\begin{equation}
    \label{eq:sub_p0}
    \begin{aligned}
        \min_{\{\mathbf{P},\bar{\mathbf{c}}\}} ~~&\frac{1}{A}\sum\nolimits_{k}\sum\nolimits_{a} \lambda_{\mathrm{p},k}^a\epsilon_{\mathrm{p},k}^a - \sum\nolimits_{k}\bar{C}_k\\
        \text{s.t.}~~~&\text{(\ref{eq:constraint_p}), (\ref{eq:constraint_c2}), (\ref{eq:constraint_c1_modified}), (\ref{eq:constraint_qos_modified})}.
    \end{aligned}
\end{equation}
