I am trying to implement a Column Generation approach, in which I separate the continuous and the discrete variables. (Bender is not possible due to different reasons)
To get an initial feasible solution, I solve the original problem without objective function. Then I use this solution and plug it into the Restricted Master Problem (Only containing this one solution) to get the dual variables of the constraints. After that, I solve the subproblems and check if the objective function is <0. If yes, I add the solution as a column to the RMP and repeat.
For some reason, I don't get an optimal solution.
The original problem has the following form ($x$ is continuous and $z$ discrete):
$min_{x,z} \vec{c}_r \cdot \vec{x} + \vec{c}_b \cdot \vec{z}$
$A_b \vec{z} = \vec{a}_b$
$B_b \vec{z} \leq \vec{b}_b$
$A_r \vec{x} = \vec{a}_r$
$B_r \vec{x} \leq \vec{b}_r$
$B_{mr} \vec{x} + B_{mb} \vec{z} \leq \vec{b}_m$
So the restricted master problem is defined as follows:
$min_{\lambda_i, \mu_j} \sum_i (\vec{c}_r \bar{x}_i) \lambda_i + \sum_j (\vec{c}_b \bar{z}_j) \mu_j$
$\sum_i(B_{mr} \bar{x}_i) \lambda_i + \sum_j (B_mb \bar{z}_j) \mu_j \leq \vec{b}_m$
$\sum_i \lambda_i = 1$, dual: $\alpha$
$\sum_j \mu_j = 1$, dual: $\beta$
$\lambda_i, \mu_j \geq 0$
For completeness, the Column Generating Subproblems are ($\rho$ is the dual of the coupling constraint of the RMP, which is always equal to zero):
$min_x \vec{c}_r \vec{x} - B_{mr} \vec{\rho} \cdot \vec{x} - \alpha$
$A_r \vec{x} = \vec{a}_r$
$B_r \vec{x} \leq \vec{b}_r$
and:
$min_z \vec{c}_b \vec{z} - B_{mb} \vec{\rho} \cdot \vec{z} - \beta$
$A_b \vec{z} = \vec{a}_b$
$B_b \vec{z} \leq \vec{b}_b$