@misc{KochAchterbergAndersenetal.2010, author = {Koch, Thorsten and Achterberg, Tobias and Andersen, Erling and Bastert, Oliver and Berthold, Timo and Bixby, Robert E. and Danna, Emilie and Gamrath, Gerald and Gleixner, Ambros and Heinz, Stefan and Lodi, Andrea and Mittelmann, Hans and Ralphs, Ted and Salvagnin, Domenico and Steffy, Daniel and Wolter, Kati}, title = {MIPLIB 2010}, doi = {10.1007/s12532-011-0025-9}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-12953}, number = {10-31}, year = {2010}, abstract = {This paper reports on the fifth version of the Mixed Integer Programming Library. The MIPLIB 2010 is the first MIPLIB release that has been assembled by a large group from academia and from industry, all of whom work in integer programming. There was mutual consent that the concept of the library had to be expanded in order to fulfill the needs of the community. The new version comprises 361 instances sorted into several groups. This includes the main benchmark test set of 87 instances, which are all solvable by today's codes, and also the challenge test set with 164 instances, many of which are currently unsolved. For the first time, we include scripts to run automated tests in a predefined way. Further, there is a solution checker to test the accuracy of provided solutions using exact arithmetic.}, language = {en} } @misc{SteffyWolter, author = {Steffy, Daniel and Wolter, Kati}, title = {Valid Linear Programming Bounds for Exact Mixed-Integer Programming}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-12332}, number = {11-08}, abstract = {Fast computation of valid linear programming (LP) bounds serves as an important subroutine for solving mixed-integer programming problems exactly. We introduce a new method for computing valid LP bounds designed for this application. The algorithm corrects approximate LP dual solutions to be exactly feasible, giving a valid bound. Solutions are repaired by performing a projection and a shift to ensure all constraints are satisfied; bound computations are accelerated by reusing structural information through the branch-and-bound tree. We demonstrate this method to be widely applicable and faster than solving a sequence of exact LPs. Several variations of the algorithm are described and computationally evaluated in an exact branch-and-bound algorithm within the mixed-integer programming framework SCIP.}, language = {en} } @misc{CookKochSteffyetal., author = {Cook, William and Koch, Thorsten and Steffy, Daniel and Wolter, Kati}, title = {A Hybrid Branch-and-Bound Approach for Exact Rational Mixed-Integer Programming}, issn = {1438-0064}, doi = {10.1007/s12532-013-0055-6}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-17171}, abstract = {We present an exact rational solver for mixed-integer linear programming that avoids the numerical inaccuracies inherent in the floating-point computations used by existing software. This allows the solver to be used for establishing theoretical results and in applications where correct solutions are critical due to legal and financial consequences. Our solver is a hybrid symbolic/numeric implementation of LP-based branch-and-bound, using numerically-safe methods for all binding computations in the search tree. Computing provably accurate solutions by dynamically choosing the fastest of several safe dual bounding methods depending on the structure of the instance, our exact solver is only moderately slower than an inexact floating-point branch-and-bound solver. The software is incorporated into the SCIP optimization framework, using the exact LP solver QSopt_ex and the GMP arithmetic library. Computational results are presented for a suite of test instances taken from the MIPLIB and Mittelmann libraries and for a new collection of numerically difficult instances.}, language = {en} } @article{CookKochSteffyetal., author = {Cook, William and Koch, Thorsten and Steffy, Daniel and Wolter, Kati}, title = {A hybrid branch-and-bound approach for exact rational mixed-integer programming}, series = {Mathematical Programming Computation}, volume = {5}, journal = {Mathematical Programming Computation}, number = {3}, doi = {10.1007/s12532-013-0055-6}, pages = {305 -- 344}, abstract = {We present an exact rational solver for mixed-integer linear programming that avoids the numerical inaccuracies inherent in the floating-point computations used by existing software. This allows the solver to be used for establishing theoretical results and in applications where correct solutions are critical due to legal and financial consequences. Our solver is a hybrid symbolic/numeric implementation of LP-based branch-and-bound, using numerically-safe methods for all binding computations in the search tree. Computing provably accurate solutions by dynamically choosing the fastest of several safe dual bounding methods depending on the structure of the instance, our exact solver is only moderately slower than an inexact floating-point branch-and-bound solver. The software is incorporated into the SCIP optimization framework, using the exact LP solver QSopt_ex and the GMP arithmetic library. Computational results are presented for a suite of test instances taken from the MIPLIB and Mittelmann libraries and for a new collection of numerically difficult instances.