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  <doc>
    <id>8128</id>
    <completedYear/>
    <publishedYear>2021</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
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    <issue/>
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    <type>article</type>
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    <title language="eng">A Computational Status Update for Exact Rational Mixed Integer Programming</title>
    <abstract language="eng">The last milestone achievement for the roundoff-error-free solution of general&#13;
mixed integer programs over the rational numbers was a hybrid-precision&#13;
branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013.&#13;
&#13;
We describe a substantial revision and extension of this framework that&#13;
integrates symbolic presolving, features an exact repair step for solutions from&#13;
primal heuristics, employs a faster rational LP solver based on LP iterative&#13;
refinement, and is able to produce independently verifiable certificates of&#13;
optimality.&#13;
&#13;
We study the significantly improved performance and give insights into the&#13;
computational behavior of the new algorithmic components.&#13;
&#13;
On the MIPLIB 2017 benchmark set, we observe an average speedup of&#13;
6.6x over the original framework and 2.8&#13;
times as many instances solved within a time limit of two hours.</abstract>
    <parentTitle language="eng">Integer Programming and Combinatorial Optimization: 22th International Conference, IPCO 2021</parentTitle>
    <identifier type="doi">10.1007/978-3-030-73879-2_12</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="AcceptedDate">2021-01-20</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <enrichment key="PreprintUrn">urn:nbn:de:0297-zib-81298</enrichment>
    <author>Leon Eifler</author>
    <submitter>Leon Eifler</submitter>
    <author>Ambros Gleixner</author>
    <collection role="projects" number="MODAL-SynLab">MODAL-SynLab</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
    <collection role="institutes" number="ais2t">AI in Society, Science, and Technology</collection>
  </doc>
  <doc>
    <id>8129</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2021-01-22</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A Computational Status Update for Exact Rational Mixed Integer Programming</title>
    <abstract language="eng">The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 6.6x over the original framework and 2.8 times as many instances solved within a time limit of two hours.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-81298</identifier>
    <enrichment key="opus.source">publish</enrichment>
    <author>Leon Eifler</author>
    <submitter>Leon Eifler</submitter>
    <author>Ambros Gleixner</author>
    <series>
      <title>ZIB-Report</title>
      <number>21-04</number>
    </series>
    <collection role="msc" number="90-XX">OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING</collection>
    <collection role="projects" number="MODAL-SynLab">MODAL-SynLab</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
    <collection role="institutes" number="ais2t">AI in Society, Science, and Technology</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/8129/paper.pdf</file>
  </doc>
  <doc>
    <id>7802</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
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    <type>reportzib</type>
    <publisherName/>
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    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2020-03-30</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">The SCIP Optimization Suite 7.0</title>
    <abstract language="eng">The SCIP Optimization Suite provides a collection of software packages for&#13;
mathematical optimization centered around the constraint integer programming frame-&#13;
work SCIP. This paper discusses enhancements and extensions contained in version 7.0&#13;
of the SCIP Optimization Suite. The new version features the parallel presolving library&#13;
PaPILO as a new addition to the suite. PaPILO 1.0 simplifies mixed-integer linear op-&#13;
timization problems and can be used stand-alone or integrated into SCIP via a presolver&#13;
plugin. SCIP 7.0 provides additional support for decomposition algorithms. Besides im-&#13;
provements in the Benders’ decomposition solver of SCIP, user-defined decomposition&#13;
structures can be read, which are used by the automated Benders’ decomposition solver&#13;
and two primal heuristics. Additionally, SCIP 7.0 comes with a tree size estimation&#13;
that is used to predict the completion of the overall solving process and potentially&#13;
trigger restarts. Moreover, substantial performance improvements of the MIP core were&#13;
achieved by new developments in presolving, primal heuristics, branching rules, conflict&#13;
analysis, and symmetry handling. Last, not least, the report presents updates to other&#13;
components and extensions of the SCIP Optimization Suite, in particular, the LP solver&#13;
