Solving Constraint Integer Programs

Pricing new variables

The task of the pricer is to search for new variables with negative reduced costs. For this, the following non-linear program is solved:

\[ \begin{equation} \min_{P \in \mathcal{RP}} \left\{1 - \sum_{t \in \mathcal{T}} \lambda_t P_t\right\}, \end{equation} \]

where \(\lambda\) is given by the dual solution of the restricted master problem. See the problem description for more details.

This problem is very hard, but can be modeled as a weighted circle packing problem for a single rectangle. Therefore, we first use a simple greedy heuristic to solve the problem. If the heuristic fails, the MINLP is solved with conventional methods on a new SCIP instance and a given time limit. If the problem can be solved and the optimal value is non-negative, the LP relaxation has been solved to optimality and what remains is ensuring integrality of the solution by the normal SCIP framework. If, on the other hand, the best solution found by both methods is negative, we have found an improving pattern, whose corresponding variable needs to be added to the restricted master problem. It is possible (and not unlikely) that neither method succeeds in finding a pattern with negative solution value. In that case, we also exit the pricing loop, just as if we had found an optimal solution, and proceed with enforcing integrality resulting in a feasible primal solution to the whole problem.

In case that the pricing problem cannot be solved to optimality, we cannot directly deduce a lower bound for the master problem. However, the following theorem by Farley shows how a valid dual bound can be computed from the LP solution and the pricing solution, see A Note on Bounding a Class of Linear Programming Problems, Including Cutting Stock Problems for more details.