User Control of the Optimization Process and Improving Efficiency

This chapter will provide information on how the solution time of an IP problem can be reduced significantly. This is important, as in contrast to ordinary LP problems, effective solution of IP problems depends critically upon good model formulation, the use of high-level branching constructs, and control of the B&B strategy. Good formulations are those whose LP feasible region is as “small as possible” not excluding any feasible MILP solution, or, to be precise, those whose LP relaxation has a feasible region which is close to the convex hull of the MILP problem’s feasible set. In practice, this means, for example, that upper bounds should be as small as possible. Formulating models in this fashion is still largely the responsibility of the modeler, although work has been done on automatically reformulating mixed zero-one problems, cf. VanRoy & Wolsey (1987), leading to tighter formulations. Preprocessing can also improve the model formulation; cf. Achterberg et al. (2008), Gamrath et al. (2015), and Achterberg et al. (2020). The merit of good formulations is evident in practical applications such as Meyer (1969), Jeroslow & Lowe (1984), and Cheshire et al. (1984).