@misc{BertholdWitzig2020, author = {Berthold, Timo and Witzig, Jakob}, title = {Conflict Analysis for MINLP}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-78964}, year = {2020}, abstract = {The generalization of MIP techniques to deal with nonlinear, potentially non-convex, constraints have been a fruitful direction of research for computational MINLP in the last decade. In this paper, we follow that path in order to extend another essential subroutine of modern MIP solvers towards the case of nonlinear optimization: the analysis of infeasible subproblems for learning additional valid constraints. To this end, we derive two different strategies, geared towards two different solution approaches. These are using local dual proofs of infeasibility for LP-based branch-and-bound and the creation of nonlinear dual proofs for NLP-based branch-and-bound, respectively. We discuss implementation details of both approaches and present an extensive computational study, showing that both techniques can significantly enhance performance when solving MINLPs to global optimality.}, language = {en} } @misc{WitzigBerthold2019, author = {Witzig, Jakob and Berthold, Timo}, title = {Conflict-Free Learning for Mixed Integer Programming}, issn = {1438-0064}, doi = {10.1007/978-3-030-58942-4_34}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-75338}, year = {2019}, abstract = {Conflict learning plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. A major step for MIP conflict learning is to aggregate the LP relaxation of an infeasible subproblem to a single globally valid constraint, the dual proof, that proves infeasibility within the local bounds. Among others, one way of learning is to add these constraints to the problem formulation for the remainder of the search. We suggest to not restrict this procedure to infeasible subproblems, but to also use global proof constraints from subproblems that are not (yet) infeasible, but can be expected to be pruned soon. As a special case, we also consider learning from integer feasible LP solutions. First experiments of this conflict-free learning strategy show promising results on the MIPLIB2017 benchmark set.}, language = {en} } @inproceedings{WitzigBerthold2020, author = {Witzig, Jakob and Berthold, Timo}, title = {Conflict-Free Learning for Mixed Integer Programming}, booktitle = {Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2020}, number = {12296}, publisher = {Springer, Cham.}, doi = {10.1007/978-3-030-58942-4_34}, pages = {521 -- 530}, year = {2020}, abstract = {Conflict learning plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. A major step for MIP conflict learning is to aggregate the LP relaxation of an infeasible subproblem to a single globally valid constraint, the dual proof, that proves infeasibility within the local bounds. Among others, one way of learning is to add these constraints to the problem formulation for the remainder of the search. We suggest to not restrict this procedure to infeasible subproblems, but to also use global proof constraints from subproblems that are not (yet) infeasible, but can be expected to be pruned soon. As a special case, we also consider learning from integer feasible LP solutions. First experiments of this conflict-free learning strategy show promising results on the MIPLIB2017 benchmark set.}, language = {en} } @inproceedings{BertholdGrimmReutheretal.2019, author = {Berthold, Timo and Grimm, Boris and Reuther, Markus and Schade, Stanley and Schlechte, Thomas}, title = {Strategic Planning of Rolling Stock Rotations for Public Tenders}, volume = {Link{\"o}ping Electronic Conference Proceedings}, booktitle = {Proceedings of the 8th International Conference on Railway Operations Modelling and Analysis - RailNorrköping 2019}, number = {069}, publisher = {Link{\"o}ping University Electronic Press, Link{\"o}pings universitet}, isbn = {978-91-7929-992-7}, issn = {1650-3686}, pages = {148 -- 159}, year = {2019}, abstract = {Since railway companies have to apply for long-term public contracts to operate railway lines in public tenders, the question how they can estimate the operating cost for long-term periods adequately arises naturally. We consider a rolling stock rotation problem for a time period of ten years, which is based on a real world instance provided by an industry partner. We use a two stage approach for the cost estimation of the required rolling stock. In the first stage, we determine a weekly rotation plan. In the second stage, we roll out this weekly rotation plan for a longer time period and incorporate scheduled maintenance treatments. We present a heuristic approach and a mixed integer programming model to implement the process of the second stage. Finally, we discuss computational results for a real world tendering scenario.}, language = {en} } @misc{BertholdGamrathSalvagnin2019, author = {Berthold, Timo and Gamrath, Gerald and Salvagnin, Domenico}, title = {Exploiting Dual Degeneracy in Branching}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-73028}, year = {2019}, abstract = {Branch-and-bound methods for mixed-integer programming (MIP) are traditionally based on solving a linear programming (LP) relaxation and branching on a variable which takes a fractional value in the (single) computed relaxation optimum. In this paper, we study branching strategies for mixed-integer programs that exploit the knowledge of multiple alternative optimal solutions (a cloud ) of the current LP relaxation. These strategies naturally extend common methods like most infeasible branching, strong branching, pseudocost branching, and their hybrids, but we also propose a novel branching rule called cloud diameter branching. We show that dual degeneracy, a requirement for alternative LP optima, is present for many instances from common MIP test sets. Computational experiments show significant improvements in the quality of branching decisions as well as reduced branching effort when using our modifications of existing branching rules. We discuss different ways to generate a cloud of solutions and present extensive computational results showing that through a careful implementation, cloud modifications can speed up full strong branching by more than 10 \% on standard test sets. Additionally, by exploiting degeneracy, we are also able to improve the state-of-the-art hybrid branching rule and reduce the solving time on affected instances by almost 20 \% on average.}, language = {en} } @misc{WitzigBertholdHeinz2019, author = {Witzig, Jakob and Berthold, Timo and Heinz, Stefan}, title = {Computational Aspects of Infeasibility Analysis in Mixed Integer Programming}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-74962}, year = {2019}, abstract = {The analysis of infeasible subproblems plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems. The first is to analyze the sequence of implications, obtained by domain propagation, that led to infeasibility. The result of this analysis is one or more sets of contradicting variable bounds from which so-called conflict constraints can be generated. This concept is called conflict graph analysis and has its origin in solving satisfiability problems and is similarly used in constraint programming. The second concept is to analyze infeasible linear programming (LP) relaxations. Every ray of the dual LP provides a set of multipliers that can be used to generate a single new globally valid linear constraint. This method is called dual proof analysis. The main contribution of this paper is twofold. Firstly, we present three enhancements of dual proof analysis: presolving via variable cancellation, strengthening by applying mixed integer rounding functions, and a filtering mechanism. Further, we provide an intense computational study evaluating the impact of every presented component regarding dual proof analysis. Secondly, this paper presents the first integrated approach to use both conflict graph and dual proof analysis simultaneously within a single MIP solution process. All experiments are carried out on general MIP instances from the standard public test set MIPLIB 2017; the presented algorithms have been implemented within the non-commercial MIP solver SCIP and the commercial MIP solver FICO Xpress.