Generalized coefficient strengthening cuts for mixed integer programming

Cutting plane methods are an important component in solving the mixed integer programming (MIP). By carefully studying the coefficient strengthening method, which is originally a presolving method, we are able to generalize this method to generate a family of valid inequalities called generalized coefficient strengthening (GCS) inequalities. The invariant property of the GCS inequalities is established under bound substitutions. Furthermore, we develop a separation algorithm for finding the violated GCS inequalities for a general mixed integer set. The separation algorithm is proved to have the polynomial time complexity. Extensive numerical experiments are made on standard MIP test sets, which demonstrate the usefulness of the resulting GCS separator.

[1]  Martin W. P. Savelsbergh,et al.  Preprocessing and Probing Techniques for Mixed Integer Programming Problems , 1994, INFORMS J. Comput..

[2]  Andrea Lodi,et al.  Strengthening Chvátal-Gomory cuts and Gomory fractional cuts , 2002, Oper. Res. Lett..

[3]  Martin W. P. Savelsbergh,et al.  Lifted Cover Inequalities for 0-1 Integer Programs: Complexity , 1999, INFORMS J. Comput..

[4]  Arie M. C. A. Koster,et al.  Algorithms to Separate {0, \frac12}\{0, \frac{1}{2}\} -Chvátal-Gomory Cuts. , 2009 .

[5]  Alper Atamtürk,et al.  On splittable and unsplittable flow capacitated network design arc–set polyhedra , 2002, Math. Program..

[6]  Tobias Achterberg,et al.  Mixed Integer Programming: Analyzing 12 Years of Progress , 2013 .

[7]  Thomas L. Magnanti,et al.  The convex hull of two core capacitated network design problems , 1993, Math. Program..

[8]  Arie M. C. A. Koster,et al.  Algorithms to Separate -Chvátal-Gomory Cuts , 2009 .

[9]  Robert E. Bixby,et al.  Presolve Reductions in Mixed Integer Programming , 2020, INFORMS J. Comput..

[10]  Roland Wunderling.  Paralleler und Objektorientierter Simplex , 1996 .

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

[12]  Tobias Achterberg,et al.  The Mcf-separator: detecting and exploiting multi-commodity flow structures in MIPs , 2010, Math. Program. Comput..

[13]  Benjamin Müller,et al.  The SCIP Optimization Suite 3.2 , 2016 .

[14]  Alper Atamtürk,et al.  On capacitated network design cut–set polyhedra , 2002, Math. Program..

[15]  Martin W. P. Savelsbergh,et al.  Lifted flow cover inequalities for mixed 0-1 integer programs , 1999, Math. Program..

[16]  L. Wolsey,et al.  Valid inequalities and separation for uncapacitated fixed charge networks , 1985 .

[17]  Tobias Achterberg,et al.  Constraint integer programming , 2007 .

[18]  Manfred W. Padberg,et al.  Improving LP-Representations of Zero-One Linear Programs for Branch-and-Cut , 1991, INFORMS J. Comput..

[19]  Laurence A. Wolsey,et al.  Valid Linear Inequalities for Fixed Charge Problems , 1985, Oper. Res..

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

[21]  Andrea Lodi,et al.  MIPLIB 2010 , 2011, Math. Program. Comput..

[22]  Arie M. C. A. Koster,et al.  Algorithms to Separate $\{0,\frac{1}{2}\}$ -Chvátal-Gomory Cuts , 2008, Algorithmica.

[23]  Robert Weismantel,et al.  On the 0/1 knapsack polytope , 1997, Math. Program..

[24]  Ellis L. Johnson,et al.  Solving Large-Scale Zero-One Linear Programming Problems , 1983, Oper. Res..

[25]  L. Wolsey,et al.  Designing Private Line Networks - Polyhedral Analysis and Computation , 1996 .

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

[27]  Martin W. P. Savelsbergh,et al.  Conflict graphs in solving integer programming problems , 2000, Eur. J. Oper. Res..

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