## 3 Network Architecture and System Model
### Transform from DRO to SDP  
We reconstruct the original problem by considering the uncertainty of computational resources and acknowledging the challenge of directly addressing the DRO problem. To solve this, we employ the Lagrangian dual to transform it into a semi-definite programming (SDP) problem. Problem (23) presents a DRO problem. Below, we further analyze the target formula and convert it into a more explicit form. \[ \begin{array}{rcl}&\sup_{\xi}\{(1-\lambda)\mathbb{E}[\theta(x,\xi)]+\lambda CVaR (\theta(x,\xi))\}\\ =&\sup_{\xi}\{\;\min_{v\in R}\{\lambda v+(1-\lambda)\mathbb{E}[\theta(x,\xi)] +\frac{\lambda}{1-\alpha}\mathbb{E}[(\theta(x,\xi)-v)_{+}]\}\}\\ =&\min_{v\in R}\{\lambda v+\sup_{\xi}\{\mathbb{E}[(1-\lambda)\theta(x,\xi)+ \frac{\lambda}{1-\alpha}(\theta(x,\xi)-v)_{+}]\}\}\end{array}  \tag{24}\]  
The ambiguous distribution set of the random parameter cannot be used directly for the solution, and we need to convert it into the common inequality form. Consider only the latter half, which involves finding the optimal value in relation to the random parameter \(\xi\). Instead of the original distribution set form, we transform it into the following semi-infinite conic linear problem form, expressed as:\[\sup_{\xi}\int_{\mathcal{M}}\left[(1-\lambda)\theta(x,\xi)+\frac{ \lambda}{1-\alpha}(\theta(x,\xi)-v)_{+}\right]\,df(\xi) \tag{25}\] \[s.t.\int_{\mathcal{M}}df(\xi)=1,\] (25a) \[\int_{\mathcal{M}}\left(\xi-\mu_{0}\right)(\xi-\mu_{0})^{T}df(\xi) \prec\gamma_{2}\sum\] (25b) \[\int_{\mathcal{M}}\left[ \begin{array}{cc}\sum&\xi-\mu_{0}\\ (\xi-\mu_{0})^{T}&\gamma_{1}\end{array} \right]df(\xi)\succcuit 0,\] (25c) \[\xi\in\mathcal{M}^{\xi}, \tag{25d}\]  
Secondly, we utilize the duality theory to convert it into an SDP problem. Consider the dual of problem (25):  
\[(Dual):\] \[ \begin{array}{l}\inf_{r,H,}\left[\begin{array}{cc}\inf_{ \begin{array}{cc}Z&z\\ z^{T}&\widehat{z}\end{array}\end{array}\right]}r+\left\langle H,\gamma_{2}\sum -\mu_{0}\mu_{0}^{T}\right\rangle+\left\langle Z,\sum\right\rangle+\widehat{z} \gamma_{1}-2z^{T}\mu_{0}\\ s.t.\hskip 14.226378 ptu(x,\xi)-r-\xi H\xi^{T}+2\xi^{T}H\mu_{0}+2z^{T}\xi\leq 0,\xi\in\mathcal{M}^{\xi},\end{array}  \tag{26a}\] \[r\in R,\] (26b) \[H\succcuit 0,\] (26c) \[\left[ \begin{array}{cc}Z&z\\ z^{T}&\widehat{z}\end{array} \right]\succcuit 0,\] (26d) \[\text{with}\hskip 5.690551 ptu(x,\xi)=(1-\lambda)\theta(x,\xi)+\frac{ \lambda}{1-\alpha}(\theta(x,\xi)-v)_{+}\]  
where \(r\), \(H\), and \(\left[ \begin{array}{cc}Z&z\\ z^{T}&\widehat{z}\end{array} \right]\) are, correspondingly, the dual variables of constraints (25a), (25b), and (25c). \(\left\langle a,b\right\rangle\) represents \(Trace(AB)\). Due to the inclusion of \(\left(\right)_{+}\), the inequality constraint (26a) has to be processed as follows:  
\[r+\xi H\xi^{T}-2\xi^{T}H\mu_{0}-2z^{T}\xi\] \[\geqslant (1-\lambda)\theta(x,\xi)+\frac{\lambda}{1-\alpha}(\theta(x,\xi)-v )_{+}\]When the random parameter \(\xi\) appears in the constraint, it cannot be directly substituted into constraint (26a). Therefore, it is necessary to consider the dual of problem \(\theta(x,\xi)\), where the random parameters are transferred into the objective function. Another problem arises, which is the coupling of variables between the dual variable \(q\) and variable \(Y\). The constraint on the dual variable \(q\) confines it to a boxed region, as indicated by the constraint. \[(Dual):\] \[\theta^{\prime}(x,\xi)=\max_{q,p}q^{T}\Big{(}(X\otimes F^{req})^{T}O_{N}- \xi\Big{)}, \tag{27}\] \[s.t.\ \ \varphi-p\geqslant q\geqslant 0,\] (27a) \[p\geqslant 0, \tag{27b}\]  
Responding to constraint (26a), taking its extreme value points, each point corresponds to two corresponding semi-definite matrix constraints. We transformed (26a) into a clearer and more concise form. \[\left\{\begin{array}{l}r+\xi H\xi^{T}-2\xi^{T}H\mu_{0}-2z^{T}\xi\\ \geqslant(1-\lambda)\theta^{\prime}(x,\xi)+\frac{\lambda}{1-\alpha}(\theta^{ \prime}(x,\xi)-v)\\ =\left(1+\frac{\lambda\alpha}{1-\alpha}  
\[\min_{X,p,r,H}\left[