Branching on Multi-aggregated Variables

In mixed-integer programming, the branching rule is a key component to a fast convergence of the branch-and-bound algorithm. The most common strategy is to branch on simple disjunctions that split the domain of a single integer variable into two disjoint intervals. Multi-aggregation is a presolving step that replaces variables by an affine linear sum of other variables, thereby reducing the problem size. While this simplification typically improves the performance of MIP solvers, it also restricts the degree of freedom in variable-based branching rules.

[1]  Thorsten Koch,et al.  Progress in presolving for mixed integer programming , 2015, Math. Program. Comput..

[2]  Domenico Salvagnin.  Detecting and Exploiting Permutation Structures in MIPs , 2014, CPAIOR.

[3]  Ted K. Ralphs,et al.  On the Complexity of Selecting Disjunctions in Integer Programming , 2010, SIAM J. Optim..

[4]  J. Christopher Beck,et al.  Recent Improvements Using Constraint Integer Programming for Resource Allocation and Scheduling , 2013, CPAIOR.

[5]  Timo Berthold,et al.  Hybrid Branching , 2009, CPAIOR.

[6]  John W. Chinneck,et al.  Active-constraint variable ordering for faster feasibility of mixed integer linear programs , 2007, Math. Program..

[7]  Martin W. P. Savelsbergh,et al.  A Computational Study of Search Strategies for Mixed Integer Programming , 1999, INFORMS J. Comput..

[8]  A. Gleixner.  Factorization and update of a reduced basis matrix for the revised simplex method , 2012 .

[9]  Matteo Fischetti,et al.  Branching on nonchimerical fractionalities , 2012, Oper. Res. Lett..

[10]  Timo Berthold,et al.  Cloud Branching , 2013, CPAIOR.

[11]  Alexander Shapiro,et al.  On complexity of multistage stochastic programs , 2006, Oper. Res. Lett..

[12]  Thorsten Koch,et al.  Constraint Integer Programming: A New Approach to Integrate CP and MIP , 2008, CPAIOR.

[13]  Tobias Achterberg,et al.  SCIP: solving constraint integer programs , 2009, Math. Program. Comput..

[14]  John W. Chinneck,et al.  Operations Research and Cyber-Infrastructure , 2009 .

[15]  Helman I. Stern,et al.  COMPUTER SCHEDULING OF PUBLIC TRANSPORT. URBAN PASSENGER VEHICLE AND CREW SCHEDULING. PAPERS BASED ON PRESENTATIONS AT THE INTERNATIONAL WORKSHOP HELD AT THE UNIVERSITY OF LEEDS, 16-18 JULY, 1980 (CONTD) , 1981 .

[16]  Gérard Cornuéjols,et al.  Branching on general disjunctions , 2011, Math. Program..

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

[18]  A. Mahajan,et al.  Experiments with Branching using General Disjunctions , 2009 .

[19]  Laurence A. Wolsey,et al.  Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 4th International Conference, CPAIOR 2007, Brussels, Belgium, May 23-26, 2007, Proceedings , 2007, CPAIOR.

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

[21]  Gérard Cornuéjols,et al.  Improved strategies for branching on general disjunctions , 2011, Math. Program..

[22]  Thorsten Koch,et al.  Branching rules revisited , 2005, Oper. Res. Lett..

[23]  Gerald Gamrath,et al.  Improving strong branching by domain propagation , 2014, EURO J. Comput. Optim..

[24]  R. J. Dakin,et al.  A tree-search algorithm for mixed integer programming problems , 1965, Comput. J..

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

[26]  Jean-Charles Régin,et al.  Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems , 2004, Lecture Notes in Computer Science.

[27]  John W. Chinneck,et al.  Achieving MILP feasibility quickly using general disjunctions , 2013, Comput. Oper. Res..

[28]  Roland Wunderling,et al.  Paralleler und objektorientierter Simplex-Algorithmus , 1996 .

[29]  John W. Chinneck,et al.  Faster integer-feasibility in mixed-integer linear programs by branching to force change , 2011, Comput. Oper. Res..

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

[31]  R. Bixby,et al.  A note on detecting simple redundancies in linear systems , 1987 .

[32]  Tuomas Sandholm,et al.  Information-theoretic approaches to branching in search , 2006, AAMAS '06.

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

[34]  Jean-Michel Gauthier,et al.  Experiments in mixed-integer linear programming using pseudo-costs , 1977, Math. Program..

[35]  Sanjay Mehrotra,et al.  Experimental Results on Using General Disjunctions in Branch-and-Bound for General-Integer Linear Programs , 2001, Comput. Optim. Appl..

[36]  Martin W. P. Savelsbergh,et al.  Information-based branching schemes for binary linear mixed integer problems , 2009, Math. Program. Comput..

[37]  Gautam Mitra,et al.  Analysis of mathematical programming problems prior to applying the simplex algorithm , 1975, Math. Program..

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

[39]  Kenneth R. Baker,et al.  Principles of Sequencing and Scheduling , 2018 .

[40]  R. Bixby,et al.  On the Solution of Traveling Salesman Problems , 1998 .

[41]  Anthony Wren,et al.  Computer Scheduling of Public Transportation: Urban Passenger Vehicle and Crew Scheduling , 1981 .

[42]  A. J. Clewett,et al.  Introduction to sequencing and scheduling , 1974 .

[43]  Matteo Fischetti,et al.  Backdoor Branching , 2011, INFORMS J. Comput..

[44]  Matteo Fischetti,et al.  Local branching , 2003, Math. Program..

[45]  Ailsa H. Land,et al.  An Automatic Method of Solving Discrete Programming Problems , 1960 .

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

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

[48]  Gautam Mitra.  Investigation of some branch and bound strategies for the solution of mixed integer linear programs , 1973, Math. Program..