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Using rank-1 lift-and-project closures to generate cuts for 0–1 MIPs, a computational investigation

P. Bonami, M. Minoux
2005 Discrete Optimization  
Besides classical techniques such as Gomory cuts, Mixed Integer Rounding cuts, lift-and-project and reformulation-linearization techniques, a new variant is also investigated: the use of the relaxation  ...  From the results obtained, the relaxation based on the elementary closure of lift-and-project cuts appears to be one of the most promising.  ...  Acknowledgements We gratefully acknowledge the referees for their careful reading and insightful and constructive comments on a first version of the paper.  ... 
doi:10.1016/j.disopt.2005.08.006 fatcat:lpi6ykulfvbhjjvyhtrsthxcmy

Strengthening Chvátal-Gomory Cuts for the Stable Set Problem [chapter]

Adam N. Letchford, Francesca Marzi, Fabrizio Rossi, Stefano Smriglio
2016 Lecture Notes in Computer Science  
We propose a new method to strengthen the LP-based upper bounds. The key idea is to take violated Chvátal-Gomory cuts and then strengthen their right-hand sides.  ...  As a result, the overall procedure proves to be capable of yielding competitive upper bounds in reasonable computing times.  ...  by separation algorithms for rank and non-rank inequalities [8] ; MKK, computed by the Lovász and Schrijver M (k, k) lifting operator applied to QSTAB(G) [14] ; BLP derived from MKK through projection  ... 
doi:10.1007/978-3-319-45587-7_18 fatcat:j24k6alfvngnbgu63h4ovlhpbu

MIPping Closures: An Instant Survey

Matteo Fischetti, Andrea Lodi
2007 Graphs and Combinatorics  
This task has been accomplished by using, in turn, mixed-integer linear models for the separation problem, and a general-purpose solver to actually find violated cuts-the so-called MIPping approach.  ...  We instantly survey these attempts by discussing their computational outcome and pointing out their practical interest for future integration in the MIP solvers.  ...  Acknowledgements This work was partially supported by MIUR, Italy, and by the EU project ADONET.  ... 
doi:10.1007/s00373-007-0711-6 fatcat:s7sntofesncmxltsddxjyvkj44

Optimizing over the First Chvàtal Closure [chapter]

Matteo Fischetti, Andrea Lodi
2005 Lecture Notes in Computer Science  
Our approach is to model the rank-1 Chvátal-Gomory separation problem, which is known to be NP-hard, through a MIP model, which is then solved through a general-purpose MIP solver.  ...  Does it pay to insist on rank-1 Chvátal-Gomory inequalities until no such inequality is violated, or one should better follow the usual strategy of generating (mixed-integer) Gomory cuts of any rank from  ...  Acknowledgements Work partially supported by MIUR and CNR, Italy, and by the EU project ADONET. Thanks are due to Jon Lee and François Margot for useful discussions.  ... 
doi:10.1007/11496915_2 fatcat:v4rvcub7vrg65dz5m4phk7ywfq

DRL*: A hierarchy of strong block-decomposable linear relaxations for 0–1 MIPs

M. Minoux, H. Ouzia
2010 Discrete Applied Mathematics  
In this paper we introduce DRL*, a new hierarchy of linear relaxations for 0-1 mixed integer linear programs (MIPs), based on the idea of Reformulation-Linearization, and explore its links with the Lift-and-Project  ...  Furthermore, a full characterization of those 0-1 MIPs for which the DRL* and RLT relaxations coincide is obtained.  ...  Acknowledgements We gratefully acknowledge three referees for their many constructive comments and suggestions.  ... 
doi:10.1016/j.dam.2010.08.020 fatcat:brwhj6ralrgjnitm5bomcyi2jm

Sparsity of Lift-and-Project Cutting Planes [chapter]

Matthias Walter
2013 Operations Research Proceedings  
Eventually, Andrea Lodi convinced me to implement the lift-and-project method in a way which Bonami discovered in 2010.  ...  The results were very typical: A significantly smaller branch & bound tree, but longer computation times. One reason was that some of the generated cuts were relatively dense.  ...  A first result which is now obvious for us is that the lift-and-project closure and the closure of basic intersection cuts are the same.  ... 
doi:10.1007/978-3-319-00795-3_2 dblp:conf/or/Walter12 fatcat:htzmxp5bfng2zhx2mtn7dygipy

Optimizing over the first Chvátal closure

Matteo Fischetti, Andrea Lodi
2006 Mathematical programming  
Our approach is to model the rank-1 Chvátal-Gomory separation problem, which is known to be NP-hard, through a MIP model, which is then solved through a general-purpose MIP solver.  ...  Does it pay to insist on rank-1 Chvátal-Gomory inequalities until no such inequality is violated, or one should better follow the usual strategy of generating (mixed-integer) Gomory cuts of any rank from  ...  Acknowledgements Work partially supported by MIUR and CNR, Italy, and by the EU project ADONET. Thanks are due to Jon Lee and François Margot for useful discussions.  ... 
doi:10.1007/s10107-006-0054-8 fatcat:hao3cmd75jej7c7uzgrllyu4uq