}, language = {en} } @article{BertholdFarmerHeinzetal.2018, author = {Berthold, Timo and Farmer, James and Heinz, Stefan and Perregaard, Michael}, title = {Parallelization of the FICO Xpress Optimizer}, volume = {33}, journal = {Optimization Methods and Software}, number = {3}, doi = {10.1080/10556788.2017.1333612}, pages = {518 -- 529}, year = {2018}, abstract = {Computing hardware has mostly thrashed out the physical limits for speeding up individual computing cores. Consequently, the main line of progress for new hardware is growing the number of computing cores within a single CPU. This makes the study of efficient parallelization schemes for computation-intensive algorithms more and more important. A natural precondition to achieving reasonable speedups from parallelization is maintaining a high workload of the available computational resources. At the same time, reproducibility and reliability are key requirements for software that is used in industrial applications. In this paper, we present the new parallelization concept for the state-of-the-art MIP solver FICO Xpress-Optimizer. MIP solvers like Xpress are expected to be deterministic. This inevitably results in synchronization latencies which render the goal of a satisfying workload a challenge in itself. We address this challenge by following a partial information approach and separating the concepts of simultaneous tasks and independent threads from each other. Our computational results indicate that this leads to a much higher CPU workload and thereby to an improved, almost linear, scaling on modern high-performance CPUs. As an added value, the solution path that Xpress takes is not only deterministic in a fixed environment, but also, to a certain extent, thread-independent. This paper is an extended version of Berthold et al. [Parallelization of the FICO Xpress-Optimizer, in Mathematical Software - ICMS 2016: 5th International Conference, G.-M. Greuel, T. Koch, P. Paule, and A. Sommere, eds., Springer International Publishing, Berlin, 2016, pp. 251-258] containing more detailed technical descriptions, illustrative examples and updated computational results.}, language = {en} } @inproceedings{BertholdPerregaardMeszaros2018, author = {Berthold, Timo and Perregaard, Michael and M{\´e}sz{\´a}ros, Csaba}, title = {Four Good Reasons to Use an Interior Point Solver Within a MIP Solver}, booktitle = {Kliewer N., Ehmke J., Bornd{\"o}rfer R. (eds) Operations Research Proceedings 2017}, doi = {10.1007/978-3-319-89920-6_22}, pages = {159 -- 164}, year = {2018}, abstract = {"Interior point algorithms are a good choice for solving pure LPs or QPs, but when you solve MIPs, all you need is a dual simplex" This is the common conception which disregards that an interior point solution provides some unique structural insight into the problem at hand. In this paper, we will discuss some of the benefits that an interior point solver brings to the solution of difficult MIPs within FICO Xpress. This includes many different components of the MIP solver such as branching variable selection, primal heuristics, preprocessing, and of course the solution of the LP relaxation.}, language = {en} } @misc{GleixnerBertholdMuelleretal.2016, author = {Gleixner, Ambros and Berthold, Timo and M{\"u}ller, Benjamin and Weltge, Stefan}, title = {Three Enhancements for Optimization-Based Bound Tightening}, issn = {1438-0064}, doi = {10.1007/s10898-016-0450-4}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-57803}, year = {2016}, abstract = {Optimization-based bound tightening (OBBT) is one of the most effective procedures to reduce variable domains of nonconvex mixed-integer nonlinear programs (MINLPs). At the same time it is one of the most expensive bound tightening procedures, since it solves auxiliary linear programs (LPs)—up to twice the number of variables many. The main goal of this paper is to discuss algorithmic techniques for an efficient implementation of OBBT. Most state-of-the-art MINLP solvers apply some restricted version of OBBT and it seems to be common belief that OBBT is beneficial if only one is able to keep its computational cost under control. To this end, we introduce three techniques to increase the efficiency of OBBT: filtering strategies to reduce the number of solved LPs, ordering heuristics to exploit simplex warm starts, and the generation of Lagrangian variable bounds (LVBs). The propagation of LVBs during tree search is a fast approximation to OBBT without the need to solve auxiliary LPs. We conduct extensive computational experiments on MINLPLib2. Our results indicate that OBBT is most beneficial on hard instances, for which we observe a speedup of 17\% to 19\% on average. Most importantly, more instances can be solved when using OBBT.}, language = {en} } @inproceedings{WitzigBertholdHeinz2019, author = {Witzig, Jakob and Berthold, Timo and Heinz, Stefan}, title = {A Status Report on Conflict Analysis in Mixed Integer Nonlinear Programming}, volume = {11494}, booktitle = {Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2019}, publisher = {Springer}, doi = {10.1007/978-3-030-19212-9_6}, pages = {84 -- 94}, year = {2019}, abstract = {Mixed integer nonlinear programs (MINLPs) are arguably among the hardest optimization problems, with a wide range of applications. MINLP solvers that are based on linear relaxations and spatial branching work similar as mixed integer programming (MIP) solvers in the sense that they are based on a branch-and-cut algorithm, enhanced by various heuristics, domain propagation, and presolving techniques. However, the analysis of infeasible subproblems, which is an important component of most major MIP solvers, has been hardly studied in the context of MINLPs. There are two main approaches for infeasibility analysis in MIP solvers: conflict graph analysis, which originates from artificial intelligence and constraint programming, and dual ray analysis. The main contribution of this short paper is twofold. Firstly, we present the first computational study regarding the impact of dual ray analysis on convex and nonconvex MINLPs. In that context, we introduce a modified generation of infeasibility proofs that incorporates linearization cuts that are only locally valid. Secondly, we describe an extension of conflict analysis that works directly with the nonlinear relaxation of convex MINLPs instead of considering a linear relaxation. This is work-in-progress, and this short paper is meant to present first theoretical considerations without a computational study for that part.