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.
A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher dimensional space by introducing variables Y ij to represent each of the products x i x j of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint Y − xx T 0 and disjunctive programming. On the other hand, their main drawback is their huge size, even for problems of moderate size. In this paper, we study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations. To do so, we use projection techniques pioneered in the context of the lift-and-project methodology. We show how the extended formulation can be algorithmically projected to the original space by solving linear programs. Furthermore, we extend the technique to project the SDP relaxation by solving SDPs. In the case of an MIQCP with a single quadratic constraint, we propose a subgradient-based heuristic to efficiently solve these SDPs. We also propose a new eigen reformulation for MIQCP, and a cut generation technique to strengthen this reformulation using polarity. We present extensive computational results to illustrate the efficiency of the proposed techniques. Our computational results have two highlights. First, on the GLOBALLib instances, we are able to generate relaxations that are almost as strong as those proposed in our companion paper even though our computing times are about 100 times smaller, on average. Second, on the box QP instances, the strengthened relaxations generated by our code are almost as strong as the well-studied SDP+RLT relaxations and can be solved in less than 2 sec even for larger instances with 100 variables; the SDP+RLT relaxations of the same set of instances can take up to a couple of hours to solve using a state-of-the-art SDP solver.
In this paper we address the following probabilistic version (PSC) of the set covering problem: min{cx | P(Ax ≥ ξ) ≥ p, x j ∈ {0, 1} N } where A is a 0-1 matrix, ξ is a random 0-1 vector and p ∈ (0, 1] is the threshold probability level. We formulate (PSC) as a mixed integer non-linear program (MINLP) and linearize the resulting (MINLP) to obtain a MIP reformulation. We introduce the concepts of p-inefficiency and polarity cuts. While the former is aimed at reducing the number of constraints in our model, the later is used as a strengthening device to obtain stronger formulations. A hierarchy of relaxations for (PSC) is introduced, and fundamental relationships between the relaxations are established culminating with a MIP reformulation of (PSC) with no additional integer constrained variables. Simplifications of the MIP model which result when one of the following conditions hold are briefly discussed: A is a balanced matrix, A has the circular ones property, the components of ξ are pairwise independent, the distribution function of ξ is a stationary distribution or has the so-called disjunctive shattering property. We corroborate our theoretical findings by an extensive computational experiment on a test-bed consisting of almost 10,000 probabilistic instances. This test-bed was created using deterministic instances from the literature and consists of probabilistic variants of the set-covering model and capacitated versions of facility location, warehouse location and k-median models. Our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to fraction of seconds.
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