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Cutting planes from two-term disjunctions

Pierre Bonami, Michele Conforti, Gérard Cornuéjols, Marco Molinaro, Giacomo Zambelli
2013 Operations Research Letters  
We consider a polyhedron intersected by a two-term disjunction, and we characterize the polyhedron resulting from taking its closed convex hull.  ...  This generalizes an earlier result of Conforti, Wolsey and Zambelli on split disjunctions.  ...  We first observe that our two-term disjunction problem is equivalent to the following complementarity problem: Ax = b, x ≥ 0 and x 1 · x 2 = 0.  ... 
doi:10.1016/j.orl.2013.05.005 fatcat:nldr6tdw2vaitd37o66fq7wsru

A computational study of the cutting plane tree algorithm for general mixed-integer linear programs

Binyuan Chen, Simge Küçükyavuz, Suvrajeet Sen
2012 Operations Research Letters  
Because the CPT algorithm is based on discovering multi-term disjunctions, this paper is the first to present computational results with multi-term disjunctions.  ...  The cutting plane tree (CPT) algorithm provides a finite disjunctive programming procedure to obtain the solution of general mixed-integer linear programs (MILP) with bounded integer variables.  ...  Lift-and-project cuts rely on two-term (simple) disjunctions of the form x k ≤ 0 or x k ≥ 1, for some binary variable x k .  ... 
doi:10.1016/j.orl.2011.10.009 fatcat:aad7yy7b5fbi5eml76zrrml7dy

Modeling of discrete/continuous optimization problems: characterization and formulation of disjunctions and their relaxations

Aldo Vecchietti, Sangbum Lee, Ignacio E. Grossmann
2003 Computers and Chemical Engineering  
A cutting plane method is also proposed that avoids the explicit generation of equations and variables of the convex hull.  ...  An interesting result is presented for improper disjunctions where results in the continuous space differ from the ones in the mixed-integer space.  ...  Acknowledgements The authors would like to acknowledge financial support from the NSF Grants ACI-0121497 and INT-0104315.  ... 
doi:10.1016/s0098-1354(02)00220-x fatcat:r4wth336wjh4dhd6bofuqe7hqy

Reduce-and-Split Cuts: Improving the Performance of Mixed-Integer Gomory Cuts

Kent Andersen, Gérard Cornuéjols, Yanjun Li
2005 Management science  
Any split cut can be derived from a basis of the linear programming relaxation and a disjunction (Andersen, Cornuéjols and Li [1]).  ...  Therefore, improvements in the performance of these cutting planes can be of great practical value.  ...  In the second cutting plane algorithm, we add the MIG cuts generated from the current optimal basis in each round of the cutting-plane algorithm.  ... 
doi:10.1287/mnsc.1050.0382 fatcat:s7h6veno4za6xoaqzfhiwzj5xi

Cutting Plane Algorithm for Convex Generalized Disjunctive Programs

Francisco Trespalacios, Ignacio E. Grossmann
2016 INFORMS journal on computing  
In this work, we propose a cutting plane algorithm to improve optimization models that are originally formulated as convex Generalized Disjunctive Programs (GDP).  ...  The (HR) reformulation can be further strengthened, by using the concept of basic step from disjunctive programming.  ...  Acknowledgments The authors would like to acknowledge financial support from the Center for Advanced Process Decisionmaking (CAPD).  ... 
doi:10.1287/ijoc.2015.0669 fatcat:miczpprrmfasbbwjvana2oylce

Complexity of branch-and-bound and cutting planes in mixed-integer optimization – II [article]

Amitabh Basu, Michele Conforti, Marco Di Summa, Hongyi Jiang
2021 arXiv   pre-print
We study the complexity of cutting planes and branching schemes from a theoretical point of view.  ...  To the best of our knowledge, our results are the first mathematically rigorous demonstration of the superiority of branch-and-cut over pure cutting planes and pure branch-and-bound.  ...  Acknowledgments Amitabh Basu and Hongyi Jiang gratefully acknowledge support from ONR Grant N000141812096, NSF Grant CCF2006587, and AFOSR Grant FA95502010341.  ... 
arXiv:2011.05474v4 fatcat:eq6yh256anbszfro2mxs6sjlse

Some thoughts on the development of integer programming during my research career – lecture delivered upon receiving the EURO Gold Medal, July 9, 2001, Rotterdam

Egon Balas
2002 European Journal of Operational Research  
On the other hand, it soon became clear that any intersection cut can be viewed as a disjunctive cut, i.e., a cutting plane derived from a disjunction.  ...  can be viewed as a cut derived from this disjunction.  ... 
doi:10.1016/s0377-2217(02)00172-8 fatcat:6qb73ktj6jhd3ewa6texfeadwe

Cutting Planes from Wide Split Disjunctions [chapter]

Pierre Bonami, Andrea Lodi, Andrea Tramontani, Sven Wiese
2017 Lecture Notes in Computer Science  
In this paper, we discuss an extension of split cuts that is based on widening the underlying disjunctions.  ...  We further discuss extensions of the existing theory with respect to cut strengthening procedures, and present some ideas on how to use the wider disjunctions also in branching.  ...  While the authors in [2] use the term general split disjunctions, we will call constructs like (1) wide split disjunctions, and the resulting cutting planes wide split cuts.  ... 
doi:10.1007/978-3-319-59250-3_9 fatcat:idyidjuhyzdgvl4o2coyennluy

