {H_{i}(x^{k},y_{i}^{k}, \omega ^{j},{\mu }^{k})} \rho (\omega ^{j})\Big) _{+} \cdot \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {{ \nabla }_{x}}{H_{i}(x^{k},y_{i}^{k},\omega ^{j},{\mu }^{k})}\rho ( \omega ^{j}) \\ \sum \limits _{i=1}^{n}2{{\sigma }}^{k}\Big(\frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {H_{i}(x^{k},y_{i}^{k}, \omega ^{j},{\mu }^{k})}\rho (\omega ^{j})\Big) _{+} \cdot \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {{ \nabla }_{y}}{H_{i}(x^{k},y_{i}^{k},\omega ^{j},{\mu }^{k})} \rho ( \omega ^{j}) \end{pmatrix} \\ &\qquad {}+ \begin{pmatrix} -\sum \limits _{i=1}^{n}{\iota _{i}^{k}}e_{i} \\ 0 \end{pmatrix} , \end{aligned}$$
(19)
$$\begin{aligned} &\iota ^{k}\ge 0,\quad -x^{k}\leq 0,\quad (\iota ^{k})^{T} x^{k}=0. \end{aligned}$$
(20)
Lemma 3
Assuming that \(\lbrace x^{k} \rbrace \) is a sequence contained in a compact set \(\mathscr{X}\subseteq R^{n}_{+}\) and x̄ is a accumulation point of the sequence \(\lbrace x^{k}\rbrace \). Then we have
$$\begin{aligned} \underset{k\to +\infty }{\mathop{\lim }}\, \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {\nabla}_{x}\left \| \Phi \big( x^{k},\mathcal{F}(x^{k},\omega ^{j} ) \big) \right \|^{2} \rho (\omega ^{j})={\nabla}_{x} g(\bar{x}). \end{aligned}$$
Proof
Without loss of generality, we assume that \(\underset{k \to +\infty }{\mathop{\lim }}\,x^{k}=\bar{x} \). According to the continuity of functions \(\nabla _{x}\left \| \Phi \big( x,\mathcal{F}(x,\omega ) \big) \right \|^{2}\), \(\mathcal{F}_{i}(x,\omega )\) and \({\nabla}_{x}\mathcal{F}_{i}(x,\omega )\), \(i\in [n]\), there exists a positive integer \(t_{2}\) for any \((x,\omega )\in \mathscr{X}\times \Omega \) such that
$$\begin{aligned} &\left \|{\nabla}_{x}\left \| \Phi \big( x,\mathcal{F}(x,\omega ) \big) \right \|^{2} \right \|\le t_{2}, \end{aligned}$$
(21)
$$\begin{aligned} &\quad \quad \left | \mathcal{F}_{i}(x,\omega ) \right |\le t_{2},\quad \quad \quad i\in [n], \end{aligned}$$
(22)
$$\begin{aligned} &\quad \left \| \nabla _{x} \mathcal{F}_{i}(x,\omega ) \right \|\le t_{2}, \quad \quad i\in [n]. \end{aligned}$$
(23)
By (21) and Theorem 16.8 in [1], it can be inferred that \({\nabla}_{x}\left \| \Phi \big( x,\mathcal{F}( x,\omega )\big) \right \|^{2}\) is integrable about ω. Therefore, following from (6), it can be inferred that, for \(\forall x\in \mathscr{X}\), there holds
$$\begin{aligned} \bigg\| \frac{1}{{{N}_{k}}}\sum \limits _{{{ \omega }^{j}}\in {{\Omega }_{k}}} {\nabla}_{x}\left \| \Phi \big( x, \mathcal{F}(x,\omega ^{j} )\big) \right \|^{2} \rho (\omega ^{j})- \mathbb{E}\left [ {\nabla}_{x}\left \| \Phi \big( x,\mathcal{F}( x, \omega ) \big) \right \|^{2} \right ]\bigg\| \xrightarrow{k\to +\infty }0. \end{aligned}$$
(24)
Next, we will prove that
$$\begin{aligned} &\bigg\| \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{ \Omega }_{k}}} {\nabla}_{x}\left \| \Phi \big( x^{k},\mathcal{F}(x^{k}, \omega ^{j} ) \big) \right \|^{2}\rho (\omega ^{j}) - \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} { \nabla}_{x}\left \| \Phi \big( \bar{x},\mathcal{F}(\bar{x},\omega ) \big) \right \|^{2}\rho (\omega ^{j})\bigg\| \\ \to &\; 0,\quad (k\to +\infty ). \end{aligned}$$
Let \({\Gamma}_{(i,\bar{x})}^{k}:=\big\{\omega ^{j}|\omega ^{j}\in{\Omega}_{k}, \big(\bar{x}_{i}, \mathcal{F}(\bar{x}_{i},{\omega}^{j})\big)=(0,0) \big\}\), \(i\in [n]\).
\(\mathrm{(i)}\) If \({\omega}^{j}\in{\Gamma}_{(i,\bar{x})}^{k}\), it is obvious that \(\phi _{i} \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ) \big)=0\), and it is easy to know that By combining (9), (10), and (23), it can be concluded that for every \(x\in \mathscr{X}\), there holds Furthermore, according to the mean value theorem, there exists \(\tau _{ikj}\in [0,1]\) for any i, \(x^{k}\) and \(\omega ^{j}\), such that
$$\begin{aligned} {\partial}_{x}\phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\cdot \phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)={0}. \end{aligned}$$
(25)
$$\begin{aligned} &\left \| {\partial}_{x}\phi \big( x_{i},\mathcal{F}_{i}( x,\omega ) \big)\right \|_{S} \\ \le &\;2\left \| {{e}^{i}}\right \|+2\big\| {{\nabla }_{x}}{{ \mathcal{F}}_{i}}(x,\omega )\big\| \\ \le &\;2+2t_{2}. \end{aligned}$$
(26)
$$\begin{aligned} \left | \mathcal{F}_{i}( {x}^{k},\omega ^{j})-\mathcal{F}_{i}( \bar{x},\omega ^{j}) \right | =&\left \| {\nabla}_{x}\mathcal{F}_{i} \left ( \tau _{ikj}{x}^{k}+(1-\tau _{ikj})\bar{x},\omega ^{j}\right )^{T}({x}^{k}- \bar{x})\right \| \\ \le & t_{2}\left \| {x}^{k}-\bar{x}\right \|. \end{aligned}$$
(27)
By (25), for \(\forall i\in [n]\), we have where \(t_{3}=4t_{2}^{2}+8t_{2}+4\), the third inequality is based on (26), the fourth inequality follows from (27).
