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Presolve Reductions in Mixed Integer Programming

Tobias Achterberg, Robert E. Bixby, Zonghao Gu, Edward Rothberg, Dieter Weninger
2019 INFORMS journal on computing  
In this paper we describe the presolve functionality in the Gurobi commercial mixed-integer programming code.  ...  A critical component in the solution of these mixed-integer programs is a set of routines commonly referred to as presolve.  ...  In mixed integer programming one needs to be more conservative to avoid presolving being too expensive.  ... 
doi:10.1287/ijoc.2018.0857 fatcat:bqpkhi6awzdv5pqqrvr43wokvq

A Computational Status Update for Exact Rational Mixed Integer Programming [article]

Leon Eifler, Ambros Gleixner
2021 arXiv   pre-print
The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch  ...  , Steffy, and Wolter in 2013.  ...  If that LP is feasible, this produces an exactly feasible solution to the mixed integer program.  ... 
arXiv:2101.09141v1 fatcat:cpg27qutxngypgljiewry6m6re

Two-row and two-column mixed-integer presolve using hashing-based pairing methods

Patrick Gemander, Wei-Kun Chen, Dieter Weninger, Leona Gottwald, Ambros Gleixner, Alexander Martin
2020 EURO Journal on Computational Optimization  
In state-of-the-art mixed-integer programming solvers, a large array of reduction techniques are applied to simplify the problem and strengthen the model formulation before starting the actual branch-and-cut  ...  Keywords Linear programming · Mixed-integer linear programming · Optimization solver · Presolve Mathematics Subject Classification 90C05 · 90C10 · 90C11 · 65Y05 P LP := {x ∈ R n | Ax • b ∧ j ≤ x j ≤ u  ...  The statements (i) and (ii) are thus a bit of a generalization for complementary slackness in the context of mixed-integer programming.  ... 
doi:10.1007/s13675-020-00129-6 fatcat:3pf2l6blvbbwzglt7gmhraggmy

Using dual presolving reductions to reformulate cumulative constraints

Stefan Heinz, Jens Schulz, J. Christopher Beck
2013 Constraints  
Presolving and dual reductions are important components of state-of-the-art mixed-integer linear programming solvers.  ...  In this paper, we introduce them both as unified, practical concepts in constraint programming solvers.  ...  An overview of presolving techniques for mixed-integer linear programs is given in [5, 6, 7] .  ... 
doi:10.1007/s10601-012-9136-9 fatcat:tyffr5vqlfcudb64tjvb6ucpim

SCIP: solving constraint integer programs

Tobias Achterberg
2009 Mathematical Programming Computation  
Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques.  ...  In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), which is free for academic and non-commercial use and can be downloaded in source code.  ...  In particular, I thank Timo Berthold, Gerald Gamrath, Stefan Heinz, Bo Jensen, Thorsten Koch, Manuel Kutschka, Alexander Martin, Michael Perregaard, Marc Pfetsch, Christian Raack, Robert Waniek, Michael  ... 
doi:10.1007/s12532-008-0001-1 fatcat:elsjugcawndjvlt2lf3e5vgtbq

MIP: Theory and Practice — Closing the Gap [chapter]

E. Robert Bixby, Mary Fenelon, Zonghao Gu, Ed Rothberg, Roland Wunderling
2000 IFIP Advances in Information and Communication Technology  
W e a r e i n terested in mixed-integer reductions.  ...  This development is critical to the wider use of mixed-integer programming in practice. Most users of mixed-integer programming are not interested in the details of how the codes work.  ... 
doi:10.1007/978-0-387-35514-6_2 fatcat:5gkphajnq5fdjbcgebneetiprm

PaPILO: A Parallel Presolving Library for Integer and Linear Programming with Multiprecision Support [article]

Ambros Gleixner, Leona Gottwald, Alexander Hoen
2022 arXiv   pre-print
To rule out conflicts between parallel presolve reductions, PaPILO uses a transaction-based design.  ...  Presolving has become an essential component of modern MIP solvers both in terms of computational performance and numerical robustness.  ...  It operates on the general class of mixed integer programs (MIPs) given in the form min c T x L ≤ Ax ≤ U ≤ x ≤ u xi ∈ Z for all i ∈ I, where A ∈ R m×n , c ∈ R n , b ∈ R m , and I ⊂ {1, . . . , n} is the  ... 
arXiv:2206.10709v1 fatcat:v57tuxnef5hyzig4o5qilhixza

Domain reduction techniques for global NLP and MINLP optimization

Yash Puranik, Nikolaos V. Sahinidis
2017 Constraints  
Domain reduction techniques are especially important in speeding up convergence to the global optimum for challenging nonconvex nonlinear programming (NLP) and mixed-integer nonlinear programming (MINLP  ...  Optimization solvers routinely utilize presolve techniques, including model simplification, reformulation and domain reduction techniques.  ...  for optimization problems and includes linear programming (LP), mixed-integer linear programming (MIP) and nonlinear programming (NLP) in its subclasses.  ... 
doi:10.1007/s10601-016-9267-5 fatcat:drn6m456x5dhteazf4uem33bki