Complexity of branch-and-bound and cutting planes in mixed-integer optimization [article]

Amitabh Basu, Michele Conforti, Marco Di Summa, Hongyi Jiang
2020 arXiv   pre-print
We investigate the theoretical complexity of branch-and-bound (BB) and cutting plane (CP) algorithms for mixed-integer optimization.  ...  We further show that if one moves away from 0/1 sets, this advantage of CP over BB disappears; there are examples where BB finishes in O(1) time, but CP takes infinitely long to prove optimality, and exponentially  ...  Daniel Dadush pointed out to us the example by Dobbs et al. from [36] that establishes an exponential lower bound on any BB proof based on general split disjunctions.  ... 
arXiv:2003.05023v2 fatcat:vs53rzg43bbidixo7fun7saony

Experiments with Two-Row Cuts from Degenerate Tableaux

Amitabh Basu, Pierre Bonami, Gérard Cornuéjols, François Margot
2011 INFORMS journal on computing  
There has been a recent interest in cutting planes generated from two or more rows of the optimal simplex tableau.  ...  Motivated by these theoretical results, we study the effect of adding a family of cutting planes generated from two rows on a set of instances from the MIPLIB library.  ...  Two-Row Cutting Planes Basic Theory In this paper, we study a class of cutting planes derived from two equations.  ... 
doi:10.1287/ijoc.1100.0437 fatcat:exuwc5xzonaz3iqzvi3ubmef64

Facial disjunctive programs and sequence of cutting-planes

Charles E. Blair
1980 Discrete Applied Mathematics  
A cutting-plane procedure for facial disjuctive programs is presented. At each step the point to be cut away and the disjunction used to generate the cut may be chosen freely.  ...  The two are joined at the point (8,&O).There are two disjunctive constraints E(l,P)={(x,y,z)~x=O or y=O or z=O}, E(2,P)={(x,y,x)lx+y=6 or x= 10 or y= 10 or t=Oj.  ...  Disjunctive programs include as special cases zero-one integer programs and linear complementarity problems. We consider cutting-plane methods of obtaining the feasible set S.  ... 
doi:10.1016/0166-218x(80)90037-2 fatcat:cmvvxchkbzdyniow2erajnwdme

Lift-and-project for Mixed 0–1 programming: recent progress

Egon Balas, Michael Perregaard
2002 Discrete Applied Mathematics  
procedure: alternative normalizations of (CGLP), complementarity of the solution components, size reduction of (CGLP), and ways of deriving multiple cuts from a disjunction.  ...  Section 5 introduces the lift-and-project cuts of Balas, Ceria and Cornuà ejols from the early nineties, and discusses the cut generating linear program (CGLP), cut lifting and cut strengthening.  ...  This strengthening procedure can also be applied to cuts derived from disjunctions other than x j ∈ {0; 1}, including disjunctions with more than two terms.  ... 
doi:10.1016/s0166-218x(01)00340-7 fatcat:mbphgisczvbilmbimrqg5elh34

Solving Integer and Disjunctive Programs by Lift and Project [chapter]

Sebastián Ceria, Gábor Pataki
1998 Lecture Notes in Computer Science  
We extend the theoretical foundations of the branch-and-cut method using lift-and-project cuts for a broader class of disjunctive constraints, and also present a new, substantially improved disjunctive  ...  cut generator.  ...  A disjunctive set is in Disjunctive Normal Form or DNF, if its terms do not contain further disjunctions.  ... 
doi:10.1007/3-540-69346-7_21 fatcat:7z5lkhp74vg7zcnvfvwvwe222e

Efficient treatment of bilinear forms in global optimization [article]

Marcia Fampa, Jon Lee
2018 arXiv   pre-print
s disjunctive cuts to generate disjunctive cuts from the inequalities (I.1 -I.4 ).  ...  The disjunctive cuts are generated by solving linear programs in cutting-plane algorithms.  ...  Extended McCormick disjunctive cuts.  ... 
arXiv:1803.07625v1 fatcat:3ovgwrp6tfewnarqu4iwjth7wy

A finitely convergent procedure for facial disjunctive programs

Hanif D. Sherali, C.M. Shetty
1982 Discrete Applied Mathematics  
This paper addresses an important class of disjunctive programs called facial disjunctive programs, examples of which include the zero-one linear integer programming problem and the linear complementarity  ...  This paper exploits another basic property of facial disjunctive programs in order to develop an alternative finitely convergent algorithm.  ...  The cut (4.15) may be generated from the disjunctive statement xE{x: jx3zl,xrO} or {x:-+x,+fx,Zl,xrO} (5.3) whence, the disjunctive cut (4.15) is (ix, + fxs 2 1) or equivalently (xs +x5 -x6 =  ... 
doi:10.1016/0166-218x(82)90071-3 fatcat:6hkx4tu3trantdgdyknejp3hdq
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