Cut Generation through Binarization

For a mixed integer linear program where all integer variables are bounded, we study a reformulation introduced by Roy that maps general integer variables to a collection of binary variables. We study theoretical properties and empirical strength of rank-2 simple split cuts of the reformulation. We show that for a pure integer problem with two integer variables, these cuts are sufficient to obtain the integer hull of the problem, but that this does not generalize to problems in higher dimensions. We also give an algorithm to compute an approximation of the rank-2 simple split cut closure. We report empirical results on 22 benchmark instances showing that the bounds obtained compare favorably with those obtained with other approximate methods to compute the split closure or lattice-free cut closure. It gives a better bound than the split closure on 6 instances while it is weaker on 9 instances, for an average gap closed 3.8% smaller than the one for the split closure.

[1]  Ralph E. Gomory,et al.  An algorithm for integer solutions to linear programs , 1958 .

[2]  Gérard Cornuéjols,et al.  On the relative strength of split, triangle and quadrilateral cuts , 2009, Math. Program..

[3]  Sanjay Mehrotra,et al.  On the Value of Binary Expansions for General Mixed-Integer Linear Programs , 2002, Oper. Res..

[4]  Gérard Cornuéjols,et al.  On the facets of mixed integer programs with two integer variables and two constraints , 2008, Math. Program..

[5]  Sanjeeb Dash,et al.  On the relative strength of different generalizations of split cuts , 2015, Discret. Optim..

[6]  Laurence A. Wolsey,et al.  Inequalities from Two Rows of a Simplex Tableau , 2007, IPCO.

[7]  Suvrajeet Sen,et al.  Finite Disjunctive Programming Characterizations for General Mixed-Integer Linear Programs , 2011, Oper. Res..

[8]  R. Bixby.  An Updated Mixed Integer Programming Library MIPLIB , 1998 .

[9]  Domenico Salvagnin,et al.  The strength of multi-row models , 2015, Math. Program. Comput..

[10]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[11]  Yanjun Li,et al.  Cook, Kannan and Schrijver's example revisited , 2008, Discret. Optim..

[12]  E. Balas,et al.  Strengthening cuts for mixed integer programs , 1980 .

[13]  Jean-Sébastien Roy.  "Binarize and Project" to Generate Cuts for General Mixed-integer Programs , 2007, Algorithmic Oper. Res..

[14]  Martin W. P. Savelsbergh,et al.  An Updated Mixed Integer Programming Library: MIPLIB 3.0 , 1998 .

[15]  William J. Cook,et al.  Chvátal closures for mixed integer programming problems , 1990, Math. Program..

[16]  Matteo Fischetti,et al.  Approximating the Split Closure , 2013, INFORMS J. Comput..

[17]  L. Wolsey,et al.  Cutting Planes form Two Rows of a Simplex Tableau , 2007 .

[18]  Egon Balas,et al.  Optimizing over the split closure , 2008, Math. Program..

[19]  Pierre Bonami.  On optimizing over lift-and-project closures , 2012, Math. Program. Comput..

[20]  Andrea Lodi,et al.  MIR closures of polyhedral sets , 2009, Math. Program..

[21]  Sanjeeb Dash,et al.  Computational Experiments with Cross and Crooked Cross Cuts , 2014, INFORMS J. Comput..

[22]  Thorsten Koch,et al.  Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin Miplib 2003 , 2022 .