SoPlex and the mixed-integer semidefinite programming solver SCIP-SDP.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-78023</identifier>
    <author>Gerald Gamrath</author>
    <submitter>Felipe Serrano</submitter>
    <author>Daniel Anderson</author>
    <author>Ksenia Bestuzheva</author>
    <author>Wei-Kun Chen</author>
    <author>Leon Eifler</author>
    <author>Maxime Gasse</author>
    <author>Patrick Gemander</author>
    <author>Ambros Gleixner</author>
    <author>Leona Gottwald</author>
    <author>Katrin Halbig</author>
    <author>Gregor Hendel</author>
    <author>Christopher Hojny</author>
    <author>Thorsten Koch</author>
    <author>Pierre Le Bodic</author>
    <author>Stephen J. Maher</author>
    <author>Frederic Matter</author>
    <author>Matthias Miltenberger</author>
    <author>Erik Mühmer</author>
    <author>Benjamin Müller</author>
    <author>Marc Pfetsch</author>
    <author>Franziska Schlösser</author>
    <author>Felipe Serrano</author>
    <author>Yuji Shinano</author>
    <author>Christine Tawfik</author>
    <author>Stefan Vigerske</author>
    <author>Fabian Wegscheider</author>
    <author>Dieter Weninger</author>
    <author>Jakob Witzig</author>
    <series>
      <title>ZIB-Report</title>
      <number>20-10</number>
    </series>
    <collection role="msc" number="65-XX">NUMERICAL ANALYSIS</collection>
    <collection role="msc" number="90-XX">OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING</collection>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="institutes" number="mip">Mathematical Optimization Methods</collection>
    <collection role="persons" number="gamrath">Gamrath, Gerald</collection>
    <collection role="persons" number="robert.gottwald">Gottwald, Robert</collection>
    <collection role="persons" number="hendel">Hendel, Gregor</collection>
    <collection role="persons" number="koch">Koch, Thorsten</collection>
    <collection role="persons" number="miltenberger">Miltenberger, Matthias</collection>
    <collection role="persons" number="shinano">Shinano, Yuji</collection>
    <collection role="persons" number="vigerske">Vigerske, Stefan</collection>
    <collection role="projects" number="ASTfSCM">ASTfSCM</collection>
    <collection role="projects" number="MIP-ZIBOPT">MIP-ZIBOPT</collection>
    <collection role="projects" number="MODAL-SynLab">MODAL-SynLab</collection>
    <collection role="projects" number="Siemens">Siemens</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
    <collection role="persons" number="schloesser">Schlösser, Franziska</collection>
    <collection role="persons" number="bestuzheva">Bestuzheva, Ksenia</collection>
    <collection role="projects" number="plan4res">Plan4res</collection>
    <collection role="institutes" number="aopt">Applied Optimization</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/7802/scipopt-70.pdf</file>
  </doc>
  <doc>
    <id>7024</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2018-08-28</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Chvátal’s Conjecture Holds for Ground Sets of Seven Elements</title>
    <abstract language="eng">We establish a general computational framework for Chvátal’s conjecture based on exact rational integer programming. As a result we prove Chvátal’s conjecture holds for all downsets whose union of sets contains seven elements or less. The computational proof relies on an exact branch-and-bound certificate that allows for elementary verification and is independent of the integer programming solver used.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-70240</identifier>
    <author>Leon Eifler</author>
    <submitter>Leon Eifler</submitter>
    <author>Ambros Gleixner</author>
    <author>Jonad Pulaj</author>
    <series>
      <title>ZIB-Report</title>
      <number>18-49</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>extremal combinatorics</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>exact rational integer programming</value>
    </subject>
    <collection role="msc" number="05-XX">COMBINATORICS (For finite fields, see 11Txx)</collection>
    <collection role="msc" number="90-XX">OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING</collection>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="institutes" number="mip">Mathematical Optimization Methods</collection>
    <collection role="persons" number="pulaj">Pulaj, Jonad</collection>
    <collection role="projects" number="MIP-ZIBOPT">MIP-ZIBOPT</collection>
    <collection role="projects" number="MODAL-SynLab">MODAL-SynLab</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
    <collection role="institutes" number="aopt">Applied Optimization</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/7024/chvatal.pdf</file>
  </doc>
  <doc>
    <id>6629</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2017-12-21</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">The SCIP Optimization Suite 5.0</title>
