Optimizing over the split closure

The polyhedron defined by all the split cuts obtainable directly (i.e. without iterated cut generation) from the LP-relaxation P of a mixed integer program (MIP) is termed the (elementary, or rank 1) split closure of P. This paper deals with the problem of optimizing over the elementary split closure. This is accomplished by repeatedly solving the following separation problem: given a fractional point, say x, find a rank-1 split cut violated by x or show that none exists. Following Caprara and Letchford [17], we formulate this separation problem as a nonlinear mixed integer program that can be treated as a parametric mixed integer linear program (PMILP) with a single parameter in the objective function and the right hand side. We develop an algorithmic framework to deal with the resulting PMILP by creating and maintaining a dynamically updated grid of parameter values, and use the corresponding mixed integer programs to generate rank 1 split cuts. Our approach was implemented in the COIN-OR framework using CPLEX 9.0 as a general purpose MIP solver. We report our computational results on well-known benchmark instances from MIPLIB 3.0 and several classes of structured integer and mixed integer problems. Our computational results show that rank-1 split cuts close more than 98% of the duality gap on 15 out of 41 mixed integer instances from MIPLIB 3.0. More than 75% of the duality gap can be closed on an additional 10 instances. The average gap closed over all 41 instances is 72.78%. In the pure integer case, rank-1 split cuts close more than 75% of the duality gap on 13 out of 24 instances from MIPLIB 3.0. On average, rank 1 split cuts close about 72% of the duality gap on these 24 instances. We also report results on several classes of structured problems: capacitated versions of warehouse location, single-source facility location, p-median, fixed charge network flow, multi-commodity network design with splittable and unsplittable flows, and lot sizing. The fraction of the integrality gap closed varies for these problem classes between 100 and 67%. We also gathered statistics on the average coefficient size (absolute value) of the disjunctions generated. They turn out to be surprisingly small.