$$\begin{aligned} & \big\| {\partial}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\cdot \phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)-{\partial}_{x}\phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\cdot \phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big) \big\| \\ =&\big\| {\partial}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\cdot \phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)-0\big\| \\ \le & \big\| {\partial}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\big\| _{S}\cdot \big|\phi \big( x_{i}^{k}, \mathcal{F}_{i}( x^{k},\omega ^{j} )\big) \big| \\ \le & \big\| {\partial}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\big\| _{S}\cdot \big|\phi \big( x_{i}^{k}, \mathcal{F}_{i}( x^{k},\omega ^{j} )\big)-\phi \big( \bar{x}_{i}, \mathcal{F}_{i}( \bar{x},\omega ^{j} )\big) \big| \\ \le &(4+4t_{2})\left (|x_{i}^{k}-\bar{x}_{i}| +|\mathcal{F}_{i}( {x}^{k}, \omega ^{j})-\mathcal{F}_{i}( \bar{x},\omega ^{j})|\right ) \\ \le &(4+4t_{2})\Big(\|x^{k}-\bar{x} \| +t_{2}\left \| {x}^{k}-\bar{x} \right \|\Big) \\ \le &t_{3}\|x^{k}-\bar{x} \|, \end{aligned}$$
(28)
\(\mathrm{(ii)}\) When \({\omega}^{j}\notin{\Gamma}_{(i,\bar{x})}^{k}\), i.e. \(\big(\bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\neq (0,0)\), there holds \(\bar {x}_{i}\neq 0\) obviously. If \(\bar {x}_{i}>0\), there exists a positive integer \(Q_{i1}\) such that sequence \({{x}_{i}^{k}}\) satisfies \({x}_{i}^{k}>\frac{\bar {x}_{i}}{2}\) when \(k \ge Q_{i1}\); If \(\bar {x}_{i}<0\), then there exists a positive integer \(Q_{i2}\), such that the sequence \({{x}_{i}^{k}}\) satisfies \({x}_{i}^{k}< \frac{\bar {x}_{i}}{2}\) when \(k\ge Q_{i2}\). Taking \(Q_{i}= \max \{Q_{i1}, Q_{i2}\}\), for any \(k\ge Q_{i}\), there holds \({x}_{i}^{k}\neq 0\). Hence, by (8), we have
$$\begin{aligned} {\partial}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\cdot \phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)={{\nabla}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\cdot \phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)}. \end{aligned}$$
(29)
On the other hand, let \(\mathscr{D}_{i}\) be a minimum compact set containing sequence \(\{ x^{k} |\ k\ge Q_{i}\} \), which means that any element x in \(\mathscr{D}_{i}\) satisfies \(x_{i}\neq 0\). Since functions \(\phi \big( x_{i},\mathcal{F}_{i}( x,\omega )\big)\), \({\nabla}_{x}\phi \big( x_{i},\mathcal{F}_{i}( x,\omega )\big)\) and \({\nabla}_{x}^{2}\phi \big( x_{i},\mathcal{F}_{i}( x,\omega )\big)\) are continuous with respect to \((x,\omega )\) on a compact set \(\mathscr{D}_{i}\times \Omega \), there exists \(t_{4}\ge 0\), such that In addition, when \(k>Q_{i}\), according to the mean value theorem, there exists \(\lambda _{ikj}\in [0,1]\), for each i, \(x^{k}\) and \(\omega ^{j}\), such that
$$\begin{aligned} &\big|\phi \big( x_{i},\mathcal{F}_{i}( x,\omega )\big)\big|\le t_{4}, \quad \quad i\in [n], \end{aligned}$$
(30)
$$\begin{aligned} & \left \|{\nabla}_{x}\phi \big( x_{i},\mathcal{F}_{i}( x,\omega )\big) \right \|\le t_{4},\quad \quad i\in [n], \end{aligned}$$
(31)
$$\begin{aligned} & \left \|{\nabla}_{x}^{2}\phi \big( x_{i},\mathcal{F}_{i}( x,\omega ) \big)\right \|_{F}\le t_{4},\quad \quad i\in [n]. \end{aligned}$$
(32)
$$\begin{aligned} &{\nabla}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k},\omega ^{j} ) \big) -{\nabla}_{x}\phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x}, \omega ^{j} )\big) \\ =& \int _{0}^{1}{\nabla}_{x}^{2}\mathcal{\phi}_{i}\left ({\lambda _{ikj}}{x}^{k}+(1-{ \lambda _{ikj}})\bar{x},\omega ^{j}\right )^{T}(x^{k}-\bar{x}){d{ \lambda _{ikj}}} \\ \le & t_{4}\left \| x^{k}-\bar{x}\right \|. \end{aligned}$$
(33)
When \(k>Q_{i}\), we have from (29) that where \(t_{5}=t_{4}^{2}+2t_{4}+2t_{2}t_{4}\), the third inequality is obtained by using (30), (31) and (33), the fourth inequality follows from (27), the fifth inequality is based on (23).
