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<export-example>
  <doc>
    <id>385</id>
    <completedYear>2021</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2021-05-13</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints</title>
    <abstract language="eng">We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.</abstract>
    <identifier type="doi">10.1007/s00186-021-00764-8</identifier>
    <enrichment key="SubmissionStatus">Appeared online</enrichment>
    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Holger Berthold</author>
    <author>Holger Heitsch</author>
    <author>René Henrion</author>
    <author>Jan Schwientek</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>probabilistic constraints</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>probust constraints</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>chance constraints</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>bilevel optimization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>semi-infinite optimization</value>
    </subject>
    <collection role="institutes" number="">Weierstraß-Institut für Angewandte Analysis und Stochastik</collection>
    <collection role="subprojects" number="">B04</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/385/wias_preprints_2835.pdf</file>
  </doc>
  <doc>
    <id>386</id>
    <completedYear>2021</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2021-05-13</completedDate>
    <publishedDate>2021-05-13</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Value at risk approach to producer's best response in electricity market with uncertain demand</title>
    <abstract language="eng">We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity demand. We find an explicit solution of the ISO problem, and use it to tackle the problem of a producer. In our model, production as well as income of a producer are determined based on the estimated electricity demand predicted by the ISO, that is unknown to producers. Thus, each producer is hedging against the uncertainty of prediction of the demand using the value-at-risk approach. To illustrate our results, a numerical study of a producer's best response given a historical distribution of both estimated and real electricity demand is provided.</abstract>
    <enrichment key="SubmissionStatus">under review</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <enrichment key="review.accepted_by">2</enrichment>
    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Martin Branda</author>
    <author>René Henrion</author>
    <author>Miroslav Pištěk</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>electricity market</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>multi-leader-common-follower game</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>stochastic demand</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>day-ahead bidding</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>chance constraints</value>
    </subject>
    <collection role="institutes" number="">Weierstraß-Institut für Angewandte Analysis und Stochastik</collection>
    <collection role="subprojects" number="">B04</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/386/wias_preprints_2831.pdf</file>
  </doc>
  <doc>
    <id>336</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2020-10-06</completedDate>
    <publishedDate>2020-10-06</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Producer’s Best Response in Pay-as-clear Day-ahead Electricity Market with Uncertain Demand</title>
    <abstract language="eng">We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity&#13;
demand. We find an explicit solution of the ISO problem, and use it to tackle the&#13;
problem of a producer. In our model, production as well as income of a producer&#13;
are determined based on the estimated electricity demand predicted by the ISO, that&#13;
is unknown to producers. Thus, each producer is hedging against the uncertainty of&#13;
prediction of the demand using the value-at-risk approach. To illustrate our results, a&#13;
numerical study of a producer’s best response given a historical distribution of both&#13;
estimated and real electricity demand is provided.</abstract>
    <enrichment key="SubmissionStatus">under review</enrichment>
    <enrichment key="review.accepted_by">2</enrichment>
    <licence>Creative Commons - CC BY-NC-ND - Namensnennung - Nicht kommerziell - Keine Bearbeitungen 4.0 International</licence>
    <author>Martin Branda</author>
    <author>René Henrion</author>
    <author>Miroslav Pistek</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>electricity market</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>multi-leader-common-follower game</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>stochastic demand</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>day-ahead bidding</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>chance constraints</value>
    </subject>
    <collection role="institutes" number="">Weierstraß-Institut für Angewandte Analysis und Stochastik</collection>
    <collection role="subprojects" number="">B04</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/336/BHP_2020.pdf</file>
  </doc>
  <doc>
    <id>215</id>
    <completedYear>2017</completedYear>
    <publishedYear>2017</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2017-11-20</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Joint model of probabilistic/robust (probust) constraints applied to gas network optimization</title>
    <abstract language="eng">Optimization tasks under uncertain conditions abound in many&#13;
real-life applications. Whereas solution approaches for probabilistic constraints&#13;
are often developed in case the uncertainties can be assumed to follow a&#13;
certain probability distribution, robust approaches are usually used in case&#13;
solutions are sought that are feasible for all realizations of uncertainties within&#13;
some pre-defined uncertainty set. As many applications contain different types&#13;
of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints as its appears in&#13;
optimization problems under uncertainty. Focusing on complex uncertain gas&#13;
network optimization problems, we show the relevance of this class of problems&#13;
for the task of maximizing free booked capacities in an algebraic model for a&#13;
stationary gas network. We furthermore present approaches for their solution.&#13;
Finally, we study the problem of controlling a transient system that is governed&#13;
by the wave equation. The task consists in determining controls such that a&#13;
certain robustness measure remains below some given upper bound, with high&#13;
probability.</abstract>
    <identifier type="doi">10.1007/s10013-020-00434-y</identifier>
    <enrichment key="SubmissionStatus">in press</enrichment>
    <author>Dennis Adelhütte</author>
    <author>Denis Aßmann</author>
    <author>Tatiana Gonzàlez Grandòn</author>
    <author>Martin Gugat</author>
    <author>Holger Heitsch</author>
    <author>Frauke Liers</author>
    <author>René Henrion</author>
    <author>Sabrina Nitsche</author>
    <author>Rüdiger Schultz</author>
    <author>Michael Stingl</author>
    <author>David Wintergerst</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>robust optimization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>chance constraints</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>optimal control</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>spheric-radial decomposition</value>
    </subject>
    <collection role="institutes" number="">Friedrich-Alexander-Universität Erlangen-Nürnberg</collection>
    <collection role="institutes" number="">Humboldt-Universität zu Berlin</collection>
    <collection role="institutes" number="">Weierstraß-Institut für Angewandte Analysis und Stochastik</collection>
    <collection role="institutes" number="">Universität Duisburg-Essen</collection>
    <collection role="subprojects" number="">B04</collection>
    <collection role="subprojects" number="">B05</collection>
    <collection role="subprojects" number="">B06</collection>
    <collection role="subprojects" number="">C03</collection>
    <collection role="subprojects" number="">Z01</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/215/uncertainty_paper.pdf</file>
  </doc>
  <doc>
    <id>617</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2026-02-04</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Continuous stochastic gradient and spherical radial decomposition</title>
    <abstract language="eng">In this paper, a new method is presented for solving chance-constrained optimization problems. The method combines the well-established Spherical-Radial Decomposition approach with the Continuous Stochastic Gradient method. While the Continuous Stochastic Gradient method has been successfully applied to chance-constrained problems in the past, only the combination with the Spherical-Radial Decomposition allows to avoid smoothing of the integrand. In this chapter, we prove this fact for a relevant class of chance-constrained problems and apply the resulting method to the capacity maximization problem for gas networks.</abstract>
    <enrichment key="opus.source">publish</enrichment>
    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Daniela Bernhard</author>
    <author>Holger Heitsch</author>
    <author>René Henrion</author>
    <author>Frauke Liers</author>
    <author>Michael Stingl</author>
    <author>Andrian Uihlein</author>
    <author>Viktor Zipf</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>chance constraints</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>continuous stochastic gradient</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>spheric-radial decomposition</value>
    </subject>
    <collection role="institutes" number="">Friedrich-Alexander-Universität Erlangen-Nürnberg</collection>
    <collection role="institutes" number="">Weierstraß-Institut für Angewandte Analysis und Stochastik</collection>
    <collection role="subprojects" number="">B04</collection>
    <collection role="subprojects" number="">B06</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/617/Preprint_3245.pdf</file>
  </doc>
</export-example>