A relax-and-cut framework for Gomory mixed-integer cuts

Matteo Fischetti, Domenico Salvagnin
2011 Mathematical Programming Computation  
In this paper we analyze a different scheme for the generation of rank-1 GMICs read from a basis of the original LP-the one before the addition of any cut.  ...  We also show how our method can be integrated with other cut generators, and successfully used in a cut-and-branch enumerative framework.  ...  Acknowledgements This work was partially supported by the "Progetto di Ateneo" on "Computational Integer Programming" of the University of Padova; the second author was also partially supported by the  ... 
doi:10.1007/s12532-011-0024-x fatcat:pc2epekcrzafnl6aao3td2sgqe

Computational Experiments with Cross and Crooked Cross Cuts

Sanjeeb Dash, Oktay Günlük, Juan Pablo Vielma
2014 INFORMS journal on computing  
In our second set of computational experiments, we investigate the strength of split, cross and crooked cross cuts using the original formulation of an MIP instead of its two row relaxations.  ...  They consider a subset of all such pairs and generate lattice-free cuts, augment the LP relaxation with these cuts and derive a new tableau (up to five times), but they do not lift the cuts with respect  ...  Acknowledgments A portion of this research was supported by grant # N000141110724 from the Office of Naval Research. We would like to thank the two referees for useful comments.  ... 
doi:10.1287/ijoc.2014.0598 fatcat:6rrvsowrwzgcfonqixncxbbnd4

Just MIP it! [chapter]

Matteo Fischetti, Andrea Lodi, Domenico Salvagnin
2009 Annals of Information Systems  
E.g., auxiliary LPs may be solved to generate powerful disjunctive cuts, or to implement a strong branching policy.  ...  This leads to the idea of "translating into a MIP model" (MIPping) some crucial decisions to be taken within a MIP algorithm (How to cut? How to improve the incumbent solution?  ...  Acknowledgments This work was supported by the Future and Emerging Technologies unit of the EC (IST priority), under contract no. FP6-021235-2 (project "ARRIVAL") and by MiUR, Italy.  ... 
doi:10.1007/978-1-4419-1306-7_2 fatcat:p2kdkn3yuzaivldgtfu5yy2tlu

Author index to volume 2

2005 Discrete Optimization  
Minoux, Using rank-1 lift-and-project closures to generate cuts for 0-1 MIPs, a computational investigation (4) 288-307 Brumelle, S., D. Granot and L.  ...  Sherali, Chvatal-Gomory-tier cuts for general integer programs Vygen, J., Geometric quadrisection in linear time, with application to VLSI placement Woeginger, G.J., A comment on scheduling two parallel  ... 
doi:10.1016/s1572-5286(05)00065-4 fatcat:bct4qsfpvjb7fpe6t5nn2lwap4

Enhanced mixed integer programming techniques and routing problems

Andrea Tramontani
2010 4OR  
Note that the MIR closure and the Split closure define the same polyhedron; see e.g., [57, 66, 67] .2 Lift-and-project cuts are a particular class of disjunctive cuts for 0-1 MIPs arising from elementary  ...  The separation of all mentioned cuts (including cliques) but lift-and-project cuts is N P-hard for a general x * .  ...  branching rules, primal-heuristic algorithms and the use of general-purpose Chvátal-Gomory cuts.  ... 
doi:10.1007/s10288-010-0140-x fatcat:x7vxwnkff5csbbuq7nqfnjbfpi

Strengthened Benders Cuts for Stochastic Integer Programs with Continuous Recourse

Merve Bodur, Sanjeeb Dash, Oktay Günlük, James Luedtke
2017 INFORMS journal on computing  
For the case of split cuts, we demonstrate that although these approaches yield equivalent relaxations when considering a single split disjunction, cut-and-project yields stronger relaxations in general  ...  We compare the effect of using cuts in two ways: (i) cut-and-project, where integrality constraints are used to derive cuts in the extended variable space, and Benders cuts are then used to project the  ...  The authors are grateful to David Morton and Feng Pan for providing the instances used in [20] .  ... 
doi:10.1287/ijoc.2016.0717 fatcat:owicwlbscnbm5iocqkjp6q5j3a

Solving Hard Mixed-Integer Programming Problems with Xpress-MP: A MIPLIB 2003 Case Study

Richard Laundy, Michael Perregaard, Gabriel Tavares, Horia Tipi, Alkis Vazacopoulos
2009 INFORMS journal on computing  
Over the past decade there has been a huge improvement in the performance of state-of-the-art MIP codes.  ...  Combined with the increased performance of computers over this period the size and complexity of the problems that can be solved has increased enormously.  ...  In conclusion, one has to use aggressive cut generation, both at the root node (including lift-and-project cuts) and during branch-and-bound, use no heuristics, and put limited effort in strong branching  ... 
doi:10.1287/ijoc.1080.0293 fatcat:npiwf3ytnzgojomz4fblczxdry

A lift-and-project cutting plane algorithm for mixed 0–1 programs

Egon Balas, Sebastián Ceria, Gérard Cornuéjols
1993 Mathematical programming  
We propose a cutting plane algorithm for mixed 0-1 programs based on a family of polyhedra which strengthen the usual LP relaxation.  ...  This cut is found through the solution of a linear program that has about twice the size of the usual LP relaxation. A lifting step is used to reduce the size of the LP's needed to generate the cuts.  ...  Section 3.3 shows how some cutting planes can be generated from the simplex tableau, and Section 3.4 discusses a lifting step used to reduce the size of the LP's needed to generate the cuts.  ... 
doi:10.1007/bf01581273 fatcat:cue65wscvfbttavx7i4i4rbtaa
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