$$\begin{aligned} & \big\| {\partial}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\cdot \phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)-{\partial}_{x}\phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\cdot \phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big) \big\| \\ =& \big\| {\nabla}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\cdot \phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)-{\nabla}_{x}\phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\cdot \phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big) \big\| \\ \le &\big\| {\nabla}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\cdot \phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)-{\nabla}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\cdot \phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\big\| \\ &+\big\| {\nabla}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\cdot \phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big) -{\nabla}_{x}\phi \big( \bar{x}_{i}, \mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\cdot \phi \big( \bar{x}_{i}, \mathcal{F}_{i}( \bar{x},\omega ^{j} )\big) \big\| \\ \le &\big\| {\nabla}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\big\| \cdot \big|\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k},\omega ^{j} )\big)-\phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\big| \\ &\quad \quad \quad \quad \quad +\big| \phi \big( \bar{x}_{i}, \mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\big|\cdot \big\| {\nabla}_{x} \phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k},\omega ^{j} )\big) -{ \nabla}_{x}\phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} ) \big) \big\| \\ \le & 2t_{4}\left (|x_{i}^{k}-\bar{x}_{i}| +|\mathcal{F}_{i}( {x}^{k}, \omega ^{j})-\mathcal{F}_{i}( \bar{x},\omega ^{j})|\right )+t_{4}^{2} \|{x}^{k}-\bar{x}\| \\ \le & 2t_{4}\left (\|x^{k}-\bar{x}\|+t_{2}\|x^{k}-\bar{x}\|\right )+t_{4}^{2} \|{x}^{k}-\bar{x}\| \\ \le & t_{5}\|x^{k}-\bar{x}\|, \end{aligned}$$
(34)
Taking (i) and (ii) into account, we have from (28) and (34) that
$$\begin{aligned} & \bigg\| \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{ \Omega }_{k}}} {\nabla}_{x}\left \| \Phi \big( x^{k},\mathcal{F}(x^{k}, \omega ^{j} ) \big) \right \|^{2}\rho (\omega ^{j}) - \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} { \nabla}_{x}\left \| \Phi \big( \bar{x},\mathcal{F}(\bar{x},\omega ) \big) \right \|^{2}\rho (\omega ^{j})\bigg\| \\ \le & \frac{2}{{{N}_{k}}}\sum \limits _{\omega ^{j}\in \Omega _{k}} \rho (\omega ^{j})\sum \limits _{i=1}^{n}\big\| {\partial}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k},\omega ^{j} )\big)\cdot \phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k},\omega ^{j} )\big)\big. \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \big. -{\partial}_{x}\phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\cdot \phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big) \big\| \\ =&\frac{2}{{{N}_{k}}}\sum \limits _{\omega ^{j}\in{{\Gamma}_{(i, \bar{x})}^{k}}}\rho (\omega ^{j})\sum \limits _{i=1}^{n}\big\| { \partial}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k},\omega ^{j} ) \big)\cdot \phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k},\omega ^{j} ) \big)\big. \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \big.-{\partial}_{x}\phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\cdot \phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big) \big\| \\ &+\frac{2}{{{N}_{k}}}\sum \limits _{\omega ^{j}\in \Omega _{k}/{{ \Gamma}_{(i,\bar{x})}^{k}}}\rho (\omega ^{j})\sum \limits _{i=1}^{n} \big\| {\partial}_{x}\phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big))\cdot \phi \big( x_{i}^{k},\mathcal{F}_{i}( x^{k}, \omega ^{j} )\big)\big. \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \big.-{\partial}_{x} \phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big)\cdot \phi \big( \bar{x}_{i},\mathcal{F}_{i}( \bar{x},\omega ^{j} )\big) \big\| \\ =& \frac{2t_{3}n}{{{N}_{k}}}\sum \limits _{\omega ^{j}\in{{\Gamma}_{(i, \bar{x})}^{k}}}\rho (\omega ^{j})\|{x}^{k}-\bar{x}\| + \frac{2t_{5}n}{{{N}_{k}}}\sum \limits _{\omega ^{j}\in \Omega _{k}/{{ \Gamma}_{(i,\bar{x})}^{k}}}\rho (\omega ^{j})\|{x}^{k}-\bar{x}\| \\ \to & 0,\quad (k\to +\infty ). \end{aligned}$$
(35)
In conclusion, by combining (24) and (35), it can be established that