Extending a CIP Framework to Solve MIQCPs [chapter]

Timo Berthold, Stefan Heinz, Stefan Vigerske
2011 IMA Volumes in Mathematics and its Applications  
This paper discusses how to build a solver for mixed integer quadratically constrained programs (MIQCPs) by extending a framework for constraint integer programming (CIP).  ...  In particular, this addresses the linear relaxation and the discrete components of the problem.  ...  We observe, that due to the reformulation (3.1), seven instances could be reformulated as mixed integer linear programs in the presolving state.  ... 
doi:10.1007/978-1-4614-1927-3_15 fatcat:pngauiqh3vfl5g3dzvsje7vyhi

ParaSCIP: A Parallel Extension of SCIP [chapter]

Yuji Shinano, Tobias Achterberg, Timo Berthold, Stefan Heinz, Thorsten Koch
2011 Competence in High Performance Computing 2010  
Mixed integer programming (MIP) has become one of the most important techniques in Operations Research and Discrete Optimization.  ...  SCIP (Solving Constraint Integer Programs) is currently one of the fastest non-commercial MIP solvers.  ...  Mixed integer programs In this paper, we only focus on mixed integer programs (MIPs), which can be defined as follows: Definition 1 (mixed integer program). LetR := R∪{±∞}.  ... 
doi:10.1007/978-3-642-24025-6_12 dblp:conf/chpc/ShinanoAB0K10 fatcat:6lgwfilfjrgydm7ptal2oidy6y

Application of mixed integer quadratic program to shortest vector problems

Keiji Kimura, Hayato Waki, Masaya Yasuda
2017 JSIAM Letters  
Keywords shortest vector problem, mixed integer quadratic program, presolve Research Activity Group Algorithmic Number Theory and Its Applications  ...  Our interest is to know how large-scale shortest vector problems can be solved by the state-of-the-art software for mixed-integer programs.  ...  Mixed integer program We briefly introduce a mixed integer program (MIP) and an algorithm to solve MIPs, i.e., a branch-andbound (B&B) algorithm.  ... 
doi:10.14495/jsiaml.9.65 fatcat:i54i6vsj7rb7jaxsa7ygmm2cdu

An Easy Way to Build Parallel State-of-the-art Combinatorial Optimization Problem Solvers: A Computational Study on Solving Steiner Tree Problems and Mixed Integer Semidefinite Programs by using ug[SCIP-*,*]-Libraries

Yuji Shinano, Daniel Rehfeldt, Tristan Gally
2019 2019 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW)  
In this paper, we present a computational study for solving Steiner tree problems and mixed integer semidefinite programs in parallel.  ...  Steiner tree test set in the literature.  ...  ): Definition 3 (mixed-integer nonlinear program).  ... 
doi:10.1109/ipdpsw.2019.00095 dblp:conf/ipps/ShinanoRG19 fatcat:k3d3t4rhsvfu3d3m7yfft2d5d4

Branching on Multi-aggregated Variables [chapter]

Gerald Gamrath, Anna Melchiori, Timo Berthold, Ambros M. Gleixner, Domenico Salvagnin
2015 Lecture Notes in Computer Science  
In mixed-integer programming, the branching rule is a key component to a fast convergence of the branch-and-bound algorithm.  ...  Our implementation within the constraint integer programming framework SCIP incorporates this into a full strong branching rule and reduces the number of branch-and-bound nodes on a general test set of  ...  Introduction Since the invention of the branch-and-bound method for solving mixed-integer linear programming in the 1960s [1, 2] , branching rules have been an important field of research, being one of  ... 
doi:10.1007/978-3-319-18008-3_10 fatcat:ah6uvfb76bh5tgrgjhsvjzzxzm

Progress in presolving for mixed integer programming

Gerald Gamrath, Thorsten Koch, Alexander Martin, Matthias Miltenberger, Dieter Weninger
2015 Mathematical Programming Computation  
Especially for mixed integer programming problems, fast and effective presolving algorithms are very important. In this paper, we report on three new presolving techniques.  ...  Then we present a presolving technique which exploits a partial order of the variables to induce fixings. Finally, we show an approach based on connected components in graphs.  ...  Presolving has been applied for solving linear and mixed integer programming problems for decades. Brearly et al.  ... 
doi:10.1007/s12532-015-0083-5 fatcat:6ielqvevojbehb3puojudkxjbe

A Structure Exploiting Branch-and-Bound Algorithm for Mixed-Integer Model Predictive Control [article]

Pedro Hespanhol, Rien Quirynen, Stefano Di Cairano
2019 arXiv   pre-print
Mixed-integer model predictive control (MI-MPC) requires the solution of a mixed-integer quadratic program (MIQP) at each sampling instant under strict timing constraints, where part of the state and control  ...  Several applications in automotive, aerospace and hybrid systems are practical examples of how such discrete-valued variables arise.  ...  Section II presents the basic idea of branch-and-bound methods for mixed-integer programming. Then, Section III presents presolve techniques in the context of mixed-integer optimal control.  ... 
arXiv:1903.09117v1 fatcat:hkheznjch5eynnjmqzltoha4b4
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