    <abstract language="eng">This article describes new features and enhanced algorithms made available in version 5.0 of the SCIP Optimization Suite. In its central component, the constraint integer programming solver SCIP, remarkable performance improvements have been achieved for solving mixed-integer linear and nonlinear programs. On MIPs, SCIP 5.0 is about 41 % faster than SCIP 4.0 and over twice as fast on instances that take at least 100 seconds to solve. For MINLP, SCIP 5.0 is about 17 % faster overall and 23 % faster on instances that take at least 100 seconds to solve. This boost is due to algorithmic advances in several parts of the solver such as cutting plane generation and management, a new adaptive coordination of large neighborhood search heuristics, symmetry handling, and strengthened McCormick relaxations for bilinear terms in MINLPs. Besides discussing the theoretical background and the implementational aspects of these developments, the report describes recent additions for the other software packages connected to SCIP, in particular for the LP solver SoPlex, the Steiner tree solver SCIP-Jack, the MISDP solver SCIP-SDP, and the parallelization framework UG.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-66297</identifier>
    <author>Ambros Gleixner</author>
    <submitter>Gregor Hendel</submitter>
    <author>Leon Eifler</author>
    <author>Tristan Gally</author>
    <author>Gerald Gamrath</author>
    <author>Patrick Gemander</author>
    <author>Robert Lion Gottwald</author>
    <author>Gregor Hendel</author>
    <author>Christopher Hojny</author>
    <author>Thorsten Koch</author>
    <author>Matthias Miltenberger</author>
    <author>Benjamin Müller</author>
    <author>Marc Pfetsch</author>
    <author>Christian Puchert</author>
    <author>Daniel Rehfeldt</author>
    <author>Franziska Schlösser</author>
    <author>Felipe Serrano</author>
    <author>Yuji Shinano</author>
    <author>Jan Merlin Viernickel</author>
    <author>Stefan Vigerske</author>
    <author>Dieter Weninger</author>
    <author>Jonas T. Witt</author>
    <author>Jakob Witzig</author>
    <series>
      <title>ZIB-Report</title>
      <number>17-61</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>constraint integer programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>linear programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>mixed-integer linear programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>mixed-integer nonlinear programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>optimization solver</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>branch-and-cut</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>branch-and-price</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>column generation framework</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>parallelization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>mixed-integer semidefinite programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Steiner tree optimization</value>
    </subject>
    <collection role="msc" number="65-XX">NUMERICAL ANALYSIS</collection>
    <collection role="msc" number="90-XX">OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING</collection>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="institutes" number="mip">Mathematical Optimization Methods</collection>
    <collection role="persons" number="gamrath">Gamrath, Gerald</collection>
    <collection role="persons" number="robert.gottwald">Gottwald, Robert</collection>
    <collection role="persons" number="hendel">Hendel, Gregor</collection>
    <collection role="persons" number="koch">Koch, Thorsten</collection>
    <collection role="persons" number="miltenberger">Miltenberger, Matthias</collection>
    <collection role="persons" number="rehfeldt">Rehfeldt, Daniel</collection>
    <collection role="persons" number="shinano">Shinano, Yuji</collection>
    <collection role="persons" number="vigerske">Vigerske, Stefan</collection>
    <collection role="projects" number="ASTfSCM">ASTfSCM</collection>
    <collection role="projects" number="MIP-ZIBOPT">MIP-ZIBOPT</collection>
    <collection role="projects" number="MODAL-SynLab">MODAL-SynLab</collection>
    <collection role="projects" number="Siemens">Siemens</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
    <collection role="persons" number="schloesser">Schlösser, Franziska</collection>
    <collection role="institutes" number="aopt">Applied Optimization</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/6629/scipopt-50.pdf</file>
  </doc>
  <doc>
    <id>6936</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2018-07-02</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">The SCIP Optimization Suite 6.0</title>
    <abstract language="eng">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 6.0 of the SCIP Optimization Suite. Besides performance improvements of the MIP and MINLP core achieved by new primal heuristics and a new selection criterion for cutting planes, one focus of this release are decomposition algorithms. Both SCIP and the automatic decomposition solver GCG now include advanced functionality for performing Benders’ decomposition in a generic framework. GCG’s detection loop for structured matrices and the coordination of pricing routines for Dantzig-Wolfe decomposition has been significantly revised for greater flexibility. Two SCIP extensions have been added&#13;
