I see following is the standard max weight bipartite matching problem,
\begin{align} \max_{X} &\sum_{i,j} C_{i,j} X_{i,j} \\ \mathrm{s.t.}& \\ &\sum_i X_{i,j} = 1 \\ &\sum_j X_{i,j} = 1 \end{align}
In my problem, the cost matrix $C$ is also a variable matrix, with following linear constraints; \begin{equation} A c \leq b \end{equation} where $c=\mathrm{vec}(C)$, is vectorized form of matrix $C$. That is, each column of matrix $C$ is taken and appended together to create vector $c$.
So, my problem is as follows,
\begin{align} \max_{X,C} &\sum_{i,j} C_{i,j} X_{i,j} \\ \mathrm{s.t.}& \\ &\sum_i X_{i,j} = 1 \\ &\sum_j X_{i,j} = 1 \\ & \hspace{0.5cm} A c \leq b \end{align}
Can anyone help me please?