}, language = {en} } @inproceedings{BertholdStuckeyWitzig2019, author = {Berthold, Timo and Stuckey, Peter and Witzig, Jakob}, title = {Local Rapid Learning for Integer Programs}, volume = {11494}, booktitle = {Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2019}, publisher = {Springer}, doi = {10.1007/978-3-030-19212-9_5}, pages = {67 -- 83}, year = {2019}, abstract = {Conflict learning algorithms are an important component of modern MIP and CP solvers. But strong conflict information is typically gained by depth-first search. While this is the natural mode for CP solving, it is not for MIP solving. Rapid Learning is a hybrid CP/MIP approach where CP search is applied at the root to learn information to support the remaining MIP solve. This has been demonstrated to be beneficial for binary programs. In this paper, we extend the idea of Rapid Learning to integer programs, where not all variables are restricted to the domain {0, 1}, and rather than just running a rapid CP search at the root, we will apply it repeatedly at local search nodes within the MIP search tree. To do so efficiently, we present six heuristic criteria to predict the chance for local Rapid Learning to be successful. Our computational experiments indicate that our extended Rapid Learning algorithm significantly speeds up MIP search and is particularly beneficial on highly dual degenerate problems.}, language = {en} } @misc{ShinanoAchterbergBertholdetal.2015, author = {Shinano, Yuji and Achterberg, Tobias and Berthold, Timo and Heinz, Stefan and Koch, Thorsten and Winkler, Michael}, title = {Solving Open MIP Instances with ParaSCIP on Supercomputers using up to 80,000 Cores}, issn = {1438-0064}, doi = {10.1109/IPDPS.2016.56}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-56404}, year = {2015}, abstract = {This paper describes how we solved 12 previously unsolved mixed-integer program- ming (MIP) instances from the MIPLIB benchmark sets. To achieve these results we used an enhanced version of ParaSCIP, setting a new record for the largest scale MIP computation: up to 80,000 cores in parallel on the Titan supercomputer. In this paper we describe the basic parallelization mechanism of ParaSCIP, improvements of the dynamic load balancing and novel techniques to exploit the power of parallelization for MIP solving. We give a detailed overview of computing times and statistics for solving open MIPLIB instances.}, language = {en} } @article{GamrathBertholdHeinzetal.2019, author = {Gamrath, Gerald and Berthold, Timo and Heinz, Stefan and Winkler, Michael}, title = {Structure-driven fix-and-propagate heuristics for mixed integer programming}, volume = {11}, journal = {Mathematical Programming Computation}, number = {4}, publisher = {Springer}, address = {Berlin Heidelberg}, doi = {10.1007/s12532-019-00159-1}, pages = {675 -- 702}, year = {2019}, abstract = {Primal heuristics play an important role in the solving of mixed integer programs (MIPs). They often provide good feasible solutions early and help to reduce the time needed to prove optimality. In this paper, we present a scheme for start heuristics that can be executed without previous knowledge of an LP solution or a previously found integer feasible solution. It uses global structures available within MIP solvers to iteratively fix integer variables and propagate these fixings. Thereby, fixings are determined based on the predicted impact they have on the subsequent domain propagation. If sufficiently many variables can be fixed that way, the resulting problem is solved first as an LP, and then as an auxiliary MIP if the rounded LP solution does not provide a feasible solution already. We present three primal heuristics that use this scheme based on different global structures. Our computational experiments on standard MIP test sets show that the proposed heuristics find solutions for about 60 \% of the instances and by this, help to improve several performance measures for MIP solvers, including the primal integral and the average solving time.}, language = {en} } @misc{BertholdStuckeyWitzig2018, author = {Berthold, Timo and Stuckey, Peter and Witzig, Jakob}, title = {Local Rapid Learning for Integer Programs}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-71190}, year = {2018}, abstract = {Conflict learning algorithms are an important component of modern MIP and CP solvers. But strong conflict information is typically gained by depth-first search. While this is the natural mode for CP solving, it is not for MIP solving. Rapid Learning is a hybrid CP/MIP approach where CP search is applied at the root to learn information to support the remaining MIP solve. This has been demonstrated to be beneficial for binary programs. In this paper, we extend the idea of Rapid Learning to integer programs, where not all variables are restricted to the domain {0, 1}, and rather than just running a rapid CP search at the root, we will apply it repeatedly at local search nodes within the MIP search tree. To do so efficiently, we present six heuristic criteria to predict the chance for local Rapid Learning to be successful. Our computational experiments indicate that our extended Rapid Learning algorithm significantly speeds up MIP search and is particularly beneficial on highly dual degenerate problems.}, language = {en} } @misc{WitzigBertholdHeinz2018, author = {Witzig, Jakob and Berthold, Timo and Heinz, Stefan}, title = {A Status Report on Conflict Analysis in Mixed Integer Nonlinear Programming}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-71170}, year = {2018}, abstract = {Mixed integer nonlinear programs (MINLPs) are arguably among the hardest optimization problems, with a wide range of applications. MINLP solvers that are based on linear relaxations and spatial branching work similar as mixed integer programming (MIP) solvers in the sense that they are based on a branch-and-cut algorithm, enhanced by various heuristics, domain propagation, and presolving techniques. However, the analysis of infeasible subproblems, which is an important component of most major MIP solvers, has been hardly studied in the context of MINLPs. There are two main approaches for infeasibility analysis in MIP solvers: conflict graph analysis, which originates from artificial intelligence and constraint programming, and dual ray analysis. The main contribution of this short paper is twofold. Firstly, we present the first computational study regarding the impact of dual ray analysis on convex and nonconvex MINLPs. In that context, we introduce a modified generation of infeasibility proofs that incorporates linearization cuts that are only locally valid. Secondly, we describe an extension of conflict analysis that works directly with the nonlinear relaxation of convex MINLPs instead of considering a linear relaxation. This is work-in-progress, and this short paper is meant to present first theoretical considerations without a computational study for that part.}, language = {en} } @article{GleixnerHendelGamrathetal.2021, author = {Gleixner, Ambros and Hendel, Gregor and Gamrath, Gerald and Achterberg, Tobias and Bastubbe, Michael and Berthold, Timo and Christophel, Philipp M. and Jarck, Kati and Koch, Thorsten and Linderoth, Jeff and L{\"u}bbecke, Marco and Mittelmann, Hans and Ozyurt, Derya and Ralphs, Ted and Salvagnin, Domenico and Shinano, Yuji}, title = {MIPLIB 2017: Data-Driven Compilation of the 6th Mixed-Integer Programming Library}, volume = {13}, journal = {Mathematical Programming Computation}, number = {3}, doi = {10.1007/s12532-020-00194-3}, pages = {443 -- 490}, year = {2021}, abstract = {We report on the selection process leading to the sixth version of the Mixed Integer Programming Library. Selected from an initial pool of over 5,000 instances, the new MIPLIB 2017 collection consists of 1,065 instances. A subset of 240 instances was specially selected for benchmarking solver performance. For the first time, the compilation of these sets was done using a data-driven selection process supported by the solution of a sequence of mixed integer optimization problems, which encoded requirements on diversity and balancedness with respect to instance features and performance data.