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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.  ...  That the formula for deriving intersection cuts based on splits can be adapted to this case has been known for a decade now.  ...  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 – 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

Helly systems and certificates in optimization [article]

Amitabh Basu, Tongtong Chen, Michele Conforti, Hongyi Jiang
2022 arXiv   pre-print
Inspired by branch-and-bound and cutting plane proofs in mixed-integer optimization and proof complexity, we develop a general approach via Hoffman's Helly systems.  ...  If we apply a cutting plane H derived from a split disjunction D on N , then the right child has K ∩ H as the corresponding set and the left child has K ∩ H c as the corresponding set, which is a subset  ...  The same holds for disjunctive cuts like those derived from split disjunctions or lattice-free sets [16] . Another example is the case of the clique inequalities for the stable set problem.  ... 
arXiv:2111.05225v3 fatcat:elfvauedlrd47ogpg22vr4tdm4

On the relative strength of different generalizations of split cuts

Sanjeeb Dash, Oktay Günlük, Marco Molinaro
2015 Discrete Optimization  
More precisely, we compare the elementary closures of split, cross, crooked cross and general multi-branch split cuts as well as cuts obtained from multi-row and basic relaxations.  ...  In addition, we also show that cross cuts, and hence crooked cross cuts, cannot always be obtained from 2-row relaxations or from basic relaxations.  ...  Cuts from relaxations. Structured relaxations of mixed-integer sets have been widely used to generate cutting planes for the original sets.  ... 
doi:10.1016/j.disopt.2014.12.003 fatcat:r2uofppizrgixa2g7htzk3wopi

On the separation of split cuts and related inequalities

Alberto Caprara, Adam N. Letchford
2003 Mathematical programming  
cuts and binary split cuts.  ...  To detect violated split cuts, one has to solve the associated separation problem. The complexity of split cut separation was recently cited as an open problem by Cornuéjols & Li [10].  ...  The same cut can be derived as a binary split cut from the disjunction (γ x ≤ δ) ∨ (γ x ≥ δ + 1).  ... 
doi:10.1007/s10107-002-0320-3 fatcat:xszkoznk6rh2vnxgjkbycjp5oy

Computational Study of Cutting Planes for a Lot-Sizing Problem in Branch-and-Cut Algorithm
Branch-and-Cut 알고리즘에서 Lot-Sizing 문제에 대한 Cutting Planes의 전산 성능 연구

Kwanghun Chung
2015 Journal of the Korean Operations Research and Management Science Society  
To use three families of cutting planes in Branch-and-Cut framework, we develop separation algorithms for each cut and implement them in CPLEX.  ...  In this paper, we evaluate the strength of three families of cutting planes for a lot-sizing problem.  ...  The split cuts derived from these disjunctions are called lift-and-project cuts; see Balas et al. [2].  ... 
doi:10.7737/jkorms.2015.40.3.023 fatcat:thqc7ilqs5berd564s74wmd24y

Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations

Anureet Saxena, Pierre Bonami, Jon Lee
2010 Mathematical programming  
We also use the convex SDP constraint Y − xx T 0 to derive convex quadratic cuts, and we combine both approaches in a cutting plane algorithm.  ...  To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology.  ...  in cutting-plane procedures.  ... 
doi:10.1007/s10107-010-0371-9 fatcat:w4c5rkcadrh4po6bsj7lgeiqrm

Branching on general disjunctions

Miroslav Karamanov, Gérard Cornuéjols
2009 Mathematical programming  
In this work, we focus on disjunctions defining the mixed integer Gomory cuts at an optimal basis of the linear programming relaxation. The procedure is tested on instances from the literature.  ...  We select promising branching disjunctions based on a heuristic measure of disjunction quality. This measure exploits the relation between branching disjunctions and intersection cuts.  ...  Any violated split disjunction can be used to define a cutting plane that cuts off points of P violating the disjunction.  ... 
doi:10.1007/s10107-009-0332-3 fatcat:nfbmlsuv65bqlf4z4w6d7jboba

Split cuts in the plane [article]