to solve the recursive circle packing problem by a problem-specific column generation scheme and to demonstrate the use of the new Benders’ framework for stochastic capacitated facility location. Last, not least, the report presents updates and additions to the other components and extensions of the SCIP Optimization Suite: the LP solver SoPlex, the modeling language Zimpl, the parallelization framework UG, the Steiner tree solver SCIP-Jack, and the mixed-integer semidefinite programming solver SCIP-SDP.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-69361</identifier>
    <submitter>Matthias Miltenberger</submitter>
    <author>Ambros Gleixner</author>
    <author>Michael Bastubbe</author>
    <author>Leon Eifler</author>
    <author>Tristan Gally</author>
    <author>Gerald Gamrath</author>
    <author>Robert Lion Gottwald</author>
    <author>Gregor Hendel</author>
    <author>Christopher Hojny</author>
    <author>Thorsten Koch</author>
    <author>Marco Lübbecke</author>
    <author>Stephen J. Maher</author>
    <author>Matthias Miltenberger</author>
    <author>Benjamin Müller</author>
    <author>Marc Pfetsch</author>
    <author>Christian Puchert</author>
    <author>Daniel Rehfeldt</author>
    <author>Franziska Schlösser</author>
    <author>Christoph Schubert</author>
    <author>Felipe Serrano</author>
    <author>Yuji Shinano</author>
    <author>Jan Merlin Viernickel</author>
    <author>Matthias Walter</author>
    <author>Fabian Wegscheider</author>
    <author>Jonas T. Witt</author>
    <author>Jakob Witzig</author>
    <series>
      <title>ZIB-Report</title>
      <number>18-26</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>constraint integer programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>linear programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>mixed-integer linear programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>mixed-integer nonlinear programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>optimization solver</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>branch-and-cut</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>branch-and-price</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>column generation framework</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>parallelization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>mixed-integer semidefinite programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Steiner tree optimization</value>
    </subject>
    <collection role="msc" number="65-XX">NUMERICAL ANALYSIS</collection>
    <collection role="msc" number="90-XX">OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING</collection>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="institutes" number="mip">Mathematical Optimization Methods</collection>
    <collection role="persons" number="gamrath">Gamrath, Gerald</collection>
    <collection role="persons" number="robert.gottwald">Gottwald, Robert</collection>
    <collection role="persons" number="hendel">Hendel, Gregor</collection>
    <collection role="persons" number="koch">Koch, Thorsten</collection>
    <collection role="persons" number="miltenberger">Miltenberger, Matthias</collection>
    <collection role="persons" number="rehfeldt">Rehfeldt, Daniel</collection>
    <collection role="persons" number="shinano">Shinano, Yuji</collection>
    <collection role="persons" number="vigerske">Vigerske, Stefan</collection>
    <collection role="projects" number="ASTfSCM">ASTfSCM</collection>
    <collection role="projects" number="MIP-ZIBOPT">MIP-ZIBOPT</collection>
    <collection role="projects" number="MODAL-SynLab">MODAL-SynLab</collection>
    <collection role="projects" number="Siemens">Siemens</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
    <collection role="persons" number="schloesser">Schlösser, Franziska</collection>
    <collection role="projects" number="plan4res">Plan4res</collection>
    <collection role="institutes" number="aopt">Applied Optimization</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/6936/scipopt-60.pdf</file>
  </doc>
  <doc>
    <id>8444</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2021-10-21</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A Safe Computational Framework for Integer Programming applied to Chvátal's Conjecture</title>
    <abstract language="eng">We describe a general and safe computational framework that provides integer programming results with the degree of certainty that is required for machine-assisted proofs of mathematical theorems.&#13;
&#13;
At its core, the framework relies on a rational branch-and-bound certificate produced by an exact integer programming solver, SCIP, in order to circumvent floating-point roundoff errors present in most state-of-the-art solvers for mixed-integer programs.&#13;
&#13;
The resulting certificates are self-contained and checker software exists that can verify their correctness independently of the integer programming solver used to produce the certificate.&#13;
&#13;
This acts as a safeguard against programming errors that may be present in complex solver software.&#13;
&#13;
