Without loss of generality, we assume that
\(\lim_{k\rightarrow\infty }(x^{k},u^{k})=(\bar{x},\bar{u})\). Let B be a compact set including the whole sequence
\(\{(x^{k},u^{k})\}\).
\(Z\subseteq R^{+}\) is a compact set including
\(\{\mu^{k}\}\). Because
\(h_{\mu}(x,u,\omega)\) is a continuously differentiable function on the compact set
\(B\times\Omega \times Z\), we can obtain that there exists a constant
\(C>0\) such that, for any
\((x,u,\omega)\in B\times\Omega\) and
\(\mu\in Z\), the following formulation holds:
$$\begin{aligned} \bigl\Vert \nabla h_{\mu}(x,u,\omega) \bigr\Vert \le C. \end{aligned}$$
(5.1)
Besides, from mean value theorem, for each
\(x^{k}\),
\(u^{k}\),
\(\omega^{i}\), and
\(\mu^{k}\), there exists
\(\alpha_{ki}\in(0,1)\) such that
\((x^{ki},u^{ki})=\alpha_{ki}(x^{k},u^{k})+(1-\alpha_{ki}) (\bar{x},\bar{u})\in B\), we then have
$$ \begin{aligned}[b] &h_{\mu^{k}} \bigl(x^{k}, u^{k},\omega^{i} \bigr)-h_{\mu^{k}} \bigl(\bar{x},\bar{u},\omega^{i} \bigr) \\ &\quad=\nabla h_{\mu^{k}} \bigl(x^{ki},u^{ki}, \omega^{i} \bigr)^{T} \bigl( \bigl(x^{k},u^{k} \bigr)-(\bar{x},\bar{u}) \bigr). \end{aligned} $$
(5.2)
Therefore, from (
5.1) and (
5.2), we have
$$ \begin{aligned}[b] & \bigl\vert \Theta^{k}_{\mu^{k}} \bigl(x^{k},u^{k} \bigr)- \Theta^{k}_{\mu^{k}}( \bar{x},\bar {u}) \bigr\vert \\ &\quad\leq \bigl\vert u^{k}-\bar{u} \bigr\vert +(1- \alpha)^{-1}\frac{1}{N_{k}}\sum_{\omega^{i}\in \Omega_{k}} \bigl\vert h_{\mu^{k}} \bigl(x^{k},u^{k}, \omega^{i} \bigr)-h_{\mu^{k}} \bigl(\bar{x},\bar {u}, \omega^{i} \bigr) \bigr\vert \\ &\quad\leq \bigl\vert u^{k}-\bar{u} \bigr\vert +(1- \alpha)^{-1}\frac{1}{N_{k}}\sum_{\omega^{i}\in \Omega_{k}} \bigl\Vert \nabla h_{\mu^{k}} \bigl(x^{ki},u^{ki}, \omega^{i} \bigr) \bigr\Vert \\ &\qquad{}\cdot \bigl\Vert \bigl(x^{k},u^{k} \bigr)-( \bar{x},\bar{u}) \bigr\Vert \\ &\quad\leq \bigl\vert u^{k}-\bar{u} \bigr\vert +\frac{C}{(1-\alpha)} \cdot\frac{1}{N_{k}}\sum_{\omega^{i}\in \Omega_{k}} \bigl\Vert \bigl(x^{k},u^{k} \bigr)-(\bar{x},\bar{u}) \bigr\Vert \\ &\quad\xrightarrow{k\rightarrow\infty}0,\quad \mbox{w.p.1}. \end{aligned} $$
(5.3)
On the other hand, from (
3.4) and (
3.6), we have
$$ \begin{aligned}[b] & \bigl\vert \Theta^{k}_{\mu^{k}}( \bar{x},\bar{u})-\Theta(\bar{x},\bar{u}) \bigr\vert \\ &\quad\leq(1-\alpha)^{-1} \biggl\vert \frac{1}{N_{k}}\sum _{\omega^{i}\in \Omega_{k}}h_{\mu^{k}} \bigl(\bar{x},\bar{u}, \omega^{i} \bigr)-{\mathbf{E}} \bigl[h(\bar{x},\bar {u},\omega) \bigr] \biggr\vert \\ &\quad\leq(1-\alpha)^{-1} \biggl[\frac{1}{N_{k}}\sum _{\omega^{i}\in \Omega_{k}} \bigl\vert h_{\mu^{k}} \bigl(\bar{x},\bar{u}, \omega^{i} \bigr)-h \bigl(\bar{x},\bar {u},\omega^{i} \bigr) \bigr\vert \\ &\qquad{}+ \biggl\vert \frac{1}{N_{k}}\sum_{\omega^{i}\in \Omega_{k}}h \bigl(\bar{x},\bar{u},\omega^{i} \bigr)-{\mathbf{E}} \bigl[h(\bar{x}, \bar {u},\omega) \bigr] \biggr\vert \biggr] \\ &\quad\leq(1-\alpha)^{-1} \biggl[\frac{1}{N_{k}}\sum _{\omega^{i}\in \Omega_{k}}\mu_{k}\ln2 \\ &\qquad{}+ \biggl\vert \frac{1}{N_{k}}\sum_{\omega^{i}\in \Omega_{k}}h \bigl(\bar{x},\bar{u},\omega^{i} \bigr)-{\mathbf{E}} \bigl[h(\bar{x}, \bar {u},\omega) \bigr] \biggr\vert \biggr] \\ &\quad\xrightarrow{k\rightarrow\infty}0, \quad \mbox{w.p.1}. \end{aligned} $$
(5.4)
Similarly, we obtain
$$\begin{aligned} \lim_{k\rightarrow\infty}\Theta^{k}_{\mu^{k}}(x,u)= \Theta(x,u),\quad \mbox{w.p.1}. \end{aligned}$$
(5.5)
Since
$$\begin{aligned} & \bigl\vert \Theta^{k}_{\mu^{k}} \bigl(x^{k},u^{k} \bigr)-\Theta(\bar{x},\bar{u}) \bigr\vert \\ &\quad\leq \bigl\vert \Theta^{k}_{\mu^{k}} \bigl(x^{k},u^{k} \bigr)-\Theta^{k}_{\mu^{k}}( \bar {x},\bar{u}) \bigr\vert + \bigl\vert \Theta^{k}_{\mu^{k}}( \bar{x},\bar{u})-\Theta(\bar{x},\bar{u}) \bigr\vert , \end{aligned}$$
from (
5.3) and (
5.4) we have
$$\begin{aligned} \lim_{k\rightarrow\infty}\Theta^{k}_{\mu^{k}} \bigl(x^{k},u^{k} \bigr)=\Theta(\bar {x},\bar{u}),\quad \mbox{w.p.1}. \end{aligned}$$
(5.6)
In fact, for each
k, due to
\((x^{k},u^{k})\) being a global optimal solution of problem (
3.7), we have
$$\begin{aligned} \Theta^{k}_{\mu^{k}} \bigl(x^{k},u^{k} \bigr)\leq\Theta^{k}_{\mu^{k}}(x,u),\quad\forall (x,u)\in R^{n}\times R_{+}. \end{aligned}$$
(5.7)
Taking limits to (
5.7), by (
5.5) and (
5.6), we have
$$\begin{aligned} \Theta(\bar{x},\bar{u})\leq\Theta(x,u),\quad \mbox{w.p.1}, \forall (x,u)\in R^{n}\times R_{+}. \end{aligned}$$
□