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 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. We show that this separation problem can be formulated 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 Capacitated Warehouse Location Problems from OrLib. Our computational results show that rank-1 split cuts close more than 98% of the duality gap on 14 out of 33 mixed integer instances from MIPLIB 3.0. More than 75% of the duality gap can be closed on an additional 11 instances. The average gap closed over all 33 instances is 82%. In the pure integer case, rank-1 split cuts close more than 75% of the duality gap on 13 out of 25 instances from MIPLIB 3.0. On average, rank 1 split cuts close about 71% of the duality gap on these 25 instances. We also report our results on two sets of real-world capacitated warehouse location problem (CWLP) instances from OrLib. The first set has 37 instances, each one having 50 customers and 16-50 warehouses. The second set has 12 instances with 1000 customers and 100 warehouses. The split closure closes 100% of the duality gap on all the 37 instances from the first set, and on average about 93% of the duality gap on the 12 instances from the second set. We also gathered statistics on the support size of the disjunctions generated (which affects the density of the resulting cut) and the average size (absolute value) of their coefficients. They turn out to be surprisingly small.