Is it possible to implement a column generation for a problem that the variables in the "complicating" constraint appear in the objective function?
Suppose the MIP is:
\begin{align} z = \min&\quad\sum_{ij} c_{ij} x_{ij} + \sum_i w_i\\ \text{s.t.}&\quad A(x) \leq a, \tag1 \\ &\quad w_i \geq c_{ij}x_{ij}, \quad \forall i, j\tag2 \\ &\quad x_{ij} \in \{0,1\},\\ &\quad w_{i} \geq 0. \end{align}
Where constraint set (2) are the complicating constraints. If we ignore (2), then what happens to the objective function?
P.S. I don't have the experience in implementing a column generation.
EDIT: I thank Rob for the great answer. For those unfamiliar with column generation, like me, there is additional information in the comments of his post as well.