$$\begin{aligned} &\bigg\| \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{ \Omega }_{k}}} {\nabla}_{x}\left \| \Phi \big( x^{k},\mathcal{F}(x^{k}, \omega ^{j} )\big) \right \|^{2} \rho (\omega ^{j})- \mathbb{E}\left [ {\nabla}_{x}\left \| \Phi \big( \bar{x},\mathcal{F}( \bar{x},\omega ) \big) \right \|^{2} \right ]\bigg\| \\ \le &\bigg\| \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{ \Omega }_{k}}} {\nabla}_{x}\left \| \Phi \big( x^{k},\mathcal{F}(x^{k}, \omega ^{j} ) \big) \right \|^{2}\rho (\omega ^{j}) - \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} { \nabla}_{x}\left \| \Phi \big( \bar{x},\mathcal{F}(\bar{x},\omega ) \big) \right \|^{2}\rho (\omega ^{j})\bigg\| \\ &+\bigg\| \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{ \Omega }_{k}}} {\nabla}_{x}\left \| \Phi \big( \bar{x},\mathcal{F}( \bar{x},\omega ^{j} )\big) \right \|^{2} \rho (\omega ^{j})- \mathbb{E}\left [ {\nabla}_{x}\left \| \Phi \big( \bar{x},\mathcal{F}( \bar{x},\omega ) \big) \right \|^{2} \right ]\bigg\| \\ \to & 0,\quad (k\to +\infty ). \end{aligned}$$
(36)
According to the continuous differentiability of \(\left \| \Phi \big( x,\mathcal{F}(x,\omega ) \big) \right \|^{2}\), it can be inferred that, for any \((x, \omega ) \in \mathscr{X} \times \Omega \), there exists a positive integer \(t_{1}\) such that By Theorem 16.8 in [1], it can be inferred that \(g (x) \) is differentiable with respect to x, and Therefore, we have from (36) and (37) that □
$$ \left \|{\nabla}_{x}\left \| \Phi \big( x,\mathcal{F}(x,\omega ) \big) \right \|^{2}\right \|\le t_{1}. $$
$$\begin{aligned} {\nabla}_{x} g(x)=\mathbb{E} \left [\nabla _{x} \left \| \Phi \big(x, \mathcal{F} (x, \omega ) \big) \right \|^{2} \right ]. \end{aligned}$$
(37)
$$ \underset{k\to +\infty }{\mathop{\lim }}\, \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {\nabla}_{x}\left \| \Phi \big( x^{k},\mathcal{F}(x^{k},\omega ^{j} ) \big) \right \|^{2} \rho (\omega ^{j})={\nabla}_{x} g(\bar{x}). $$
Lemma 4
For any \(i\in [n]\), \((\bar{x},\bar{y})\in R_{+}^{n}\times R^{n} \), and almost every \(\omega \in \Omega \), we have
$$ {\mathrm{{conv}}}\Big\{ \underset{{{x}^{k}}\to \bar{x},{y}^{k}\to \bar{y},\mu ^{k} \to 0}{\mathop{\overline{\lim }}} \,{{\nabla }_{xy}}{{H}_{i}}({{x}^{k}},{{y}_{i}^{k}},\omega , \mu ^{k} ) \Big\} \subseteq{{\partial }_{xy}}{{H}_{i}}(\bar{x},\bar {{y}_{i}}, \omega ). $$
Proof
For any \(i\in [n]\) and \((\bar{x},\bar{y})\in R_{+}^{n}\times R^{n} \), we define indicator sets Obviously, for any \(i\in [n]\) and \((\bar{x},\bar{y})\in R_{+}^{n}\times R^{n} \), there holds
$$\begin{aligned}& \mathcal{I}_{\{i,(\bar{x},\bar{y})\}}:= \{ \omega |\ \bar{y}_{i}>-{{ \mathcal{F}}_{i}}(\bar{x},\omega ) \},\\& \mathcal{L}_{\{i,(\bar{x},\bar{y})\}}:= \{ \omega |\ \bar{y}_{i}=-{{ \mathcal{F}}_{i}}(\bar{x},\omega ) \},\\& \mathcal{N}_{\{i,(\bar{x},\bar{y})\}}:= \{ \omega |\ \bar{y}_{i}< -{{ \mathcal{F}}_{i}}(\bar{x},\omega ) \}. \end{aligned}$$
$$ \mathcal{I}_{\{i,(\bar{x},\bar{y})\}}\cup \mathcal{L}_{\{i,(\bar{x}, \bar{y})\}}\cup \mathcal{N}_{\{i,(\bar{x},\bar{y})\}}=\Omega . $$
(i) Firstly, we give the expression of \({{\partial }_{xy}}{{H}_{i}}(\bar{x},\bar {{y}_{i}},\omega )\). Obviously, for any \(i\in [n]\) and \(\omega \in \Omega \), functions \(y_{i}\alpha _{i}\) and \(-{{\mathcal{F}}_{i}}( x,\omega )+\left ( {{\alpha }_{i}}-1 \right ){{y}_{i}}\) are continuously differentiable with respect to \((x,y)\in R_{+}^{n}\times R^{n}\). According to the Corollary on page 32 in [4] and Proposition 2.3.6 in [4], it can be concluded that for any \(i\in [n]\) and \((x,y)\in R_{+}^{n}\times R^{n}\), \(y_{i}\alpha _{i}\) and \(-{{\mathcal{F}}_{i}}(x,\omega )+\left ( {{\alpha }_{i}}-1 \right ) {{y}_{i}}\) are Clarke regular.
Therefore, according to the proposition 2.3.12 in [4], it can be inferred that when \(\omega \in \mathcal{L}_{\{i,(\bar{x},\bar{y})\}}\), i.e. \(\bar{y}_{i}=-{{\mathcal{F}}_{i}}(\bar{x},\omega )\), there holds
Therefore, we have
(38)
(ii) Next, we will provide the relationship between \({{\partial }_{xy}}{{H}_{i}}(\bar{x},\bar {{y}_{i}},\omega )\) and \(\mathrm{conv}\Big\{ \underset{{{x}^{k}}\to \bar{x},y^{k}\to \bar{y},\mu ^{k} \to 0}{\mathop{\overline{\lim }}} \,{{\nabla }_{xy}}{{H}_{i}}({{x}^{k}},{{y}^{k}_{i}},\omega ,\mu ^{k}) \Big\}\). Obviously, for any \(i\in [n]\), \(\mu \ne 0\), \(( x, y)\in R_{+}^{n} \times R^{n} \) and \(\omega \in \Omega \), there holds Note that for any \(i\in [n]\), \((x,y,\mu )\in R_{+}^{n}\times {R}^{n} \times {R}_{+}\) and almost every \(\omega \in \Omega \), we have This means that for any \(i\in [n]\), \((\bar{x},\bar{y})\in R_{+}^{n}\times R^{n} \) and almost every \(\omega \in \Omega \), there holds
(39)
$$ -1\le \frac{{{\mathcal{F}}_{i}}(x,\omega )+{{y}_{i}}}{\sqrt{4{{\mu }^{2}}+{{\left ( -{{\mathcal{F}}_{i}}(x,\omega )-{{y}_{i}} \right )}^{2}}}} \le 1. $$