}, language = {en} } @article{ShinanoAchterbergBertholdetal.2013, author = {Shinano, Yuji and Achterberg, Tobias and Berthold, Timo and Heinz, Stefan and Koch, Thorsten and Vigerske, Stefan and Winkler, Michael}, title = {制約整数計画ソルバ SCIP の並列化}, volume = {61}, journal = {統計数理}, number = {1}, pages = {47 -- 78}, year = {2013}, abstract = {制約整数計画(CIP: Constraint Integer Programs)は,制約プログラミング(CP: Constraint Programming),混合整数計画(MIP: Mixed Integer Programming),充足可能性問題(SAT: Satisfability Problem)の研究分野におけるモデリング技術と解法を統合している.その結果,制約整数計画は,広いクラスの最適化問題を扱うことができる.SCIP(Solving Constraint Integer Programs)は,CIP を解くソルバとして実装され,Zuse Institute Berlin(ZIB)の研究者を中心として継続的に拡張が続けられている.本論文では,著者らによって開発された SCIP に対する2 種類の並列化拡張を紹介する.一つは,複数計算ノード間で大規模に並列動作する ParaSCIPである.もう一つは,複数コアと共有メモリを持つ 1 台の計算機上で(スレッド)並列で動作する FiberSCIP である.ParaSCIP は,HLRN II スーパーコンピュータ上で,一つのインスタンスを解くために最大 7,168 コアを利用した動作実績がある.また,統計数理研究所の Fujitsu PRIMERGY RX200S5 上でも,最大 512 コアを利用した動作実績がある.統計数理研究所のFujitsu PRIMERGY RX200S5 上では,これまでに最適解が得られていなかった MIPLIB2010のインスタンスである dg012142 に最適解を与えた.}, language = {ja} } @misc{TurnerBertholdBesanconetal.2022, author = {Turner, Mark and Berthold, Timo and Besan{\c{c}}on, Mathieu and Koch, Thorsten}, title = {Cutting Plane Selection with Analytic Centers and Multiregression}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-89065}, year = {2022}, abstract = {Cutting planes are a crucial component of state-of-the-art mixed-integer programming solvers, with the choice of which subset of cuts to add being vital for solver performance. We propose new distance-based measures to qualify the value of a cut by quantifying the extent to which it separates relevant parts of the relaxed feasible set. For this purpose, we use the analytic centers of the relaxation polytope or of its optimal face, as well as alternative optimal solutions of the linear programming relaxation. We assess the impact of the choice of distance measure on root node performance and throughout the whole branch-and-bound tree, comparing our measures against those prevalent in the literature. Finally, by a multi-output regression, we predict the relative performance of each measure, using static features readily available before the separation process. Our results indicate that analytic center-based methods help to significantly reduce the number of branch-and-bound nodes needed to explore the search space and that our multiregression approach can further improve on any individual method.}, language = {en} } @misc{BertholdMexiSalvagnin2022, author = {Berthold, Timo and Mexi, Gioni and Salvagnin, Domenico}, title = {Using Multiple Reference Vectors and Objective Scaling in the Feasibility Pump}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-88513}, year = {2022}, abstract = {The Feasibility Pump (FP) is one of the best-known primal heuristics for mixed-integer programming (MIP): more than 15 papers suggested various modifications of all of its steps. So far, no variant considered information across multiple iterations, but all instead maintained the principle to optimize towards a single reference integer point. In this paper, we evaluate the usage of multiple reference vectors in all stages of the FP algorithm. In particular, we use LP-feasible vectors obtained during the main loop to tighten the variable domains before entering the computationally expensive enumeration stage. Moreover, we consider multiple integer reference vectors to explore further optimizing directions and introduce alternative objective scaling terms to balance the contributions of the distance functions and the original MIP objective. Our computational experiments demonstrate that the new method can improve performance on general MIP test sets. In detail, our modifications provide a 29.3\% solution quality improvement and 4.0\% running time improvement in an embedded setting, needing 16.0\% fewer iterations over a large test set of MIP instances. In addition, the method's success rate increases considerably within the first few iterations. In a standalone setting, we also observe a moderate performance improvement, which makes our version of FP suitable for the two main use-cases of the algorithm.}, language = {en} } @misc{ShinanoAchterbergBertholdetal.2020, author = {Shinano, Yuji and Achterberg, Tobias and Berthold, Timo and Heinz, Stefan and Koch, Thorsten and Winkler, Michael}, title = {Solving Previously Unsolved MIP Instances with ParaSCIP on Supercomputers by using up to 80,000 Cores}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-78393}, year = {2020}, abstract = {Mixed-integer programming (MIP) problem is arguably among the hardest classes of optimization problems. This paper describes how we solved 21 previously unsolved MIP instances from the MIPLIB benchmark sets. To achieve these results we used an enhanced version of ParaSCIP, setting a new record for the largest scale MIP computation: up to 80,000 cores in parallel on the Titan supercomputer. In this paper, we describe the basic parallelization mechanism of ParaSCIP, improvements of the dynamic load balancing and novel techniques to exploit the power of parallelization for MIP solving. We give a detailed overview of computing times and statistics for solving open MIPLIB instances.}, language = {en} } @incollection{GamrathBertholdHeinzetal.2015, author = {Gamrath, Gerald and Berthold, Timo and Heinz, Stefan and Winkler, Michael}, title = {Structure-Based Primal Heuristics for Mixed Integer Programming}, volume = {13}, booktitle = {Optimization in the Real World}, publisher = {Springer Japan}, isbn = {978-4-431-55419-6}, doi = {10.1007/978-4-431-55420-2_3}, pages = {37 -- 53}, year = {2015}, abstract = {Primal heuristics play an important role in the solving of mixed integer programs (MIPs). They help to reach optimality faster and provide good feasible solutions early in the solving process. In this paper, we present two new primal heuristics which take into account global structures available within MIP solvers to construct feasible solutions at the beginning of the solving process. These heuristics follow a large neighborhood search (LNS) approach and use global structures to define a neighborhood that is with high probability significantly easier to process while (hopefully) still containing good feasible solutions. The definition of the neighborhood is done by iteratively fixing variables and propagating these fixings. Thereby, fixings are determined based on the predicted impact they have on the subsequent domain propagation. The neighborhood is solved as a sub-MIP and solutions are transferred back to the original problem. Our computational experiments on standard MIP test sets show that the proposed heuristics find solutions for about every third instance and therewith help to improve the average solving time.