I have the following multiobjective optimization problem
Here, $a^{(l)}_{c,u}$ is an binary variable, $f^{(l)}_{c,u}$ is a bounded continuous variable and $z_{l,t}$ is also a binary variable.
TheThe objectives are non-conflicting.
$\mathcal{C}^{(l)}=\left\{1,2,\cdots,N_C^{(l)}\right\}$ and $\mathcal{U}^{(l)}=\left\{1,2,\cdots,N_U^{(l)}\right\}$
$\textbf{The Optimization problem:}$
\begin{equation}
\label{main:psfinal}
\begin{array}{*{35}{l}}
\underset{a^{(l)}_{c,u},f^{(l)}_{c,u},z_{l,t},l\in\mathcal{L}}{\max}\hspace{6mm}\left\{t^{(l)}_{\rm U},l=1,2,\cdots,N_{\rm L},t_{\rm L}\right\}\\
\text{}\text{subject to }\text{ C1:} \hspace{2mm}\sum_{c=1}^{N^{(l)}_{\rm C}} a^{(l)}_{c,u}\le \Delta_{\max},\forall u,u\in\mathcal{U}^{(l)}, \\
\text{}\hspace{15mm}\text{ C2:} \hspace{2mm}\sum_{u=1}^{N^{(l)}_{\rm U}} f^{(l)}_{c,u}\le 1, \forall c,c\in\mathcal{C}^{(l)}, \\
%\text{}\hspace{15mm}\text{ C4:} \hspace{2mm}\frac{f^{(l)}_{c,u}}{\sum_{c=1}^{N^{(l)}_{\rm C}}f^{(l)}_{c,u}}\ge \Theta_{\max},\hspace{1mm} u=1,2,\cdots,N^{(l)}_{\rm U}\\
%\text{}\hspace{15mm}\text{ C6:} \hspace{2mm}0\le f^{(l)}_{c,u}\le 1, u=1,\cdots, N^{(l)}_{\rm U}, c=1,\cdots, N^{(l)}_{\rm C} \\
%\text{}\hspace{15mm}\text{ C1:} \hspace{2mm} s_{u,l}\ge t_{\rm U} d_{u,l}\\
%\text{}\hspace{15mm}\text{ C1:} \hspace{2mm}s_{l}\ge t_{\rm C} d_{l}\\
\text{}\hspace{15mm}\text{ C3:} \hspace{2mm}\sum_{l=1}^{N_{\rm L}}z_{l,t}\le N_{\rm T},\forall t,t=1,2,\cdots, N_{\rm TS}\\
\text{}\hspace{15mm}\text{ C4:} \hspace{2mm} s_{u,l}\ge t^{(l)}_{\rm U} d_{u,l},\forall u,u\in\mathcal{U}^{(l)}, l=1,2,\cdots,{N_{\rm L}}\\
\text{}\hspace{15mm}\text{ C5:} \hspace{2mm} s_{l}\ge t_{\rm L} d_{l},l=1,2,\cdots,{N_{\rm L}}\\
\text{}\hspace{15mm} \text{ C6}:\left\{ \begin{array}{*{35}{l}}
\text{C6-a}: f^{(l)}_{c,u} \le a^{(l)}_{c,u}\\
\text{C6-b}: f^{(l)}_{c,u} \ge \epsilon - (1-a^{(l)}_{c,u})
\end{array}\right.\\
\text{}\hspace{15mm} \text{ C7}:\left\{ \begin{array}{*{35}{l}}
\text{C7-a}:q^{(l)}_{c,u,t}\ge 0\\
\text{C7-b}:q^{(l)}_{c,u,t}\le z_{l,t}\\
\text{C7-c}:q^{(l)}_{c,u,t}\le f^{(l)}_{c,u}\\
\text{C7-d}:q^{(l)}_{c,u,t}\ge f^{(l)}_{c,u}-(1- z_{l,t})
\end{array}\right.
%\text{}\hspace{15mm}\text{ C4:} \hspace{2mm} z_{n_1,t} + z_{n_2,t} \leq 1, \forall (n_1, n_2)\in \mathcal{P} \text{ with } \mathcal{P}_{n_1 n_2}=1
\end{array}
\end{equation}
with
\begin{equation}
s_{u,l}=\sum_{t=1}^{N_{\rm TS}}\sum_{c=1}^{N^{(l)}_{\rm C}}q^{(l)}_{c,u,t}r^{(l)}_{c,u},,\forall u,u\in\mathcal{U}^{(l)}
\end{equation}
and
\begin{equation}
s_{l}=\sum_{u\in\mathcal{U}^{(l)}}s_{u,l}
\end{equation}
I also have
\begin{equation}
d_{l}=\sum_{u\in\mathcal{U}^{(l)}}d_{u,l}
\end{equation}
Let, $N_U^{(1)}=N_U^{(2)}=\cdots=N_U^{(N_L)}=30$
and $N_C^{(1)}=N_C^{(2)}=\cdots=N_C^{(N_L)}=8$, $N_{TS}=264$
$N_L=17$, $N_T=3$, $\Delta_{\max}=2$.
$r^{(l)}_{c,u}$ are known ($>0$).\begin{equation}
\label{main:psfinal}
\underset{a^{(l)}_{c,u},f^{(l)}_{c,u},z_{l,t},l\in\mathcal{L}}{\max}\hspace{6mm}\left\{t^{(l)}_{\rm U},l=1,2,\cdots,N_{\rm L},t_{\rm L}\right\}
\end{equation}
Since there is no preference (all the objectives have the same priority), I tried with the following objective.
$\underset{a^{(l)}_{c,u},f^{(l)}_{c,u},z_{l,t},l\in\mathcal{L}}{\max}\hspace{6mm}\left\{\sum_{l=1}^{N_L}t^{(l)}_{\rm U}+t_{\rm L}\right\}$
I am not sure if I am doing it right. The program runs for ever. When I interrupt it, I see that the constraint C5 is being working while the constraints C4 are not being maintained. (I tried a smaller version of it).
I am using Matlab, CVX with Gurobi. Even MOSEK behaves the same.
So, how can I resolve this issue. Is the single objective that I am using correct?