Specifically, when \(\omega \in \mathcal{I}_{\{i,(\bar{x},\bar{y})\}}\), i.e. \(\bar{y}_{i}>-{{\mathcal{F}}_{i}}(\bar{x},\omega )\), for the sufficiently large k, we have \(y_{i}^{k}>-{{\mathcal{F}}_{i}}(x_{i}^{k},\omega )\). Combined with (39), it can be inferred that
When \(\omega \in \mathcal{N}_{\{i,(\bar{x},\bar{y})\}}\), i.e. \(\bar{y}_{i}<-{{\mathcal{F}}_{i}}(\bar{x},\omega )\), for the sufficiently large k, we have \(y_{i}^{k}<-{{\mathcal{F}}_{i}}(x_{i}^{k},\omega )\). Taking (39) into account, we can deduce that
In summary, it can be seen that for any \(i\in [n]\), \((\bar{x},\bar{y})\in R_{+}^{n}\times R^{n} \) and almost every \(\omega \in \Omega \), there holds Combined with the convexity of \({{\partial}_{xy}}{{H}_{i}}(\bar{x},\bar {{y}_{i}},\omega )\), it is evident that for any \(i\in [n]\), \((\bar{x},\bar{y})\in R_{+}^{n}\times R^{n} \), and almost every \(\omega \in \Omega \), we have □
$$\begin{aligned} \underset{{{x}^{k}}\to \bar{x},{{y}^{k}}\to \bar{y},\mu ^{k} \to 0}{\mathop{\overline{\lim }}} \,{{\nabla }_{xy}}{{H}_{i}}({{x}^{k}},{{y}^{k}_{i}},\omega ,\mu ^{k}) \subseteq {{\partial }_{xy}}{{H}_{i}}(\bar{x},\bar {{y}_{i}},\omega ). \end{aligned}$$
$$ {\mathrm{{conv}}}\Big\{ \underset{{{x}^{k}}\to \bar{x},y^{k}\to \bar{y},\mu ^{k} \to 0}{\mathop{\overline{\lim }}} \,{{\nabla }_{xy}}{{H}_{i}}({{x}^{k}},{{y}^{k}_{i}},\omega ,\mu ^{k}) \Big\} \subseteq{{\partial }_{(x,y)}}{{H}_{i}}(\bar{x},\bar {{y}_{i}}, \omega ). $$
Lemma 5
[11] Let Z be a compact set and \(\mathcal{J}(z, \omega ): Z \times \Omega \rightarrow 2^{R^{n}}\) be a measurable and compact set-valued mapping that is upper semi-continuous with respect to z on Z for almost every ω. Let \(\Omega _{k}=\{\omega ^{1}, \ldots , \omega ^{N_{k}}\}\) be a set with independently and identically distributed random samples and \(\mathcal{J}^{k}(z):=\frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}} \in {{\Omega }_{k}}} \mathcal{J}(z, \omega ^{j})\rho (\omega ^{j})\) where \(\Omega _{k}\subseteq \Omega \) and \(N_{k}\to +\infty \) when \(k\to +\infty \). Suppose that \(\mathcal{J}(z, \omega )\) is dominated by an integrable function. Let \(\{z^{k}\}\) be an arbitrary sequence in Z and z̄ be an accumulation point of \(\{z^{k}\}\). Then
$$ \lim _{k \rightarrow +\infty} \mathbb{D}\Big(\mathcal{J}^{k}(z^{k}), \mathbb{E}[\operatorname{conv} \mathcal{J}(\bar{z}, \omega )]\Big)=0. $$
Lemma 6
Assume that \(\mu _{0}> 0\) is a fixed constant, \(\mathscr{X}\times \mathscr{Y}\times \Omega \times [0,\mu _{0}] \subset R_{+}^{n}\times R^{n}\times R^{n}\times R_{+}\) is a compact set containing \(\lbrace x^{k},y^{k},\omega ,\mu ^{k}\rbrace \) and \(\underset{k\to +\infty }{\mathop{\lim }}\,( x^{k},y^{k},\mu ^{k}) =( \bar{x},\bar{y},0) \). Then, for each \(i\in [n]\), we have
$$\begin{aligned} \underset{k\to +\infty}{\mathop{\overline{\lim }}}\, \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {{ \nabla }_{xy}}{H_{i}(x^{k},y_{i}^{k},\omega ^{j},{\mu }^{k})}\rho ( \omega ^{j})\subseteq \mathbb{E}\left [ {{\partial }_{xy}}{{H}_{i}}( \bar{x},\bar{{y}_{i}},\omega ) \right ]. \end{aligned}$$
Proof
For each fixed \(i \in [n] \), we define a set value mapping:
which is a stochastic compact set-valued mapping obviously.
To apply Lemma 5, we need to prove that for any \(\omega \in \Omega \), the set valued mapping \(\mathcal{H}_{i}\left ( \cdot ,\cdot ,\omega ,\cdot \right )\) is upper semicontinuous with respect to the variable \((x, y, \mu ) \) on set \(\mathscr{X}\times \mathscr{Y}\times [0,\mu _{0}]\).
(i) If \(\mu \ne 0\), for every point \((x',y',\mu ')\) in a sufficiently small neighborhood of \((x,y,\mu )\), there holds \(\mathcal{H}_{i}\left ( x',y',\omega ,\mu ' \right )={{\nabla }_{xy}}{{H}_{i}}(x',{{y}_{i}'}, \omega ,\mu ' )\). We have from (39) that \({{\nabla }_{xy}}{{H}_{i }}(x,{{y}_{i}},\omega ,\mu )\) is continuous at \(\left (x,y, \mu \right )\).
(ii) If \(\mu =0\), it follows from the definition of \(\mathcal{H}_{i}\) that \(\mathcal{H}_{i}\left ( \cdot ,\cdot ,\cdot ,\omega \right )\) is closed at \(\left ( x,y,0\right ) \). Obviously, we know that \(\left (x',y', \mu '\right )\), \({{\partial }_{xy}}{{H}_{i }}(x',{{y}_{i}'},\omega , \mu ' )\) is bounded for every point \(\left (x',y', \mu '\right )\) in a sufficiently small neighborhood of \(\left (x,y, 0\right ) \). Thus, the closure of \(\bigcup \limits _{\left ( x',y',\mu ' \right )\in B\left ( (x,y,0) \right )}\mathcal{H}_{i}\bigl( x',y', \omega ,\mu ' \bigr)\) is compact. Using Lemma 1, we conclude that \(\mathcal{H}_{i}\left ( \cdot ,\cdot ,\omega , \cdot \right )\) is upper semi-continuous at \((x,y,0)\).
