Abstract
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).
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Notes
- 1.
The terms preprocessing and presolve are often used synonymously. Sometimes the term presolve is used for those procedures which try to reduce the problem size and to discover whether the problem is unbounded or infeasible. Preprocessing involves the presolving phase but includes all other techniques which try, for instance, to improve the MILP formulation. It might be interesting to point out here that transferring a solution back to the space of the optimal solution is called postsolving and is a non-trivial step in some cases.
- 2.
In 2020, coefficient reduction still matters, but there are a number of more advanced techniques to tighten coefficients, such as those that consider other rows in the problem or cliques.
- 3.
This was still true in the 1990s. Nowadays, 2020, branching control, pseudo-costs, etc. are calculated once and updated automatically — there is nothing a user needs to “specify” anymore.
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Kallrath, J. (2021). User Control of the Optimization Process and Improving Efficiency. In: Business Optimization Using Mathematical Programming. International Series in Operations Research & Management Science, vol 307. Springer, Cham. https://doi.org/10.1007/978-3-030-73237-0_9
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