}, language = {en} } @misc{WitzigBertholdHeinz2016, author = {Witzig, Jakob and Berthold, Timo and Heinz, Stefan}, title = {Experiments with Conflict Analysis in Mixed Integer Programming}, issn = {1438-0064}, doi = {10.1007/978-3-319-59776-8_17}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61087}, year = {2016}, abstract = {The analysis of infeasible subproblems plays an import role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems. The first is to analyze the sequence of implications obtained by domain propagation that led to infeasibility. The result of the analysis is one or more sets of contradicting variable bounds from which so-called conflict constraints can be generated. This concept has its origin in solving satisfiability problems and is similarly used in constraint programming. The second concept is to analyze infeasible linear programming (LP) relaxations. The dual LP solution provides a set of multipliers that can be used to generate a single new globally valid linear constraint. The main contribution of this short paper is an empirical evaluation of two ways to combine both approaches. Experiments are carried out on general MIP instances from standard public test sets such as Miplib2010; the presented algorithms have been implemented within the non-commercial MIP solver SCIP. Moreover, we present a pool-based approach to manage conflicts which addresses the way a MIP solver traverses the search tree better than aging strategies known from SAT solving.}, language = {en} } @article{BertholdMexiSalvagnin2023, author = {Berthold, Timo and Mexi, Gioni and Salvagnin, Domenico}, title = {Using Multiple Reference Vectors and Objective Scaling in the Feasibility Pump}, volume = {11}, journal = {EURO Journal on Computational Optimization}, doi = {10.1016/j.ejco.2023.100066}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-87392}, year = {2023}, abstract = {The Feasibility Pump (FP) is one of the best-known primal heuristics for mixed-integer programming (MIP): more than 15 papers suggested various modifications of all of its steps. So far, no variant considered information across multiple iterations, but all instead maintained the principle to optimize towards a single reference integer point. In this paper, we evaluate the usage of multiple reference vectors in all stages of the FP algorithm. In particular, we use LP-feasible vectors obtained during the main loop to tighten the variable domains before entering the computationally expensive enumeration stage. Moreover, we consider multiple integer reference vectors to explore further optimizing directions and introduce alternative objective scaling terms to balance the contributions of the distance functions and the original MIP objective. Our computational experiments demonstrate that the new method can improve performance on general MIP test sets. In detail, our modifications provide a 29.3\% solution quality improvement and 4.0\% running time improvement in an embedded setting, needing 16.0\% fewer iterations over a large test set of MIP instances. In addition, the method's success rate increases considerably within the first few iterations. In a standalone setting, we also observe a moderate performance improvement, which makes our version of FP suitable for the two main use-cases of the algorithm.}, language = {en} } @misc{RalphsShinanoBertholdetal.2018, author = {Ralphs, Ted and Shinano, Yuji and Berthold, Timo and Koch, Thorsten}, title = {Parallel Solvers for Mixed Integer Linear Optimization}, journal = {Handbook of Parallel Constraint Reasoning}, editor = {Hamadi, Youssef}, publisher = {Springer Nature}, doi = {10.1007/978-3-319-63516-3_8}, pages = {283 -- 336}, year = {2018}, language = {en} } @article{BolusaniBesanconGleixneretal.2024, author = {Bolusani, Suresh and Besan{\c{c}}on, Mathieu and Gleixner, Ambros and Berthold, Timo and D'Ambrosio, Claudia and Mu{\~n}oz, Gonzalo and Paat, Joseph and Thomopulos, Dimitri}, title = {The MIP workshop 2023 computational competition on reoptimization}, volume = {16}, journal = {Mathematical Programming Computation}, doi = {10.1007/s12532-024-00256-w}, pages = {255 -- 266}, year = {2024}, abstract = {This paper describes the computational challenge developed for a computational competition held in 2023 for the 20th anniversary of the Mixed Integer Programming Workshop. The topic of this competition was reoptimization, also known as warm starting, of mixed integer linear optimization problems after slight changes to the input data for a common formulation. The challenge was to accelerate the proof of optimality of the modified instances by leveraging the information from the solving processes of previously solved instances, all while creating high-quality primal solutions. Specifically, we discuss the competition's format, the creation of public and hidden datasets, and the evaluation criteria. Our goal is to establish a methodology for the generation of benchmark instances and an evaluation framework, along with benchmark datasets, to foster future research on reoptimization of mixed integer linear optimization problems.}, language = {en} } @inproceedings{MexiBertholdGleixneretal.2023, author = {Mexi, Gioni and Berthold, Timo and Gleixner, Ambros and Nordstr{\"o}m, Jakob}, title = {Improving Conflict Analysis in MIP Solvers by Pseudo-Boolean Reasoning}, volume = {280}, booktitle = {29th International Conference on Principles and Practice of Constraint Programming (CP 2023)}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum f{\"u}r Informatik}, doi = {10.4230/LIPIcs.CP.2023.27}, pages = {27:1 -- 27:19}, year = {2023}, abstract = {Conflict analysis has been successfully generalized from Boolean satisfiability (SAT) solving to mixed integer programming (MIP) solvers, but although MIP solvers operate with general linear inequalities, the conflict analysis in MIP has been limited to reasoning with the more restricted class of clausal constraint. This is in contrast to how conflict analysis is performed in so-called pseudo-Boolean solving, where solvers can reason directly with 0-1 integer linear inequalities rather than with clausal constraints extracted from such inequalities. In this work, we investigate how pseudo-Boolean conflict analysis can be integrated in MIP solving, focusing on 0-1 integer linear programs (0-1 ILPs). Phrased in MIP terminology, conflict analysis can be understood as a sequence of linear combinations and cuts. We leverage this perspective to design a new conflict analysis algorithm based on mixed integer rounding (MIR) cuts, which theoretically dominates the state-of-the-art division-based method in pseudo-Boolean solving. We also report results from a first proof-of-concept implementation of different pseudo-Boolean conflict analysis methods in the open-source MIP solver SCIP. When evaluated on a large and diverse set of 0-1 ILP instances from MIPLIB2017, our new MIR-based conflict analysis outperforms both previous pseudo-Boolean methods and the clause-based method used in MIP. Our conclusion is that pseudo-Boolean conflict analysis in MIP is a promising research direction that merits further study, and that it might also make sense to investigate the use of such conflict analysis to generate stronger no-goods in constraint programming.}, language = {en} } @article{WitzigBerthold2021, author = {Witzig, Jakob and Berthold, Timo}, title = {Conflict Analysis for MINLP}, volume = {33}, journal = {INFORMS Journal on Computing}, number = {2}, doi = {10.1287/ijoc.2020.1050}, pages = {421 -- 435}, year = {2021}, abstract = {The generalization of MIP techniques to deal with nonlinear, potentially non-convex, constraints have been a fruitful direction of research for computational MINLP in the last decade. In this paper, we follow that path in order to extend another essential subroutine of modern MIP solvers towards the case of nonlinear optimization: the analysis of infeasible subproblems for learning additional valid constraints. To this end, we derive two different strategies, geared towards two different solution approaches. These are using local dual proofs of infeasibility for LP-based branch-and-bound and the creation of nonlinear dual proofs for NLP-based branch-and-bound, respectively. We discuss implementation details of both approaches and present an extensive computational study, showing that both techniques can significantly enhance performance when solving MINLPs to global optimality.