Continuous cutting plane algorithms in integer programming

Cutting planes for mixed-integer linear programs (MILPs) are typically computed in rounds by iteratively solving optimization problems. Instead, we reframe the problem of finding good cutting planes as a continuous optimization problem over weights parametrizing families of valid inequalities. We propose a practical two-step algorithm, and demonstrate empirical gains when optimizing Gomory mixed-integer inequalities over various families of MILPs. Finally, a reinterpretation of our approach as optimization of the subadditive dual using a deep neural network is provided. Code for reproducing the experiments can be found at https://github.com/dchetelat/subadditive

Two row mixed integer cuts via lifting

Recently, Andersen et al. [1], Borozan and Cornuéjols [6] and Cornuéjols and Margot [9] characterized extreme inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these inequalities to obtain cuts from two rows of a general simplex tableau, one approach is to extend the system to include all possible nonnegative integer variables (giving the two-row mixed integer infinite-group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we study the characteristics of these lifting functions. We begin by observing that functions giving valid coefficients for the nonnegative integer variables can be constructed by lifting a subset of the integer variables and then applying the fill-in procedure presented in Johnson [23]. We present conditions for these 'general fill-in functions" to be extreme for the two-row mixed integer infinite-group problem. We then show that there exists a unique 'trivial' lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal lattice-free triangle with integral vertices and one integer point in the relative interior of each side. In all other cases (maximal lattice-free triangle with one integer point in the relative interior of each side and non-integral vertices, and maximal lattice-free quadrilaterals), non-unique lifting functions may yield distinct extreme inequalities. For the case of a triangle with one integer point in the relative interior of each side and non-integral vertices, we present sufficient conditions to yield an extreme inequality for the two-row mixed integer infinite-group problem.

Computational Experiments with Cross and Crooked Cross Cuts

In this paper, we study whether cuts obtained from two simplex tableau rows at a time can strengthen the bounds obtained by Gomory mixed-integer (GMI) cuts based on single tableau rows. We also study whether cross and crooked cross cuts, which generalize split cuts, can be separated in an effective manner for practical mixed-integer programs (MIPs) and can yield a nontrivial improvement over the bounds obtained by split cuts. We give positive answers to both these questions for MIPLIB 3.0 problems. Cross cuts are a special case of the t-branch split cuts studied by Li and Richard [Li Y, Richard J-PP (2008) Cook, Kannan and Schrijvers's example revisited. Discrete Optim. 5:724–734]. Split cuts are 1-branch split cuts, and cross cuts are 2-branch split cuts. Crooked cross cuts were introduced by Dash, Günlük, and Lodi [Dash S, Günlük O, Lodi A (2010) MIR closures of polyhedral sets. Math Programming 121:33–60] and were shown to dominate cross cuts by Dash, Günlük, and Molinaro [Dash S, Günlük O, Molinaro M (2012b) On the relative strength of different generalizations of split cuts. IBM Technical Report RC25326, IBM, Yorktown Heights, NY].United States. Office of Naval Research (Grant N000141110724

Computing with Multi-Row Intersection Cuts

Cutting planes are one of the main techniques currently used to solve large-scale Mixed-Integer Linear Programming (MIP) models. Many important cuts used in practice, such as Gomory Mixed-Integer (GMI) cuts, are obtained by solving the linear relaxation of the MIP, extracting a single row of the simplex tableau, then applying integrality arguments to it. A natural extension, which has received renewed attention, is to consider cuts that can only be generated when considering multiple rows of the simplex tableau simultaneously. Although the theoretical importance of such multi-row cutting planes has been proved in a number of works, their effective use in practice remains a challenge. Since the entire class of multi-row cuts proves challenging to separate, one approach to obtain them is the following. First, the integral non-basic variables are fixed to zero. Then, a lattice-free set, which induces an intersection cut, is generated. Finally, the cut coefficients for the integral non-basic variables are computed by the so-called trivial lifting procedure. In this thesis, we address some computational aspects of this approach, and we make three novel contributions. In our first contribution, we describe a small subset of multi-row intersection cuts based on the infinity norm, which works for relaxations with arbitrary numbers of rows. We present an algorithm to generate them and run extensive computational experiments to evaluate their effectiveness. We conclude that these cuts yield benefits comparable to using the entire class of multi-row cuts, but at a small fraction of the computational cost. In our second contribution, we describe a practical method for performing the trivial lifting step on relaxations with two rows. Unlike previous methods, our method is applicable to intersection cuts derived from any lattice-free set, and, for maximal lattice-free sets, it is guaranteed to run in constant time. Computational experiments confirm that the algorithm is at least two orders of magnitudes faster than current alternatives. In our final contribution, we revisit single-row relaxations containing a single integral non-basic variable, with the goal of obtaining inequalities that are not dominated by GMI cuts. The novelty in our approach is that we use the framework of intersection cuts and trivial lifting, which allows us to obtain a geometric interpretation of our cuts, a fast algorithm for generating them, and an upper bound on their split rank. We run computational experiments and conclude that, for a few instances, they close considerably more gap than GMI cuts alone

Mixed-integer cuts from cyclic groups

We analyze a separation procedure for Mixed-Integer Programs related to the work of Gomory and Johnson on interpolated subadditive functions. This approach has its roots in the Gomory-Johnson characterization on the master cyclic group polyhedron. To our knowledge, the practical benefit that can be obtained by embedding interpolated subadditive cuts in a cutting plane algorithm was not investigated computationally by previous authors. In this paper we compute, for the first time, the lower bound value obtained when adding (implicitly) all the interpolated subadditive cuts that can be derived from the individual rows of an optimal LP tableau, thus approximating the optimization over the intersection of the Gomory corner polyhedron with the LP relaxation of the original problem formulation. The computed bound is compared with that obtained when only Gomory mixed-integer cuts are used, on a very large test-bed of MIP instances