A heuristic to generate rank-1 GMI cuts

Gomory mixed-integer (GMI) cuts are among the most effective cutting planes for general mixed-integer programs (MIP). They are traditionally generated from an optimal basis of a linear programming (LP) relaxation of a MIP. In this paper we propose a heuristic to generate useful GMI cuts from additional bases of the initial LP relaxation. The cuts we generate have rank one, i.e., they do not use previously generated GMI cuts. We demonstrate that for problems in MIPLIB 3.0 and MIPLIB 2003, the cuts we generate form an important subclass of all rank-1 mixed-integer rounding cuts. Further, we use our heuristic to generate globally valid rank-1 GMI cuts at nodes of a branch-and-cut tree and use these cuts to solve a difficult problem from MIPLIB 2003, namely timtab2, without using problem-specific cuts.

[1]  Manfred W. Padberg,et al.  The Sharpest Cut (MPS-Siam Series on Optimization) , 2004 .

[2]  Egon Balas,et al.  Gomory cuts revisited , 1996, Oper. Res. Lett..

[3]  Sanjeeb Dash,et al.  On the strength of Gomory mixed-integer cuts as group cuts , 2008, Math. Program..

[4]  Uwe H. Suhl,et al.  Computing Sparse LU Factorizations for Large-Scale Linear Programming Bases , 1990, INFORMS J. Comput..

[5]  Milind Dawande,et al.  Combining and Strengthening Gomory Cuts , 1995, IPCO.

[6]  R. Gomory.  AN ALGORITHM FOR THE MIXED INTEGER PROBLEM , 1960 .

[7]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

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

[9]  Michael C. Ferris,et al.  Grid-Enabled Optimization with GAMS , 2009, INFORMS J. Comput..

[10]  Egon Balas,et al.  A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer gomory cuts for 0-1 programming , 2003, Math. Program..

[11]  Alberto Caprara,et al.  On the separation of split cuts and related inequalities , 2003, Math. Program..

[12]  Robert E. Bixby,et al.  MIP: Theory and Practice - Closing the Gap , 1999, System Modelling and Optimization.

[13]  Gérard Cornuéjols,et al.  Elementary closures for integer programs , 2001, Oper. Res. Lett..

[14]  Robert E. Bixby,et al.  Mixed-Integer Programming: A Progress Report , 2004, The Sharpest Cut.

[15]  Matteo Fischetti,et al.  Optimizing over the first Chvátal closure , 2005, Math. Program..

[16]  Mathematik Und.  Information on MIPLIB's timetab-instances , 2003 .

[17]  M. Minoux,et al.  Using rank-1 lift-and-project closures to generate cuts for 0-1 MIPs, a computational investigation , 2005, Discret. Optim..

[18]  Edmond Chow,et al.  Combinatorial Algorithms for Computing Column Space Bases That Have Sparse Inverses , 2005 .

[19]  Gerhard J. Woeginger,et al.  Operations Research Letters , 2011 .

[20]  Ricardo Fukasawa,et al.  On the Exact Separation of Mixed Integer Knapsack Cuts , 2007, IPCO.

[21]  Egon Balas,et al.  Generating lift-and-project cuts from the LP simplex tableau: open source implementation and testing of new variants , 2009, Math. Program. Comput..

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

[23]  François Margot,et al.  Testing cut generators for mixed-integer linear programming , 2009, Math. Program. Comput..

[24]  Kent Andersen,et al.  Reduce-and-Split Cuts: Improving the Performance of Mixed-Integer Gomory Cuts , 2005, Manag. Sci..

[25]  Jack Edmonds,et al.  Matroids and the greedy algorithm , 1971, Math. Program..

[26]  H. Markowitz.  The Elimination form of the Inverse and its Application to Linear Programming , 1957 .

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

[28]  A. M. Geoffrion,et al.  Multicommodity Distribution System Design by Benders Decomposition , 1974 .

[29]  Laurence A. Wolsey,et al.  Aggregation and Mixed Integer Rounding to Solve MIPs , 2001, Oper. Res..

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

[31]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[32]  Sanjeeb Dash,et al.  Two-Step MIR Inequalities for Mixed Integer Programs , 2010, INFORMS J. Comput..

[33]  William J. Cook,et al.  Numerically Safe Gomory Mixed-Integer Cuts , 2009, INFORMS J. Comput..

[34]  Matteo Fischetti,et al.  Mixed-Integer Cuts from Cyclic Groups , 2007, Math. Program..

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