}, language = {en} } @article{GamrathBertholdSalvagnin2020, author = {Gamrath, Gerald and Berthold, Timo and Salvagnin, Domenico}, title = {An exploratory computational analysis of dual degeneracy in mixed-integer programming}, journal = {EURO Journal on Computational Optimization}, number = {8}, doi = {10.1007/s13675-020-00130-z}, pages = {241 -- 246}, year = {2020}, abstract = {Dual degeneracy, i.e., the presence of multiple optimal bases to a linear programming (LP) problem, heavily affects the solution process of mixed integer programming (MIP) solvers. Different optimal bases lead to different cuts being generated, different branching decisions being taken and different solutions being found by primal heuristics. Nevertheless, only a few methods have been published that either avoid or exploit dual degeneracy. The aim of the present paper is to conduct a thorough computational study on the presence of dual degeneracy for the instances of well-known public MIP instance collections. How many instances are affected by dual degeneracy? How degenerate are the affected models? How does branching affect degeneracy: Does it increase or decrease by fixing variables? Can we identify different types of degenerate MIPs? As a tool to answer these questions, we introduce a new measure for dual degeneracy: the variable-constraint ratio of the optimal face. It provides an estimate for the likelihood that a basic variable can be pivoted out of the basis. Furthermore, we study how the so-called cloud intervals—the projections of the optimal face of the LP relaxations onto the individual variables—evolve during tree search and the implications for reducing the set of branching candidates.}, language = {en} } @misc{GamrathMelchioriBertholdetal.2015, author = {Gamrath, Gerald and Melchiori, Anna and Berthold, Timo and Gleixner, Ambros and Salvagnin, Domenico}, title = {Branching on multi-aggregated variables}, issn = {1438-0064}, doi = {10.1007/978-3-319-18008-3_10}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-53829}, year = {2015}, abstract = {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. We present a novel branching scheme that tries to overcome the above drawback by considering general disjunctions defined by multi-aggregated variables in addition to the standard disjunctions based on single variables. This natural idea results in a hybrid between variable- and constraint-based branching rules. 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 publicly available benchmark instances. For a specific class of problems, we show that the solving time decreases significantly.}, language = {en} } @misc{BertholdHendel2013, author = {Berthold, Timo and Hendel, Gregor}, title = {Shift-And-Propagate}, issn = {1438-0064}, doi = {10.1007/s10732-014-9271-0}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-17596}, year = {2013}, abstract = {For mixed integer programming, recent years have seen a growing interest in the design of general purpose primal heuristics for use inside complete solvers. Many of these heuristics rely on an optimal LP solution. Finding this may itself take a significant amount of time. The presented paper addresses this issue by the introduction of the Shift-And-Propagate heuristic. Shift-And-Propagate is a pre-root primal heuristic that does not require a previously found LP solution. It applies domain propagation techniques to quickly drive a variable assignment towards feasibility. Computational experiments indicate that this heuristic is a powerful supplement of existing rounding and propagation heuristics.}, language = {en} } @inproceedings{ShinanoBertholdHeinz2016, author = {Shinano, Yuji and Berthold, Timo and Heinz, Stefan}, title = {A First Implementation of ParaXpress: Combining Internal and External Parallelization to Solve MIPs on Supercomputers}, booktitle = {Mathematical Software - ICMS 2016, 5th International Conference Berlin, Germany, July 11-14, 2016 Proceedings}, doi = {10.1007/978-3-319-42432-3_38}, pages = {308 -- 316}, year = {2016}, language = {en} } @inproceedings{BertholdFarmerHeinzetal.2016, author = {Berthold, Timo and Farmer, James and Heinz, Stefan and Perregaard, Michael}, title = {Parallelization of the FICO Xpress-Optimizer}, booktitle = {Mathematical Software - ICMS 2016, 5th International Conference Berlin, Germany, July 11-14, 2016 Proceedings}, doi = {10.1007/978-3-319-42432-3_31}, pages = {251 -- 258}, year = {2016}, language = {en} } @article{GleixnerBertholdMuelleretal.2017, author = {Gleixner, Ambros and Berthold, Timo and M{\"u}ller, Benjamin and Weltge, Stefan}, title = {Three Enhancements for Optimization-Based Bound Tightening}, volume = {67}, journal = {Journal of Global Optimization}, number = {4}, doi = {10.1007/s10898-016-0450-4}, pages = {731 -- 757}, year = {2017}, abstract = {Optimization-based bound tightening (OBBT) is one of the most effective procedures to reduce variable domains of nonconvex mixed-integer nonlinear programs (MINLPs). At the same time it is one of the most expensive bound tightening procedures, since it solves auxiliary linear programs (LPs)—up to twice the number of variables many. The main goal of this paper is to discuss algorithmic techniques for an efficient implementation of OBBT. Most state-of-the-art MINLP solvers apply some restricted version of OBBT and it seems to be common belief that OBBT is beneficial if only one is able to keep its computational cost under control. To this end, we introduce three techniques to increase the efficiency of OBBT: filtering strategies to reduce the number of solved LPs, ordering heuristics to exploit simplex warm starts, and the generation of Lagrangian variable bounds (LVBs). The propagation of LVBs during tree search is a fast approximation to OBBT without the need to solve auxiliary LPs. We conduct extensive computational experiments on MINLPLib2. Our results indicate that OBBT is most beneficial on hard instances, for which we observe a speedup of 17\% to 19\% on average. Most importantly, more instances can be solved when using OBBT.}, language = {en} } @article{ShinanoBertholdHeinz2018, author = {Shinano, Yuji and Berthold, Timo and Heinz, Stefan}, title = {ParaXpress: An Experimental Extension of the FICO Xpress-Optimizer to Solve Hard MIPs on Supercomputers}, volume = {33}, journal = {Optimization Methods \& Software}, number = {3}, doi = {10.1080/10556788.2018.1428602}, pages = {530 -- 539}, year = {2018}, abstract = {The Ubiquity Generator (UG) is a general framework for the external parallelization of mixed integer programming (MIP) solvers. In this paper, we present ParaXpress, a distributed memory parallelization of the powerful commercial MIP solver FICO Xpress. Besides sheer performance, an important feature of Xpress is that it provides an internal parallelization for shared memory systems. When aiming for a best possible performance of ParaXpress on a supercomputer, the question arises how to balance the internal Xpress parallelization and the external parallelization by UG against each other. We provide computational experiments to address this question and we show computational results for running ParaXpress on a Top500 supercomputer, using up to 43,344 cores in parallel.