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  <doc>
    <id>6782</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
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    <contributingCorporation/>
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    <completedDate>--</completedDate>
    <publishedDate>2018-03-08</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Strong Convex Nonlinear Relaxations of the Pooling Problem</title>
    <abstract language="eng">We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which input materials are mixed in intermediate pools, with the outputs of these pools further mixed to make output products meeting given attribute percentage requirements. Our relaxations are derived by considering a set which arises from the formulation by considering a single product, a single attibute, and a single pool. The convex hull of the resulting nonconvex set is not polyhedral. We derive valid linear and convex nonlinear inequalities for the convex hull, and demonstrate that different subsets of these inequalities define the convex hull of the nonconvex set in three cases determined by the parameters of the set. Computational results on literature instances and newly created larger test instances demonstrate that the inequalities can significantly strengthen the convex relaxation of the pq-formulation of the pooling problem, which is the relaxation known to have the strongest bound.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-67824</identifier>
    <author>James Luedtke</author>
    <submitter>Jonas Schweiger</submitter>
    <author>Claudia D'Ambrosio</author>
    <author>Jeff Linderoth</author>
    <author>Jonas Schweiger</author>
    <series>
      <title>ZIB-Report</title>
      <number>18-12</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Pooling Problem</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Relaxation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Valid Inequalities</value>
    </subject>
    <collection role="msc" number="90C20">Quadratic programming</collection>
    <collection role="msc" number="90C26">Nonconvex programming, global optimization</collection>
    <collection role="msc" number="90C35">Programming involving graphs or networks [See also 90C27]</collection>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="projects" number="MODAL-GasLab">MODAL-GasLab</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
    <collection role="persons" number="schweiger">Schweiger, Jonas</collection>
    <collection role="institutes" number="enernet">Energy Network Optimization</collection>
    <collection role="institutes" number="aopt">Applied Optimization</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/6782/ZR-18-12.pdf</file>
  </doc>
  <doc>
    <id>6780</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>2018-03-08</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Strong Convex Nonlinear Relaxations of the Pooling Problem: Extreme Points</title>
    <abstract language="eng">We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which products are mixed in intermediate pools in order to meet quality targets at their destinations. In this technical report, we characterize the extreme points of the convex hull of our non-convex set, and show that they are not finite, i.e., the convex hull is not polyhedral. This analysis was used to derive valid nonlinear convex inequalities and show that, for a specific case, they characterize the convex hull of our set. The new valid inequalities and computational results are presented in ZIB Report 18-12.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-67801</identifier>
    <author>James Luedtke</author>
    <submitter>Jonas Schweiger</submitter>
    <author>Claudia D'Ambrosio</author>
    <author>Jeff Linderoth</author>
    <author>Jonas Schweiger</author>
    <series>
      <title>ZIB-Report</title>
      <number>18-13</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Pooling Problem</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Relaxation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Extreme Points</value>
    </subject>
    <collection role="msc" number="90C20">Quadratic programming</collection>
    <collection role="msc" number="90C26">Nonconvex programming, global optimization</collection>
    <collection role="msc" number="90C35">Programming involving graphs or networks [See also 90C27]</collection>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="projects" number="MODAL-GasLab">MODAL-GasLab</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
    <collection role="persons" number="schweiger">Schweiger, Jonas</collection>
    <collection role="institutes" number="enernet">Energy Network Optimization</collection>
    <collection role="institutes" number="aopt">Applied Optimization</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/6780/ZR-18-13.pdf</file>
  </doc>
</export-example>