The viability of this approach is tested by applying it to finite cases of Chvátal's conjecture, a long-standing open question in extremal combinatorics.&#13;
&#13;
We take particular care to verify also the correctness of the input for this specific problem, using the Coq formal proof assistant.&#13;
As a result we are able to provide a first machine-assisted proof that&#13;
&#13;
Chvátal's conjecture holds for all downsets whose union of sets contains seven elements or less.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-84444</identifier>
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    <author>Leon Eifler</author>
    <submitter>Leon Eifler</submitter>
    <author>Ambros Gleixner</author>
    <author>Jonad Pulaj</author>
    <series>
      <title>ZIB-Report</title>
      <number>21-35</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>exact rational mixed integer programming</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>extremal combinatorics</value>
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    <title language="eng">A Safe Computational Framework for Integer Programming applied to Chvátal's Conjecture</title>
    <abstract language="eng">We describe a general and safe computational framework that provides integer programming results with the degree of certainty that is required for machine-assisted proofs of mathematical theorems. At its core, the framework relies on a rational branch-and-bound certificate produced by an exact integer programming solver, SCIP, in order to circumvent floating-point roundoff errors present in most state-of-the-art solvers for mixed-integer programs.The resulting certificates are self-contained and checker software exists that can verify their correctness independently of the integer programming solver used to produce the certificate. This acts as a safeguard against programming errors that may be present in complex solver software. The viability of this approach is tested by applying it to finite cases of Chvátal's conjecture, a long-standing open question in extremal combinatorics. We take particular care to verify also the correctness of the input for this specific problem, using the Coq formal proof assistant. As a result we are able to provide a first machine-assisted proof that&#13;
Chvátal's conjecture holds for all downsets whose union of sets contains seven elements or less.</abstract>
    <parentTitle language="eng">ACM Transactions on Mathematical Software</parentTitle>
    <identifier type="doi">10.1145/3485630</identifier>
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    <enrichment key="AcceptedDate">2021-17-09</enrichment>
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    <author>Leon Eifler</author>
    <submitter>Leon Eifler</submitter>
    <author>Ambros Gleixner</author>
    <author>Jonad Pulaj</author>
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    <title language="eng">A computational status update for exact rational mixed integer programming</title>
    <abstract language="eng">The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 10.7x over the original framework and 2.9 times as many instances solved within a time limit of two hours.</abstract>
    <parentTitle language="eng">Mathematical Programming</parentTitle>
    <identifier type="doi">10.1007/s10107-021-01749-5</identifier>
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    <author>Leon Eifler</author>
    <submitter>Leon Eifler</submitter>
    <author>Ambros Gleixner</author>
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    <title language="eng">Combining Precision Boosting with LP Iterative Refinement for Exact Linear Optimization</title>
    <abstract language="eng">This article studies a combination of the two state-of-the-art algorithms for the exact solution of linear programs (LPs) over the rational numbers, i.e., without any roundoff errors or numerical tolerances. By integrating the method of precision boosting inside an LP iterative refinement loop, the combined algorithm is able to leverage the strengths of both methods: the speed of LP iterative refinement, in particular in the majority of cases when a double-precision floating-point solver is able to compute approximate solutions with small errors, and the robustness of precision boosting whenever extended levels of precision become necessary.&#13;
We compare the practical performance of the resulting algorithm with both puremethods on a large set of LPs and mixed-integer programs (MIPs). The results show that the combined algorithm solves more instances than a pure LP iterative refinement approach, while being faster than pure precision boosting. When embedded in an exact branch-and-cut framework for MIPs, the combined algorithm is able to reduce the number of failed calls to the exact LP solver to zero, while maintaining the speed of the pure LP iterative refinement approach.</abstract>
    <parentTitle language="eng">INFORMS Journal on Computing</parentTitle>
    <identifier type="doi">10.1287/ijoc.2023.0409</identifier>
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    <author>Leon Eifler</author>
    <submitter>Leon Eifler</submitter>