}, language = {en} } @phdthesis{Berthold2014, author = {Berthold, Timo}, title = {Heuristic algorithms in global MINLP solvers}, publisher = {Dr. Hut Verlag}, isbn = {978-3-8439-1931-9}, pages = {366}, year = {2014}, abstract = {In the literature for mixed integer programming, heuristic algorithms (particularly primal heuristics) are often considered as stand-alone procedures; in that context, heuristics are treated as an alternative to solving a problem to proven optimality. This conceals the fact that heuristic algorithms are a fundamental component of state-of-the-art global solvers for mixed integer linear programming (MIP) and mixed integer nonlinear programming (MINLP). In the present thesis, we focus on this latter aspect; we study heuristic algorithms that are tightly integrated within global MINLP solvers and analyze their impact on the overall solution process. Our contributions comprise generalizations of primal heuristics for MIP towards MINLP as well as novel ideas for MINLP primal heuristics and for heuristic algorithms to take branching decisions and to collect global information in MIP. These are: - Shift-and-Propagate, a novel propagation heuristic for MIP that does not require the solution to an LP relaxation, - a generic way to generalize large neighborhood search (LNS) heuristics from MIP to MINLP, - an Objective Feasibility Pump heuristic for nonconvex MINLP that uses second-order information and a dynamic selection of rounding procedures, - RENS, an LNS start heuristic for MINLP that optimizes over the set of feasible roundings of an LP solution, - Undercover, an LNS start heuristic for MINLP that solves a largest sub-MIP of a given MINLP, - Rapid Learning, a heuristic algorithm to generate globally valid conflict constraints for MIPs, - Cloud Branching, a heuristic algorithm that exploits dual degeneracy to reduce the number of candidates for branching variable selection. Additionally, we propose a new performance measure, the primal integral, that captures the benefits of primal heuristics better than traditional methods. In our computational study, we compare the performance of the MIP and MINLP solver SCIP with and without primal heuristics on six test sets with altogether 983 instances from academic and industrial sources, including our project partners ForNe, SAP, and Siemens. We observe that heuristics improve the solver performance regarding all measures that we used - by different orders of magnitude. We further see that the harder a problem is to solve to global optimality, the more important the deployment of primal heuristics becomes. The algorithms presented in this thesis are available in source code as part of the solver SCIP, of which the author has been a main developer for the last years. Methods described in this thesis have also been re-implemented within several commercial and noncommercial MIP and MINLP software packages, including Bonmin, CBC, Cplex, Gams, Sulum, and Xpress.}, language = {en} } @misc{BertholdPerregaardMeszaros2017, author = {Berthold, Timo and Perregaard, Michael and M{\´e}sz{\´a}ros, Csaba}, title = {Four good reasons to use an Interior Point solver within a MIP solver}, issn = {1438-0064}, doi = {10.1007/978-3-319-89920-6_22}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64599}, year = {2017}, abstract = {"Interior point algorithms are a good choice for solving pure LPs or QPs, but when you solve MIPs, all you need is a dual simplex." This is the common conception which disregards that an interior point solution provides some unique structural insight into the problem at hand. In this paper, we will discuss some of the benefits that an interior point solver brings to the solution of difficult MIPs within FICO Xpress. This includes many different components of the MIP solver such as branching variable selection, primal heuristics, preprocessing, and of course the solution of the LP relaxation.}, language = {en} } @inproceedings{ShinanoAchterbergBertholdetal.2016, author = {Shinano, Yuji and Achterberg, Tobias and Berthold, Timo and Heinz, Stefan and Koch, Thorsten and Winkler, Michael}, title = {Solving Open MIP Instances with ParaSCIP on Supercomputers using up to 80,000 Cores}, booktitle = {Proc. of 30th IEEE International Parallel \& Distributed Processing Symposium}, doi = {10.1109/IPDPS.2016.56}, year = {2016}, abstract = {This paper describes how we solved 12 previously unsolved mixed-integer program- ming (MIP) instances from the MIPLIB benchmark sets. To achieve these results we used an enhanced version of ParaSCIP, setting a new record for the largest scale MIP computation: up to 80,000 cores in parallel on the Titan supercomputer. In this paper we describe the basic parallelization mechanism of ParaSCIP, improvements of the dynamic load balancing and novel techniques to exploit the power of parallelization for MIP solving. We give a detailed overview of computing times and statistics for solving open MIPLIB instances.}, language = {en} } @article{BertholdHendelKoch2017, author = {Berthold, Timo and Hendel, Gregor and Koch, Thorsten}, title = {From feasibility to improvement to proof: three phases of solving mixed-integer programs}, volume = {33}, journal = {Optimization Methods and Software}, number = {3}, publisher = {Taylor \& Francis}, doi = {10.1080/10556788.2017.1392519}, pages = {499 -- 517}, year = {2017}, abstract = {Modern mixed-integer programming (MIP) solvers employ dozens of auxiliary algorithmic components to support the branch-and-bound search in finding and improving primal solutions and in strengthening the dual bound. Typically, all components are tuned to minimize the average running time to prove optimality. In this article, we take a different look at the run of a MIP solver. We argue that the solution process consists of three distinct phases, namely achieving feasibility, improving the incumbent solution, and proving optimality. We first show that the entire solving process can be improved by adapting the search strategy with respect to the phase-specific aims using different control tunings. Afterwards, we provide criteria to predict the transition between the individual phases and evaluate the performance impact of altering the algorithmic behaviour of the non-commercial MIP solver Scip at the predicted phase transition points.}, language = {en} } @inproceedings{GamrathMelchioriBertholdetal.2015, author = {Gamrath, Gerald and Melchiori, Anna and Berthold, Timo and Gleixner, Ambros and Salvagnin, Domenico}, title = {Branching on Multi-aggregated Variables}, volume = {9075}, booktitle = {Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2015}, doi = {10.1007/978-3-319-18008-3_10}, pages = {141 -- 156}, year = {2015}, abstract = {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. We present a novel branching scheme that tries to overcome the above drawback by considering general disjunctions defined by multi-aggregated variables in addition to the standard disjunctions based on single variables. This natural idea results in a hybrid between variable- and constraint-based branching rules. 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 publicly available benchmark instances. For a specific class of problems, we show that the solving time decreases significantly.