Depth-Optimized Convexity Cuts

This paper presents a general, self-contained treatment of convexity or intersection cuts. It describes two equivalent ways of generating a cut — via a convex set or a concave function — and a notion of strong cuts. We then characterize the structure of the sets and functions that generate strong cuts. We then specialize the framework to the case of mixed-integer linear programming (MIP). For this case, we formulate two kinds of the deepest cut generation problem, via sets or via functions. We then consider some special cases of the deepest cut generation problem which are amenable to efficient computation. We conclude with computational tests of one of these procedures on a large set of MIPLIB problems. Acknowledgements: This research was supported in part by National Science Foundation grant CCR-9902092. The second author gratefully acknowledges partial support from DIMACS and SHARCNET. We would also like to thank John Forrest of IBM for his assistance in using the COIN software library and anonymous referees for helpful comments and suggestions.