    <author>Jules Nicolas-Thouvenin</author>
    <author>Ambros Gleixner</author>
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    <publishedDate>2023-11-14</publishedDate>
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    <title language="eng">Combining Precision Boosting with LP Iterative Refinement for Exact Linear Optimization</title>
    <abstract language="eng">This article studies a combination of the two state-of-the-art algorithms for the exact solution of linear programs (LPs) over the rational numbers, i.e., without any roundoff errors or numerical tolerances. By integrating the method of precision boosting inside an LP iterative refinement loop, the combined algorithm is able to leverage the strengths of both methods: the speed of LP iterative refinement, in particular in the majority of cases when a double-precision floating-point solver is able to compute approximate solutions with small errors, and the robustness of precision boosting whenever extended levels of precision become necessary.&#13;
We compare the practical performance of the resulting algorithm with both puremethods on a large set of LPs and mixed-integer programs (MIPs). The results show that the combined algorithm solves more instances than a pure LP iterative refinement approach, while being faster than pure precision boosting. When embedded in an exact branch-and-cut framework for MIPs, the combined algorithm is able to reduce the number of failed calls to the exact LP solver to zero, while maintaining the speed of the pure LP iterative refinement approach.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-92573</identifier>
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    <author>Leon Eifler</author>
    <submitter>Leon Eifler</submitter>
    <author>Jules Nicolas-Thouvenin</author>
    <author>Ambros Gleixner</author>
    <series>
      <title>ZIB-Report</title>
      <number>23-26</number>
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    <pageFirst>97</pageFirst>
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    <title language="eng">Branch and cut for partitioning a graph into a cycle of clusters</title>
    <abstract language="eng">In this paper we study formulations and algorithms for the cycle clustering problem, a partitioning problem over the vertex set of a directed graph with nonnegative arc weights that is used to identify cyclic behavior in simulation data generated from nonreversible Markov state models. Here, in addition to partitioning the vertices into a set of coherent clusters, the resulting clusters must be ordered into a cycle such as to maximize the total net flow in the forward direction of the cycle. We provide a problem-specific binary programming formulation and compare it to a formulation based on the reformulation-linearization technique (RLT). We present theoretical results on the polytope associated with our custom formulation and develop primal heuristics and separation routines for both formulations. In computational experiments on simulation data from biology we find that branch and cut based on the problem-specific formulation outperforms the one based on RLT.</abstract>
    <parentTitle language="eng">Combinatorial Optimization. ISCO 2024</parentTitle>
    <identifier type="doi">10.1007/978-3-031-60924-4_8</identifier>
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    <enrichment key="Series">Lecture Notes in Computer Science</enrichment>
    <author>Leon Eifler</author>
    <submitter>Christoph Spiegel</submitter>
    <author>Jakob Witzig</author>
    <author>Ambros Gleixner</author>
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    <title language="eng">Safe and Verified Gomory Mixed Integer Cuts in a Rational MIP Framework</title>
    <abstract language="eng">This paper is concerned with the exact solution of mixed-integer programs (MIPs) over the rational numbers, i.e., without any roundoff errors and error tolerances. Here, one computational bottleneck that should be avoided whenever possible is to employ large-scale symbolic computations. Instead it is often possible to use safe directed rounding methods, e.g., to generate provably correct dual bounds. In this work, we continue to leverage this paradigm and extend an exact branch-and-bound framework by separation routines for safe cutting planes, based on the approach first introduced by Cook, Dash, Fukasawa, and Goycoolea in 2009. Constraints are aggregated safely using approximate dual multipliers from an LP solve, followed by mixed-integer rounding to generate provably valid, although slightly weaker inequalities. We generalize this approach to problem data that is not representable in floating-point arithmetic, add routines for controlling the encoding length of the resulting cutting planes, and show how these cutting planes can be verified according to the VIPR certificate standard. Furthermore, we analyze the performance impact of these cutting planes in the context of an exact MIP framework, showing that we can solve 21.5% more instances and reduce solving times by 26.8% on the MIPLIB 2017 benchmark test set.</abstract>