Amitabh Basu, Michele Conforti, Marco Di Summa, Hongyi Jiang
2020 arXiv   pre-print
We provide a polynomial time cutting plane algorithm based on split cuts to solve integer programs in the plane. We also prove that the split closure of a polyhedron in the plane has polynomial size.  ...  Acknowledgments We are very grateful for insightful comments on content and presentation from two anonymous reviewers. These helped a lot to improve the paper.  ...  In the following, whenever we say "the split cut derived from a given disjunction" we refer to this specific split cut.  ... 
arXiv:2003.05022v2 fatcat:bwz62c7udzcvjnaayppjhmw3iy

A mixed-integer branching approach for very small formulations of disjunctive constraints [article]

Joey Huchette, Juan Pablo Vielma
2017 arXiv   pre-print
Our main technical result gives an explicit linear inequality description for both traditional MIP and mixed-integer branching formulations for a wide range of disjunctive constraints.  ...  functions and other disjunctive constraints.  ...  The crucial observation of Bonami et al. is that wide split disjunctions also readily admit standard cutting plane techniques, such as the intersection cut [1] .  ... 
arXiv:1709.10132v2 fatcat:4k7lezx6u5hcbfkhjgbqre6xvy

Mixed Integer Programming Computation [chapter]

Andrea Lodi
2009 50 Years of Integer Programming 1958-2008  
We run over these 50 exciting years by showing some crucial milestones and we highlight the building blocks that are making nowadays solvers effective from both a performance and an application viewpoint  ...  Cutting planes exploitation As shown in Section 16.2, cutting plane generation has been a key step for the success of MIP solvers and their capability of being effective for a wide variety of problems.  ...  Both papers above investigate the accuracy issue of cutting plane generation mainly from a research viewpoint.  ... 
doi:10.1007/978-3-540-68279-0_16 fatcat:ud2aajvmzbdebpis6pm2sribma

An interior point cutting plane heuristic for mixed integer programming

Joe Naoum-Sawaya, Samir Elhedhli
2011 Computers & Operations Research  
General cutting planes such as Gomory cuts, disjunctive cuts, lift-and-project cuts, split cuts, and mixed-integer-rounding cuts are all used in branch-and-cut frameworks in the state of the art software  ...  [16] uses reduce-and-split inequalities to improve the quality of the GMI disjunctions.  ... 
doi:10.1016/j.cor.2010.12.008 fatcat:t5xe3ohdpjfiviqeqdmcs376xq

Branch-and-cut for linear programs with overlapping SOS1 constraints

Tobias Fischer, Marc E. Pfetsch
2017 Mathematical Programming Computation  
The corresponding conflict graph can algorithmically be exploited, for instance, for improved branching rules, preprocessing, primal heuristics, and cutting planes.  ...  In this article, we investigate a branch-and-cut algorithm to solve linear programs with SOS1 constraints. We focus on the case in which the SOS1 constraints overlap.  ...  A further class of cutting planes are disjunctive cuts, which can be directly generated from the simplex tableau.  ... 
doi:10.1007/s12532-017-0122-5 fatcat:3rxna375dfgi3o5l53u5b3zira

Disjunctive cuts for cross-sections of the second-order cone

Sercan Yıldız, Gérard Cornuéjols
2015 Operations Research Letters  
There has been a lot of recent interest in extending disjunctive cutting-plane theory from the domain of mixed-integer linear programming to that of mixed-integer conic programming [2, 7, 9, 11, 12, 18  ...  convex hull of all two-term disjunctions on ellipsoids and paraboloids and a wide class of two-term disjunctions-including split disjunctions-on hyperboloids.  ...  Consider C 1 and C 2 defined by a split disjunction on C as in (2) . Suppose Assumptions 3 and 4 hold. Let c 1 and c 2 be defined as in (10) . Then (16) holds.  ... 
doi:10.1016/j.orl.2015.06.001 fatcat:5jzzbgaofvbwzpiwzf54jqtqb4

Complexity of optimizing over the integers [article]

Amitabh Basu
2022 arXiv   pre-print
Acknowledgement The author benefited greatly from discussions with Daniel Dadush at CWI, Amsterdam and Timm Oertel at FAU, Erlangen-Nürmberg.  ...  Comments from two anonymous referees helped the author significantly to consolidate the material, improve its presentation and make tighter connections to the existing literature on the complexity of optimization  ...  Split cutting planes (disjunctive cutting planes based on split disjunctions -see point 2. in Example 5.14) and split disjunctions are not a complementary pair for general pure integer convex problems.  ... 
arXiv:2110.06172v6 fatcat:gwv6pq4pbfea3cwfixcgpzfety
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