}, language = {en} } @article{BertholdHendel2014, author = {Berthold, Timo and Hendel, Gregor}, title = {Shift-and-Propagate}, volume = {21}, journal = {Journal of Heuristics}, number = {1}, doi = {10.1007/s10732-014-9271-0}, pages = {73 -- 106}, year = {2014}, abstract = {In recent years, there has been a growing interest in the design of general purpose primal heuristics for use inside complete mixed integer programming solvers. Many of these heuristics rely on an optimal LP solution, which may take a significant amount of time to find. In this paper, we address this issue by introducing a pre-root primal heuristic that does not require a previously found LP solution. This heuristic, named Shift-and-Propagate , applies domain propagation techniques to quickly drive a variable assignment towards feasibility. Computational experiments indicate that this heuristic is a powerful supplement to existing rounding and propagation heuristics.}, language = {en} } @misc{RalphsShinanoBertholdetal.2016, author = {Ralphs, Ted and Shinano, Yuji and Berthold, Timo and Koch, Thorsten}, title = {Parallel Solvers for Mixed Integer Linear Programming}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-62239}, year = {2016}, abstract = {In this article, we introduce parallel mixed integer linear programming (MILP) solvers. MILP solving algorithms have been improved tremendously in the last two decades. Currently, commercial MILP solvers are known as a strong optimization tool. Parallel MILP solver development has started in 1990s. However, since the improvements of solving algorithms have much impact to solve MILP problems than application of parallel computing, there were not many visible successes. With the spread of multi-core CPUs, current state-of-the-art MILP solvers have parallel implementations and researches to apply parallelism in the solving algorithm also getting popular. We summarize current existing parallel MILP solver architectures.}, language = {en} } @inproceedings{WitzigBertholdHeinz2017, author = {Witzig, Jakob and Berthold, Timo and Heinz, Stefan}, title = {Experiments with Conflict Analysis in Mixed Integer Programming}, volume = {10335}, booktitle = {Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2017}, publisher = {Springer}, doi = {10.1007/978-3-319-59776-8_17}, pages = {211 -- 222}, year = {2017}, abstract = {The analysis of infeasible subproblems plays an import role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems. The first is to analyze the sequence of implications obtained by domain propagation that led to infeasibility. The result of the analysis is one or more sets of contradicting variable bounds from which so-called conflict constraints can be generated. This concept has its origin in solving satisfiability problems and is similarly used in constraint programming. The second concept is to analyze infeasible linear programming (LP) relaxations. The dual LP solution provides a set of multipliers that can be used to generate a single new globally valid linear constraint. The main contribution of this short paper is an empirical evaluation of two ways to combine both approaches. Experiments are carried out on general MIP instances from standard public test sets such as Miplib2010; the presented algorithms have been implemented within the non-commercial MIP solver SCIP. Moreover, we present a pool-based approach to manage conflicts which addresses the way a MIP solver traverses the search tree better than aging strategies known from SAT solving.}, language = {en} } @misc{GamrathBertholdHeinzetal.2015, author = {Gamrath, Gerald and Berthold, Timo and Heinz, Stefan and Winkler, Michael}, title = {Structure-based primal heuristics for mixed integer programming}, issn = {1438-0064}, doi = {http://dx.doi.org/10.1007/978-4-431-55420-2_3}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-55518}, year = {2015}, abstract = {Primal heuristics play an important role in the solving of mixed integer programs (MIPs). They help to reach optimality faster and provide good feasible solutions early in the solving process. In this paper, we present two new primal heuristics which take into account global structures available within MIP solvers to construct feasible solutions at the beginning of the solving process. These heuristics follow a large neighborhood search (LNS) approach and use global structures to define a neighborhood that is with high probability significantly easier to process while (hopefully) still containing good feasible solutions. The definition of the neighborhood is done by iteratively fixing variables and propagating these fixings. Thereby, fixings are determined based on the predicted impact they have on the subsequent domain propagation. The neighborhood is solved as a sub-MIP and solutions are transferred back to the original problem. Our computational experiments on standard MIP test sets show that the proposed heuristics find solutions for about every third instance and therewith help to improve the average solving time.}, language = {en} } @misc{GamrathBertholdHeinzetal.2017, author = {Gamrath, Gerald and Berthold, Timo and Heinz, Stefan and Winkler, Michael}, title = {Structure-driven fix-and-propagate heuristics for mixed integer programming}, issn = {1438-0064}, doi = {10.1007/s12532-019-00159-1}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-65387}, year = {2017}, abstract = {Primal heuristics play an important role in the solving of mixed integer programs (MIPs). They often provide good feasible solutions early in the solving process and help to solve instances to optimality faster. In this paper, we present a scheme for primal start heuristics that can be executed without previous knowledge of an LP solution or a previously found integer feasible solution. It uses global structures available within MIP solvers to iteratively fix integer variables and propagate these fixings. Thereby, fixings are determined based on the predicted impact they have on the subsequent domain propagation. If sufficiently many variables can be fixed that way, the resulting problem is solved as an LP and the solution is rounded. If the rounded solution did not provide a feasible solution already, a sub-MIP is solved for the neighborhood defined by the variable fixings performed in the first phase. The global structures help to define a neighborhood that is with high probability significantly easier to process while (hopefully) still containing good feasible solutions. We present three primal heuristics that use this scheme based on different global structures. Our computational experiments on standard MIP test sets show that the proposed heuristics find solutions for about three out of five instances and therewith help to improve several performance measures for MIP solvers, including the primal integral and the average solving time.}, language = {en} } @misc{BertholdHendelKoch2016, author = {Berthold, Timo and Hendel, Gregor and Koch, Thorsten}, title = {The Three Phases of MIP Solving}, issn = {1438-0064}, doi = {10.1080/10556788.2017.1392519}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61607}, year = {2016}, abstract = {Modern MIP solvers employ dozens of auxiliary algorithmic components to support the branch-and-bound search in finding and improving primal solutions and in strengthening the dual bound. Typically, all components are tuned to minimize the average running time to prove optimality. In this article, we take a different look at the run of a MIP solver. We argue that the solution process consists of three different phases, namely achieving feasibility, improving the incumbent solution, and proving optimality. We first show that the entire solving process can be improved by adapting the search strategy with respect to the phase-specific aims using different control tunings. Afterwards, we provide criteria to predict the transition between the individual phases and evaluate the performance impact of altering the algorithmic behavior of the MIP solver SCIP at the predicted phase transition points.}, language = {en} }