    <parentTitle language="eng">SIAM Journal on Optimization</parentTitle>
    <identifier type="doi">10.1137/23M156046X</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
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    <enrichment key="PreprintUrn">urn:nbn:de:0297-zib-90159</enrichment>
    <author>Leon Eifler</author>
    <submitter>Leon Eifler</submitter>
    <author>Ambros Gleixner</author>
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  <doc>
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    <publishedDate>2023-03-22</publishedDate>
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    <title language="eng">Safe and Verified Gomory Mixed Integer Cuts in a Rational MIP Framework</title>
    <abstract language="eng">This paper is concerned with the exact solution of mixed-integer programs (MIPs) over the rational numbers, i.e., without any roundoff errors and error tolerances. Here, one computational bottleneck that should be avoided whenever possible is to employ large-scale symbolic computations. Instead it is often possible to use safe directed rounding methods, e.g., to generate provably correct dual bounds. In this work, we continue to leverage this paradigm and extend an exact branch-and-bound framework by separation routines for safe cutting planes, based on the approach first introduced by Cook, Dash, Fukasawa, and Goycoolea in 2009. Constraints are aggregated safely using approximate dual multipliers from an LP solve, followed by mixed-integer rounding to generate provably valid, although slightly weaker inequalities. We generalize this approach to problem data that is not representable in floating-point arithmetic, add routines for controlling the encoding length of the resulting cutting planes, and show how these cutting planes can be verified according to the VIPR certificate standard. Furthermore, we analyze the performance impact of these cutting planes in the context of an exact MIP framework, showing that we can solve 21.5% more instances and reduce solving times by 26.8% on the MIPLIB 2017 benchmark test set.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-90159</identifier>
    <enrichment key="opus.source">publish</enrichment>
    <author>Leon Eifler</author>
    <submitter>Leon Eifler</submitter>
    <author>Ambros Gleixner</author>
    <series>
      <title>ZIB-Report</title>
      <number>23-09</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Mixed integer programming, Exact computation, Rational arithmetic, Cutting Planes, Symbolic Computations, Certificate of correctness</value>
    </subject>
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    <title language="eng">Safe and verified Gomory mixed integer cuts in a rational MIP framework</title>
    <abstract language="eng">This paper is concerned with the exact solution of mixed-integer programs (MIPs) over the rational numbers, i.e., without any roundoff errors and error tolerances. Here, one computational bottleneck that should be avoided whenever possible is to employ large-scale symbolic computations. Instead it is often possible to use safe directed rounding methods, e.g., to generate provably correct dual bounds. In this work, we continue to leverage this paradigm and extend an exact branch-and-bound framework by separation routines for safe cutting planes, based on the approach first introduced by Cook, Dash, Fukasawa, and Goycoolea in 2009 [INFORMS J. Comput., 21 (2009), pp. 641–649]. Constraints are aggregated safely using approximate dual multipliers from an LP solve, followed by mixed-integer rounding to generate provably valid, although slightly weaker inequalities. We generalize this approach to problem data that is not representable in floating-point arithmetic, add routines for controlling the encoding length of the resulting cutting planes, and show how these cutting planes can be verified according to the VIPR certificate standard. Furthermore, we analyze the performance impact of these cutting planes in the context of an exact MIP framework, showing that we can solve 21.5% more instances to exact optimality and reduce solving times by 26.8% on the MIPLIB 2017 benchmark test set.</abstract>
    <parentTitle language="eng">SIAM Journal on Optimization</parentTitle>
    <identifier type="doi">10.1137/23M156046X</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <author>Leon Eifler</author>
    <submitter>Christoph Spiegel</submitter>
    <author>Ambros Gleixner</author>
    <collection role="projects" number="MODAL-SynLab">MODAL-SynLab</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
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  </doc>
  <doc>
    <id>9407</id>
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    <publishedYear>2023</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>793</pageFirst>
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    <edition/>
    <issue/>
    <volume>197</volume>
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    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A computational status update for exact rational mixed integer programming</title>