Since \(\mathcal{H}_{i}\left ( \cdot ,\cdot ,\omega ,\cdot \right )\) is upper semicontinuous with respect to variable \(\left (x,y, \mu \right ) \) on \(\mathscr{X}\times \mathscr{Y}\times [0,\mu _{0}]\), and \(\mathcal{H}_{i}\left ( x,y,\omega ,\mu \right )\) is clearly bounded on the compact set \(\mathscr{X}\times \mathscr{Y}\times \Omega \times [0,\mu _{0}]\), the condition in Lemma 5 hold. Combining Lemma 4 and Lemma 5, we have □
$$\begin{aligned} &\underset{k\to +\infty}{\mathop{\overline{\lim }}}\, \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} \mathcal{H}_{i}( x^{k},y^{k},\omega ^{j},{\mu }^{k} )\rho (\omega ^{j}) \\ \subseteq &\;\mathbb{E}\left [\mathcal{H}_{i}\left ( \bar{x},\bar{y}, \omega ,0 \right ) \right ] \\ =&\;\mathbb{E}\left [{\mathrm{{conv}}}\Big\{ \underset{{{x}'}\to \bar{x},{{y}'}\to \bar{y},\mu ' \to 0}{\mathop{\overline{\lim }}} \,{{\nabla }_{xy}}{{H}_{i}}({x}',{{y}_{i}}',\omega , \mu ' )\Big\} \right ] \\ \subseteq &\; \mathbb{E}\left [ {{\partial }_{xy}}{{H}_{i}}(\bar{x}, \bar{{y}_{i}},\omega ) \right ]. \end{aligned}$$
Theorem 1
Suppose that \(\mathrm{(A1)-(A3)}\) hold. Let \((x^{k},y^{k})\) be a stationary point of problem (15) for each k and \((\bar{x},\bar{y})\) be an accumulation point of \((x^{k},y^{k})\). Suppose that there exists a constant P such that \({{\Theta }^{k}}(x^{k},y^{k})\leq P\) for every k, then \((\bar{x},\bar{y})\) is a stationary point of problem (13).
Proof
Without loss of generality, we assume that \(\underset{k \to +\infty }{\mathop{\lim }}\,(x^{k},y^{k})=(\bar{x}, \bar{y})\). Let \(\mathscr{X}\times \mathscr{Y}\subseteq R_{+}^{n}\times R^{n}\) be a compact set containing the sequence \(\left \lbrace ( x^{k},y^{k}) \right \rbrace \). From the continuity of function \(\nabla _{x}{\mathcal{F}}({{x}},{{\omega }})\), it can be inferred that there exists a positive integer M such that
$$\begin{aligned} \|\nabla _{x}{\mathcal{F}}({{x}},{{\omega }})\|_{F}\le M,\quad \quad \forall (x,\omega )\in \mathscr{X}\times \Omega . \end{aligned}$$
(40)
According to \({{\Theta }^{k}}(x^{k},y^{k})\le P\) and (15), it is easy to verify that Based on [9], we have \({\big( {h^{k}(x^{k},y^{k},{\mu }^{k})}\big) _{+}}\to \big( {h(\bar{x}, \bar{y},\bar{\mu })}\big) _{+}\) and \({{g}^{k}}(x^{k})\to {{g}}(\bar{x})\) on the compact set \(\mathscr{X}\times \mathscr{Y}\times \Omega \subseteq R_{+}^{n} \times R^{n}\times R^{n}\) when \(k\to +\infty \). So taking the limits on both sides of (41), it can be inferred that \({{h(\bar{x},\bar{y})}_{+}}=0\), which implies \(h(\bar{x},\bar{y})\le 0\). Therefore, \((\bar{x},\bar{y})\) is a feasible point of problem (13).
$$\begin{aligned} {{\left \| {\big( {h^{k}(x^{k},y^{k},{\mu }^{k})}\big) _{+}} \right \|}^{2}}\le ( {{\sigma }^{k}}) ^{-1} (P- {{g}^{k}}(x^{k})) . \end{aligned}$$
(41)
On the other hand, for any \(i\in [n]\), we have where the third inequality is obtained from \(\sqrt{ {{a}^{2}}+{{b}^{2}} }\le \left | a \right |+\left | b \right |\) and \(\big|\left | a \right |-\left | b \right |\big|\le \left | a-b \right |\), the fifth inequality follows from the mean value theorem and (40).
$$\begin{aligned} & \left | {{H}_{i}}( {{x}^{k}},y_{i}^{k},{{\omega }^{j}},{\mu }^{k} )-{{H}_{i}}( \bar{x},{{{\bar{y}}}_{i}},{{\omega }^{j}} ) \right | \\ = &\Big| y_{i}^{k}{{\alpha }_{i}}+ \frac{\sqrt{4{{( {{\mu }^{k}} )}^{2}}+{{\left ( {{\mathcal{F}}_{i}}({{x}^{k}},{{\omega }^{j}})+y_{i}^{k} \right )}^{2}}}- {{\mathcal{F}}_{i}}({{x}^{k}},{{\omega }^{j}}) -y_{i}^{k} }{2} \Big. \\ &\quad \quad \quad \quad \quad \quad \Big.-{{{\bar{y}}}_{i}}{{ \alpha }_{i}}- \frac{\left | {{\mathcal{F}}_{i}}({\bar{x}},{{\omega }^{j}})+{{{\bar{y}}}_{i}} \right | -{{\mathcal{F}}_{i}}({\bar{x}},{\omega }^{j})-{{{\bar{y}}}_{i}} }{2} \Big| \\ \le &\frac{3}{2}\left | y_{i}^{k}-{{{\bar{y}}}_{i}} \right |+ \frac{1}{2}\Big( \sqrt{4{{\left ( {{\mu }^{k}} \right )}^{2}}+{{ \left ( {{\mathcal{F}}_{i}}({{x}^{k}},{{\omega }^{j}})+y_{i}^{k} \right )}^{2}}}-\left |{{\mathcal{F}}_{i}} (x^{k},{{\omega }^{j}})+{{y}^{k}_{i}} \right |\Big) \\ &\quad \quad \quad \quad \quad \quad +\frac{1}{2}\Big|\left |{{ \mathcal{F}}_{i}}(x^{k},{{\omega }^{j}})+{{y}^{k}_{i}} \right | - \left | {{\mathcal{F}}_{i}}(\bar{x},{{\omega }^{j}})+{{{\bar{y}}}_{i}} \right | \Big|+\frac{1}{2}\left | {{\mathcal{F}}_{i}}({{x}^{k}},{{ \omega }^{j}})-{{\mathcal{F}}_{i}}(\bar{x},{{\omega }^{j}}) \right | \\ \le & {{\mu }^{k}}+2\left | y_{i}^{k}-{{{\bar{y}}}_{i}} \right |+ \left | {{\mathcal{F}}_{i}}({{x}^{k}},{{\omega }^{j}})-{{\mathcal{F}}_{i}}({ \bar {x}},{{\omega }^{j}}) \right |, \\ \le & {{\mu }^{k}}+2\left | y_{i}^{k}-{{{\bar{y}}}_{i}} \right |+ \left \| {{\mathcal{F}}}({{x}^{k}},{{\omega }^{j}})-{{\mathcal{F}}}({ \bar {x}},{{\omega }^{j}}) \right \| \\ \le & {{\mu }^{k}}+2\left | y_{i}^{k}-{{{\bar{y}}}_{i}} \right |+M \left \| {{x}^{k}}- \bar {x} \right \|, \end{aligned}$$
(42)
Since \((\bar{x},\bar{y})\) is a feasible point of problem (13), then for any \(i \in [n]\), there holds \(\left ( \mathbb{E}\left [ {H_{i}(\bar{x},\bar{y}_{i},\omega )}\right ] \right ) _{+}=0\). Further, we have where the first inequality is obtained from \(\left | {{(a)}_{+}}-{{(b)}_{+}} \right |\le \left | a-b \right |\), the third inequality follows from (42) and the limit can be established by (A3) and the fact of \(\underset{k\to +\infty }{\mathop{\lim }}\,\frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} \rho (\omega ^{j})= \mathbb{E}\left [\rho (\omega ) \right ]=1\).
