On the Practical Strength of Two-Row Tableau Cuts

Abstract

Following the flurry of recent theoretical work on cutting planes from two-row mixed integer group relaxations of an LP tableau, we report on computational tests to evaluate the strength of two-row cuts based on lattice-free triangles having more than one integer point on one side. A heuristic procedure to generate such triangles (referred to in the literature as "type 2" triangles) is presented, and then the coefficients of the integer variables are tightened by lifting.To test the effectiveness of triangle cuts, we compare the gap closed using Gomory mixed integer cuts for one round, the gap closed in one round using all the triangle cuts generated by our heuristic and the gap closed by a small number of two-row split cuts. Our tests are carried out on randomly generated instances designed to represent different problem features by varying the number of integer non-basic variables, bounds, non-negativity constraints and density, as well as on the classical MIPLIB instances.The outcome of this computational analysis is some insight into key characteristics of MIP instances whose presence makes two-row triangle cuts computationally effective. In particular it appears to be necessary that the tableau row pairs are dense, and more subjectively that the non-basic continuous variables are "important". Unfortunately these characteristics seem rarely to be present among real life instances, and more specifically the tableau rows of the MIPLIB instances are far from dense. A preliminary version of this work has been published in [18].

Analysis of Sparse Cutting Planes for Sparse MILPs with Applications to Stochastic MILPs

“…Indeed, one way to view the results we obtain in this paper is to attempt a mathematical explanation for the empirical observations of quality of sparse cutting-planes obtained in [8]. Finally, we mention here in passing that the quality of Gomory mixed integer cuts were found empirically to be related to the sparsity of LP relaxation optimal tableaux in the paper [9]; however we do not explore particular families of sparse cutting-planes in this paper.…”

“…Now consider optimizing over the weak specific-scenario cuts closure P V,C (where the row support list is V = {{v 1 }, {v 2 }}), i.e., the closure corresponding to the cuts on (x, y 1 ) variables and on (x, y 2 ) variables. Since the y i variable can be used to satisfy the ith set of covering constraints, it is easy to see that the only undominated (x, y 1 )-cuts are the ones implied only the first set of covering constraints (8), and similarly the only undominated (x, y 2 )-cuts are the ones implied only by the second set of covering constraints (9). Thus, the point x 1 = x 2 = x 3 = x 4 = 1 2 , y 1 = y 2 = 0 belongs to P V,C , giving z * = z V,C ≤ 2.…”

On the relative strength of families of intersection cuts arising from pairs of tableau constraints in mixed integer programs

“…We proved that Type 2 triangle closure is within 50% of the convex hull of integer points, R, and no single family (among the five families) can guarantee better than a 12.5% approximation to R. Moreover, the inclusion lattice Figure 2 together with the facts that Split and Type 1 closures can give arbitrarily bad approximations of R, and to close in on R with a tighter than 12% approximation, one needs both Type 3 triangle closure and the quadrilateral closure, indicate that Type 2 triangles provide a natural compromise for implementation. The additional fact that one needs fewer parameters to describe Type 2 triangles compared to the union of Type 3 triangles and quadrilaterals, adds to the argument for focusing on Type 2 triangles for implementations [4], [12], [17].…”

“…For a proof that such triangle inequalities always define a polyhedron, see [6]. For computational work, see [4], [12] and [17]. For a probabilistic analysis, see [5], [11], [15].…”

When Lift-and-Project Cuts Are Different

“…The elegance and convenience of generating cuts from the simplex tableau has motivated a recent stream of theoretical work on generating cuts from two rows of the simplex tableau (Andersen et al 2007, Cornuéjols andMargot 2008) and subsequently more rows (Borozan andCornuéjols 2009, Basu et al 2010), on how these cuts can be strengthened when nonbasic variables are integer (Dey and Wolsey 2010, Conforti et al 2011b, Basu et al 2013, Fukasawa et al 2016, and several other variants. The reader is referred to Conforti et al (2011a) and Basu et al (2015) for a broader review of this line of work, which has been accompanied by extensive computational experimentation (Espinoza 2008, Basu et al 2011, Dey et al 2014, Louveaux et al 2015.…”