When generating multirow intersection cuts for mixed-integer linear optimization problems, an important practical question is deciding which intersection cuts to use. Even when restricted to cuts that are facet defining for the corner relaxation, the number of potential candidates is still very large, especially for instances of large size. In this paper, we introduce a subset of intersection cuts based on the infinity norm that is very small, works for relaxations having arbitrary number of rows and, unlike many subclasses studied in the literature, takes into account the entire data from the simplex tableau. We describe an algorithm for generating these inequalities and run extensive computational experiments in order to evaluate their practical effectiveness in real-world instances. We conclude that this subset of inequalities yields, in terms of gap closure, around 50% of the benefits of using all valid inequalities for the corner relaxation simultaneously, but at a small fraction of the computational cost, and with a very small number of cuts. Summary of Contribution: Cutting planes are one of the most important techniques used by modern mixed-integer linear programming solvers when solving a variety of challenging operations research problems. The paper advances the state of the art on general-purpose multirow intersection cuts by proposing a practical and computationally friendly method to generate them.

The basis matrices corresponding to consecutive iterations of the simplex method only differ in a single column. This fact is commonly exploited in current linear programming solvers to avoid having to compute a new factorization of the basis at every iteration. Instead, a previous factorization is updated to reflect the modified column. Several methods are known for performing the update, most prominently the Forrest–Tomlin method. We present an alternative algorithm for the special case where the update can be performed purely by permuting rows and columns of the factors. In our experiments, this occurred for about half of the basis updates, and the new algorithm provides a modest reduction in computation time for the dual simplex method.

The resolution of integer programming problems is typically performed via branch and bound. Nodes of the branch-and-bound tree are pruned whenever the corresponding subproblem is proven not to contain a solution better than the best solution found so far. This is a key ingredient for achieving reasonable solution times. However, since subproblems are solved in floating-point arithmetic, numerical errors can occur and may lead to inappropriate pruning. As a consequence, optimal solutions may be cut off. We propose several methods for avoiding this issue, in the special case of a branch-cut-and-price formulation for the capacitated vehicle routing problem. The methods are based on constructing dual feasible solutions for the linear programming relaxations of the subproblems and obtaining, by weak duality, bounds on their objective function value. Such approaches have been proposed before for formulations with a small number of variables (dual constraints), but the problem becomes more complex when the number of variables is exponentially large, which is the case in consideration. We show that, in practice, along with being safe, our bounds are stronger than those usually employed, obtained with unsafe floating-point arithmetic plus some heuristic tolerance, and all of this at a negligible computational cost. We also discuss some potential advantages and other uses of our safe bounds derivation. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0747 .