$$\begin{aligned} &\Big| {{\sigma }}^{k}\Big(\frac{1}{{{N}_{k}}}\sum \limits _{{{ \omega }^{j}}\in {{\Omega }_{k}}} {H_{i}(x^{k},y_{i}^{k},\omega ^{j},{ \mu }^{k})}\rho (\omega ^{j})\Big) _{+}\Big| \\ =& {{\sigma }}^{k}\Big|\Big(\frac{1}{{{N}_{k}}}\sum \limits _{{{ \omega }^{j}}\in {{\Omega }_{k}}} {H_{i}(x^{k},y_{i}^{k},\omega ^{j},{ \mu }^{k})}\rho (\omega ^{j})\Big) _{+}-\left ( \mathbb{E}\left [ {H_{i}( \bar{x},\bar{y}_{i},\omega )}\right ]\right ) _{+}\Big| \\ \le & {{\sigma }}^{k}\Big|\frac{1}{{{N}_{k}}}\sum \limits _{{{ \omega }^{j}}\in {{\Omega }_{k}}} {H_{i}(x^{k},y_{i}^{k},\omega ^{j},{ \mu }^{k})} \rho (\omega ^{j})-\mathbb{E}\left [ {H_{i}(\bar{x},\bar{y}_{i}, \omega )}\right ]\Big| \\ \le & \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} \rho (\omega ^{j}){{\sigma }}^{k}\Big|{{H_{i}(x^{k},y_{i}^{k},\omega ^{j},{ \mu}^{k})}}-{{H_{i}(\bar{x},\bar{y}_{i},\omega ^{j})}}\Big| \\ &\quad \quad \quad \quad \quad \quad \quad + {{\sigma }}^{k}\Big|{{ \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {H_{i}( \bar{x},\bar{y}_{i},\omega ^{j})}}}\rho (\omega ^{j})-\mathbb{E}\left [ {H_{i}( \bar{x},\bar{y}_{i},\omega )}\right ]\Big| \\ \le & {{\sigma }}^{k} {{\mu }^{k}}\cdot \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} \rho (\omega ^{j})+2{{ \sigma }}^{k}\left | y_{i}^{k}-{{{\bar{y}}}_{i}} \right | \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} \rho (\omega ^{j})+ M{{\sigma }}^{k} \left \| x^{k}-\bar{x}\right \| \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} \rho (\omega ^{j}) \\ & \quad \quad \quad \quad \quad \quad \quad + {{\sigma }}^{k}\Big|{{ \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {H_{i}( \bar{x},\bar{y}_{i},\omega ^{j})}}}\rho (\omega ^{j}) -\mathbb{E}\left [ {H_{i}(\bar{x},\bar{y}_{i},\omega )}\right ]\Big| \\ \to & 0,\quad (k\to +\infty ), \end{aligned}$$
(43)
(i) Firstly, we prove the boundedness of sequence \(\left \lbrace \iota ^{k} \right \rbrace \). On the contrary, we assume that the sequence \(\left \lbrace \iota ^{k}\right \rbrace \) is unbounded, and then we can find a subcolumn that tends to infinity. For sign simplicity, we write it as \(\underset{k \to +\infty }{\mathop{\lim }}\,\left \| \iota ^{k} \right \|= +\infty \). Dividing the two sides of (19) by \(\left \| \iota ^{k}\right \|\), we obtain We have from Lemma 3 that Therefore, when \(k\to +\infty \), the first term on the right-hand side of (44) converges to 0. On the other hand, according to Lemma 6, for any \(i \in [n] \), there holds According to (38), it is easy to know that for \(\forall i\in [n]\), \(\mathbb{E}\left [ {{\partial}_{xy}}H_{i}(\bar{x},\bar{y}_{i}, \omega )\right ]\) is bounded on a compact set Ω. Combined with (43), i.e. it can be seen that when \(k\to +\infty \), the third term on the right-hand side of (44) converges to 0.
