Column generation when intractable variables appear in the objective function

Suppose $K$ is the set of columns, where the $k$th column $x^k in {0,1}^n$ satisfies $A(x^k) le a$. Now express $x$ as a convex combination of the columns $x^k$. Explicitly, substitute $x_{i,j} = sum_{kin K} lambda_k x_{i,j}^k$, where $sum_{kin K} lambda_k = 1$ and $lambda_k ge 0$ for all $kin K$. You then obtain the following master problem over $lambda$ and $w$, with dual variables in parentheses: begin{align} &text{minimize} &sum_{kin K} left(sum_{i,j} c_{i,j} x_{i,j}^kright) lambda_k + sum_i w_i \ &text{subject to} &w_i - c_{i,j} sum_{kin K} x_{i,j}^k lambda_k &ge 0 &&text{for all $i,j$} &&(pi_{i,j} ge 0)\ &&sum_{kin K} lambda_k &= 1 &&&&(text{$alpha$ free})\ &&lambda_k &ge 0 &&text{for all $k$} \ &&w_i &ge 0 &&text{for all $i$} end{align}

The column generation subproblem over $x$ is then to minimize the reduced cost of $lambda_k$. That is, minimize $$sum_{i,j} c_{i,j} (1+pi_{i,j}) x_{i,j} - alpha$$ subject to $A(x) le a$ and $x in{0,1}^n$.