@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} } @misc{BertholdGleixner2012, author = {Berthold, Timo and Gleixner, Ambros}, title = {Undercover: a primal MINLP heuristic exploring a largest sub-MIP}, issn = {1438-0064}, doi = {10.1007/s10107-013-0635-2}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14631}, number = {12-07}, year = {2012}, abstract = {We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programming (MINLP) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a minimal set of variables that need to be fixed in order to linearize each constraint, a so-called cover. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. We apply domain propagation and conflict analysis to try to avoid infeasibilities and learn from them, respectively. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and general MINLPs from MINLPLib. It turns out that the majority of these instances allow for small covers. Although general in nature, the heuristic appears most promising for MIQCPs, and complements nicely with existing root node heuristics in different state-of-the-art solvers.}, 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} }