$$\begin{aligned} 0&=\frac{1}{\left \| \iota ^{k}\right \|} \begin{pmatrix} \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} { \nabla}_{x}\left \| \Phi \big( x^{k},\mathcal{F}(x^{k},\omega ) \big) \right \|^{2} \rho (\omega ^{j}) \\ 0 \end{pmatrix} + \begin{pmatrix} \frac{\sum \limits _{i=1}^{n}{-\iota _{i}^{k}}e_{i} }{\left \| \iota ^{k}\right \|} \\ 0 \end{pmatrix} \\ &\quad {}+\frac{1}{\left \| \iota ^{k}\right \|} \begin{pmatrix} \sum \limits _{i=1}^{n}2{{\sigma }}^{k} \Big(\frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {H_{i}(x^{k},y_{i}^{k}, \omega ^{j},{\mu }^{k})}\rho (\omega ^{j})\Big) _{+} \cdot \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {{ \nabla }_{x}}{H_{i}(x^{k},y_{i}^{k},\omega ^{j},{\mu }^{k})}\rho ( \omega ^{j}) \\ \sum \limits _{i=1}^{n}2{{\sigma }}^{k}\Big(\frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {H_{i}(x^{k},y_{i}^{k}, \omega ^{j},{\mu }^{k})}\rho (\omega ^{j})\Big) _{+} \cdot \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {{ \nabla }_{y}}{H_{i}(x^{k},y_{i}^{k},\omega ^{j},{\mu }^{k})}\rho ( \omega ^{j}) \end{pmatrix} . \end{aligned}$$
(44)
$$ \underset{k\to + \infty }{\mathop{\lim }}\, \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {\nabla}_{x}\left \| \Phi \big( x^{k},\mathcal{F}(x^{k},\omega ^{j} )\big) \right \|^{2} \rho (\omega ^{j}) = {\nabla}_{x} g(\bar{x}). $$
$$ \underset{k \to +\infty }{\mathop{\overline{\lim }}}\, \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}}{{ \nabla }_{xy}}{H_{i}(x^{k},y_{i}^{k},\omega ^{j},{\mu}^{k})}\rho ( \omega ^{j})\subseteq \mathbb{E}[{{\partial }_{xy}}H_{i}(\bar{x}, \bar{y}_{i},\omega )]. $$
$$ {{\sigma }}^{k}\Big(\frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}} \in {{\Omega }_{k}}} {H_{i}(x^{k},y_{i}^{k},\omega ^{j},{\mu }^{k})} \rho (\omega ^{j})\Big) _{+}\xrightarrow{k\to +\infty }0, $$
In summary, when we take the limit on (44), \(\underset{k \to +\infty }{\mathop{\lim }}\, \frac{\sum \limits _{i=1}^{n}{-\iota _{i}^{k}}e_{i} }{\left \| \iota ^{k}\right \|} =0\) holds. This is clearly contradictory to \(\underset{k \to +\infty }{\mathop{\lim }}\, \frac{\left \| \sum \limits _{i=1}^{n}{-\iota _{i}^{k}}e_{i} \right \| }{\left \| \iota ^{k}\right \|} =1\). Therefore, the sequence \(\left \lbrace \iota ^{k} \right \rbrace \) is bounded.
(ii) Next, we prove that \((\bar{x},\bar{y})\) is a stationary point of problem (13).
Since \(\left \lbrace \iota ^{k} \right \rbrace \) is bounded, at least one subcolumn has the limit. For symbol simplicity, we set By Lemma 3, we have Therefore, taking the limit on (19), the first term on the right-hand side converges to the first term on the right-hand side of (16). Furthermore, according to (38), for \(\forall i\in [n]\), \(\mathbb{E}\left [ {{\partial }_{xy}}H_{i}(\bar{x},\bar{y}_{i}, \omega )\right ]\) is bounded on a compact set Ω. By Lemma 6, i.e. it is easy to know that \(\frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}}{{ \nabla }_{xy}}{H_{i}(x^{k},y_{i}^{k},\omega ^{j},{\mu}^{k})}\rho ( \omega ^{j})\) is bounded. From (43), i.e. it can be seen that the second term on the right-hand side of (19) converges to 0, corresponding to \(\bar{\upsilon}=0\) in (16). Finally, the third term on the right-hand side of (19) converges to the third term on the right-hand side of (16), which can be directly obtained by (45).
$$\begin{aligned} \bar{\iota} = \underset{k \to +\infty }{\mathop{\lim }}\,\iota ^{k}. \end{aligned}$$
(45)
$$\begin{aligned} \underset{k\to + \infty }{\mathop{\lim }}\, \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}} {\nabla}_{x}\left \| \Phi \big( x^{k},\mathcal{F}(x^{k},\omega ^{j} )\big) \right \|^{2} \rho (\omega ^{j}) = {\nabla}_{x} g(\bar{x}). \end{aligned}$$
$$ \underset{k \to +\infty }{\mathop{\overline{\lim }}}\, \frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}}\in {{\Omega }_{k}}}{{ \nabla }_{xy}}{H_{i}(x^{k},y_{i}^{k},\omega ^{j},{\mu}^{k})}\rho ( \omega ^{j})\subseteq \mathbb{E}[{{\partial }_{xy}}H_{i}(\bar{x}, \bar{y}_{i},\omega )],\quad \quad i\in [n], $$
$$ {{\sigma }}^{k}\Big(\frac{1}{{{N}_{k}}}\sum \limits _{{{\omega }^{j}} \in {{\Omega }_{k}}} {H_{i}(x^{k},y_{i}^{k},\omega ^{j},{\mu }^{k})} \rho (\omega ^{j})\Big) _{+}\xrightarrow{k\to +\infty }0, $$
4 Conclusions
In this paper, we study the CVaR-ERM model for solving STCP. Due to the difficulties associated with using existing algorithms to solve this model, we give the approximation problem of this model by the smoothing method, the penalty function method and the sample average approximation method. In this paper, we focus on the convergence of the stationary points of the approximation problem. Owing to the complexity of the gradient of \(\left \| \Phi \big( x,\mathcal{F}(x,\omega ) \big) \right \|^{2}\), traditional methods for proving the convergence of stationary points are not applicable; instead, we successfully prove the convergence of the stationary points for the approximation problem by leveraging the special properties of \(\nabla _{x}\left \| \Phi \big( x,\mathcal{F}(x,\omega ) \big) \right \|^{2}\).
Acknowledgements
The authors wish to thank the referees and the editor for their insightful comments and valuable suggestions for improvement of the paper.
Declarations
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
