@misc{BertholdGamrathSalvagnin, 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}, 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} } @article{GamrathBertholdHeinzetal., author = {Gamrath, Gerald and Berthold, Timo and Heinz, Stefan and Winkler, Michael}, title = {Structure-driven fix-and-propagate heuristics for mixed integer programming}, series = {Mathematical Programming Computation}, volume = {11}, journal = {Mathematical Programming Computation}, number = {4}, publisher = {Springer}, address = {Berlin Heidelberg}, doi = {10.1007/s12532-019-00159-1}, pages = {675 -- 702}, 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} } @article{GamrathBertholdSalvagnin, author = {Gamrath, Gerald and Berthold, Timo and Salvagnin, Domenico}, title = {An exploratory computational analysis of dual degeneracy in mixed-integer programming}, series = {EURO Journal on Computational Optimization}, journal = {EURO Journal on Computational Optimization}, number = {8}, doi = {10.1007/s13675-020-00130-z}, pages = {241 -- 246}, 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{GamrathAndersonBestuzhevaetal., author = {Gamrath, Gerald and Anderson, Daniel and Bestuzheva, Ksenia and Chen, Wei-Kun and Eifler, Leon and Gasse, Maxime and Gemander, Patrick and Gleixner, Ambros and Gottwald, Leona and Halbig, Katrin and Hendel, Gregor and Hojny, Christopher and Koch, Thorsten and Le Bodic, Pierre and Maher, Stephen J. and Matter, Frederic and Miltenberger, Matthias and M{\"u}hmer, Erik and M{\"u}ller, Benjamin and Pfetsch, Marc and Schl{\"o}sser, Franziska and Serrano, Felipe and Shinano, Yuji and Tawfik, Christine and Vigerske, Stefan and Wegscheider, Fabian and Weninger, Dieter and Witzig, Jakob}, title = {The SCIP Optimization Suite 7.0}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-78023}, abstract = {The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming frame- work SCIP. This paper discusses enhancements and extensions contained in version 7.0 of the SCIP Optimization Suite. The new version features the parallel presolving library PaPILO as a new addition to the suite. PaPILO 1.0 simplifies mixed-integer linear op- timization problems and can be used stand-alone or integrated into SCIP via a presolver plugin. SCIP 7.0 provides additional support for decomposition algorithms. Besides im- provements in the Benders' decomposition solver of SCIP, user-defined decomposition structures can be read, which are used by the automated Benders' decomposition solver and two primal heuristics. Additionally, SCIP 7.0 comes with a tree size estimation that is used to predict the completion of the overall solving process and potentially trigger restarts. Moreover, substantial performance improvements of the MIP core were achieved by new developments in presolving, primal heuristics, branching rules, conflict analysis, and symmetry handling. Last, not least, the report presents updates to other components and extensions of the SCIP Optimization Suite, in particular, the LP solver SoPlex and the mixed-integer semidefinite programming solver SCIP-SDP.}, language = {en} } @article{GleixnerHendelGamrathetal., 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}, series = {Mathematical Programming Computation}, volume = {13}, journal = {Mathematical Programming Computation}, number = {3}, doi = {10.1007/s12532-020-00194-3}, pages = {443 -- 490}, 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} } @misc{BestuzhevaBesanconChenetal., author = {Bestuzheva, Ksenia and Besan{\c{c}}on, Mathieu and Chen, Wei-Kun and Chmiela, Antonia and Donkiewicz, Tim and van Doornmalen, Jasper and Eifler, Leon and Gaul, Oliver and Gamrath, Gerald and Gleixner, Ambros and Gottwald, Leona and Graczyk, Christoph and Halbig, Katrin and Hoen, Alexander and Hojny, Christopher and van der Hulst, Rolf and Koch, Thorsten and L{\"u}bbecke, Marco and Maher, Stephen J. and Matter, Frederic and M{\"u}hmer, Erik and M{\"u}ller, Benjamin and Pfetsch, Marc E. and Rehfeldt, Daniel and Schlein, Steffan and Schl{\"o}sser, Franziska and Serrano, Felipe and Shinano, Yuji and Sofranac, Boro and Turner, Mark and Vigerske, Stefan and Wegscheider, Fabian and Wellner, Philipp and Weninger, Dieter and Witzig, Jakob}, title = {The SCIP Optimization Suite 8.0}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-85309}, abstract = {The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP. This paper discusses enhancements and extensions contained in version 8.0 of the SCIP Optimization Suite. Major updates in SCIP include improvements in symmetry handling and decomposition algorithms, new cutting planes, a new plugin type for cut selection, and a complete rework of the way nonlinear constraints are handled. Additionally, SCIP 8.0 now supports interfaces for Julia as well as Matlab. Further, UG now includes a unified framework to parallelize all solvers, a utility to analyze computational experiments has been added to GCG, dual solutions can be postsolved by PaPILO, new heuristics and presolving methods were added to SCIP-SDP, and additional problem classes and major performance improvements are available in SCIP-Jack.}, language = {en} } @article{BestuzhevaBesanconChenetal., author = {Bestuzheva, Ksenia and Besan{\c{c}}on, Mathieu and Chen, Wei-Kun and Chmiela, Antonia and Donkiewicz, Tim and Doornmalen, Jasper and Eifler, Leon and Gaul, Oliver and Gamrath, Gerald and Gleixner, Ambros and Gottwald, Leona and Graczyk, Christoph and Halbig, Katrin and Hoen, Alexander and Hojny, Christopher and Hulst, Rolf and Koch, Thorsten and L{\"u}bbecke, Marco and Maher, Stephen J. and Matter, Frederic and M{\"u}hmer, Erik and M{\"u}ller, Benjamin and Pfetsch, Marc and Rehfeldt, Daniel and Schlein, Steffan and Schl{\"o}sser, Franziska and Serrano, Felipe and Shinano, Yuji and Sofranac, Boro and Turner, Mark and Vigerske, Stefan and Wegscheider, Fabian and Wellner, Philipp and Weninger, Dieter and Witzig, Jakob}, title = {Enabling research through the SCIP optimization suite 8.0}, series = {ACM Transactions on Mathematical Software}, volume = {49}, journal = {ACM Transactions on Mathematical Software}, number = {2}, doi = {10.1145/3585516}, pages = {1 -- 21}, abstract = {The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP. The focus of this article is on the role of the SCIP Optimization Suite in supporting research. SCIP's main design principles are discussed, followed by a presentation of the latest performance improvements and developments in version 8.0, which serve both as examples of SCIP's application as a research tool and as a platform for further developments. Furthermore, this article gives an overview of interfaces to other programming and modeling languages, new features that expand the possibilities for user interaction with the framework, and the latest developments in several extensions built upon SCIP.}, language = {en} }