}, language = {en} } @inproceedings{MiltenbergerRalphsSteffy, author = {Miltenberger, Matthias and Ralphs, Ted and Steffy, Daniel}, title = {Exploring the Numerics of Branch-and-Cut for Mixed Integer Linear Optimization}, series = {Operations Research Proceedings 2017}, booktitle = {Operations Research Proceedings 2017}, publisher = {Springer International Publishing}, doi = {10.1007/978-3-319-89920-6}, pages = {151 -- 157}, abstract = {We investigate how the numerical properties of the LP relaxations evolve throughout the solution procedure in a solver employing the branch-and-cut algorithm. The long-term goal of this work is to determine whether the effect on the numerical conditioning of the LP relaxations resulting from the branching and cutting operations can be effectively predicted and whether such predictions can be used to make better algorithmic choices. In a first step towards this goal, we discuss here the numerical behavior of an existing solver in order to determine whether our intuitive understanding of this behavior is correct.}, language = {en} } @misc{MiltenbergerRalphsSteffy, author = {Miltenberger, Matthias and Ralphs, Ted and Steffy, Daniel}, title = {Exploring the Numerics of Branch-and-Cut for Mixed Integer Linear Optimization}, series = {Operations Research Proceedings 2017}, journal = {Operations Research Proceedings 2017}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64645}, abstract = {We investigate how the numerical properties of the LP relaxations evolve throughout the solution procedure in a solver employing the branch-and-cut algorithm. The long-term goal of this work is to determine whether the effect on the numerical conditioning of the LP relaxations resulting from the branching and cutting operations can be effectively predicted and whether such predictions can be used to make better algorithmic choices. In a first step towards this goal, we discuss here the numerical behavior of an existing solver in order to determine whether our intuitive understanding of this behavior is correct.}, language = {en} } @misc{CheungGleixnerSteffy, author = {Cheung, Kevin K. H. and Gleixner, Ambros and Steffy, Daniel}, title = {Verifying Integer Programming Results}, issn = {1438-0064}, doi = {10.1007/978-3-319-59250-3_13}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61044}, abstract = {Software for mixed-integer linear programming can return incorrect results for a number of reasons, one being the use of inexact floating-point arithmetic. Even solvers that employ exact arithmetic may suffer from programming or algorithmic errors, motivating the desire for a way to produce independently verifiable certificates of claimed results. Due to the complex nature of state-of-the-art MILP solution algorithms, the ideal form of such a certificate is not entirely clear. This paper proposes such a certificate format, illustrating its capabilities and structure through examples. The certificate format is designed with simplicity in mind and is composed of a list of statements that can be sequentially verified using a limited number of simple yet powerful inference rules. We present a supplementary verification tool for compressing and checking these certificates independently of how they were created. We report computational results on a selection of mixed-integer linear programming instances from the literature. To this end, we have extended the exact rational version of the MIP solver SCIP to produce such certificates.}, language = {en} } @inproceedings{GleixnerSteffyWolter2012, author = {Gleixner, Ambros and Steffy, Daniel and Wolter, Kati}, title = {Improving the Accuracy of Linear Programming Solvers with Iterative Refinement}, series = {ISSAC '12. Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation}, booktitle = {ISSAC '12. Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation}, doi = {10.1145/2442829.2442858}, pages = {187 -- 194}, year = {2012}, language = {en} } @article{KochAchterbergAndersenetal.2011, author = {Koch, Thorsten and Achterberg, Tobias and Andersen, Erling and Bastert, Oliver and Berthold, Timo and Bixby, Robert E. and Danna, Emilie and Gamrath, Gerald and Gleixner, Ambros and Heinz, Stefan and Lodi, Andrea and Mittelmann, Hans and Ralphs, Ted and Salvagnin, Domenico and Steffy, Daniel and Wolter, Kati}, title = {MIPLIB 2010}, series = {Mathematical Programming Computation}, volume = {3}, journal = {Mathematical Programming Computation}, number = {2}, doi = {10.1007/s12532-011-0025-9}, pages = {103 -- 163}, year = {2011}, language = {en} } @inproceedings{CookKochSteffyetal.2011, author = {Cook, William and Koch, Thorsten and Steffy, Daniel and Wolter, Kati}, title = {An Exact Rational Mixed-Integer Programming Solver}, series = {IPCO 2011}, volume = {6655}, booktitle = {IPCO 2011}, editor = {G{\"u}nl{\"u}k, Oktay and Woeginger, Gerhard}, doi = {10.1007/978-3-642-20807-2_9}, pages = {104 -- 116}, year = {2011}, language = {en} }