    <abstract language="eng">The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 10.7x over the original framework and 2.9 times as many instances solved within a time limit of two hours.</abstract>
    <parentTitle language="eng">Mathematical Programming</parentTitle>
    <identifier type="doi">10.1007/s10107-021-01749-5</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <author>Leon Eifler</author>
    <submitter>Christoph Spiegel</submitter>
    <author>Ambros Gleixner</author>
    <collection role="projects" number="MODAL-SynLab">MODAL-SynLab</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
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  </doc>
  <doc>
    <id>6035</id>
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    <edition/>
    <issue/>
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    <completedDate>--</completedDate>
    <publishedDate>2016-05-09</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Mixed-Integer Programming for Cycle Detection in Non-reversible Markov Processes</title>
    <abstract language="eng">In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-60353</identifier>
    <identifier type="doi">10.1137/16M1091162</identifier>
    <author>Jakob Witzig</author>
    <submitter>Jakob Witzig</submitter>
    <author>Isabel Beckenbach</author>
    <author>Leon Eifler</author>
    <author>Konstantin Fackeldey</author>
    <author>Ambros Gleixner</author>
    <author>Andreas Grever</author>
    <author>Marcus Weber</author>
    <series>
      <title>ZIB-Report</title>
      <number>16-39</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Non-reversible Markov Processes</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>NESS</value>
    </subject>
    <subject>
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      <language>eng</language>
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      <value>Markov State Models</value>
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    <title language="eng">Mixed-Integer Programming for Cycle Detection in Non-reversible Markov Processes</title>
    <abstract language="eng">In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.</abstract>
    <parentTitle language="eng">Multiscale Modeling and Simulation</parentTitle>
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    <author>Jakob Witzig</author>
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    <author>Leon Eifler</author>
    <author>Konstantin Fackeldey</author>
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    <author>Andreas Grever</author>
    <author>Marcus Weber</author>
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      <value>NESS</value>
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      <value>Non-reversible Markov Processes</value>
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    <title language="eng">The SCIP Optimization Suite 8.0</title>
    <abstract language="eng">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.</abstract>
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    <author>Ksenia Bestuzheva</author>
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    <author>Antonia Chmiela</author>
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    <author>Alexander Hoen</author>
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    <author>Benjamin Müller</author>
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    <author>Mark Turner</author>
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    <author>Jakob Witzig</author>
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      <value>Mixed-integer semidefinite programming</value>
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    <title language="eng">The SCIP Optimization Suite 9.0</title>
    <abstract language="eng">The SCIP Optimization Suite provides a collection of software packages for mathematical optimization, centered around the constraint integer programming framework SCIP. This report discusses the enhancements and extensions included in the SCIP Optimization Suite 9.0. The updates in SCIP 9.0 include improved symmetry handling, additions and improvements of nonlinear handlers and primal heuristics, a new cut generator and two new cut selection schemes, a new branching rule, a new LP interface, and several bug fixes. The SCIP Optimization Suite 9.0 also features new Rust and C++ interfaces for SCIP, new Python interface for SoPlex, along with enhancements to existing interfaces. The SCIP Optimization Suite 9.0 also includes new and improved features in the LP solver SoPlex, the presolving library PaPILO, the parallel framework UG, the decomposition framework GCG, and the SCIP extension SCIP-SDP. These additions and enhancements have resulted in an overall performance improvement of SCIP in terms of solving time, number of nodes in the branch-and-bound tree, as well as the reliability of the solver.</abstract>
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    <title language="eng">Enabling research through the SCIP optimization suite 8.0</title>
    <abstract language="eng">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.</abstract>
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