---
_id: '8750'
abstract:
- lang: eng
  text: In this article we propose a descent method for equality and inequality constrained
    multiobjective optimization problems (MOPs) which generalizes the steepest descent
    method for unconstrained MOPs by Fliege and Svaiter to constrained problems by
    using two active set strategies. Under some regularity assumptions on the problem,
    we show that accumulation points of our descent method satisfy a necessary condition
    for local Pareto optimality. Finally, we show the typical behavior of our method
    in a numerical example.
author:
- first_name: Bennet
  full_name: Gebken, Bennet
  id: '32643'
  last_name: Gebken
- first_name: Sebastian
  full_name: Peitz, Sebastian
  id: '47427'
  last_name: Peitz
  orcid: https://orcid.org/0000-0002-3389-793X
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
citation:
  ama: 'Gebken B, Peitz S, Dellnitz M. A Descent Method for Equality and Inequality
    Constrained Multiobjective Optimization Problems. In: <i>Numerical and Evolutionary
    Optimization – NEO 2017</i>. Cham; 2018. doi:<a href="https://doi.org/10.1007/978-3-319-96104-0_2">10.1007/978-3-319-96104-0_2</a>'
  apa: Gebken, B., Peitz, S., &#38; Dellnitz, M. (2018). A Descent Method for Equality
    and Inequality Constrained Multiobjective Optimization Problems. In <i>Numerical
    and Evolutionary Optimization – NEO 2017</i>. Cham. <a href="https://doi.org/10.1007/978-3-319-96104-0_2">https://doi.org/10.1007/978-3-319-96104-0_2</a>
  bibtex: '@inproceedings{Gebken_Peitz_Dellnitz_2018, place={Cham}, title={A Descent
    Method for Equality and Inequality Constrained Multiobjective Optimization Problems},
    DOI={<a href="https://doi.org/10.1007/978-3-319-96104-0_2">10.1007/978-3-319-96104-0_2</a>},
    booktitle={Numerical and Evolutionary Optimization – NEO 2017}, author={Gebken,
    Bennet and Peitz, Sebastian and Dellnitz, Michael}, year={2018} }'
  chicago: Gebken, Bennet, Sebastian Peitz, and Michael Dellnitz. “A Descent Method
    for Equality and Inequality Constrained Multiobjective Optimization Problems.”
    In <i>Numerical and Evolutionary Optimization – NEO 2017</i>. Cham, 2018. <a href="https://doi.org/10.1007/978-3-319-96104-0_2">https://doi.org/10.1007/978-3-319-96104-0_2</a>.
  ieee: B. Gebken, S. Peitz, and M. Dellnitz, “A Descent Method for Equality and Inequality
    Constrained Multiobjective Optimization Problems,” in <i>Numerical and Evolutionary
    Optimization – NEO 2017</i>, 2018.
  mla: Gebken, Bennet, et al. “A Descent Method for Equality and Inequality Constrained
    Multiobjective Optimization Problems.” <i>Numerical and Evolutionary Optimization
    – NEO 2017</i>, 2018, doi:<a href="https://doi.org/10.1007/978-3-319-96104-0_2">10.1007/978-3-319-96104-0_2</a>.
  short: 'B. Gebken, S. Peitz, M. Dellnitz, in: Numerical and Evolutionary Optimization
    – NEO 2017, Cham, 2018.'
conference:
  name: 'NEO 2017: Numerical and Evolutionary Optimization'
date_created: 2019-03-29T13:26:47Z
date_updated: 2022-01-06T07:04:00Z
department:
- _id: '101'
doi: 10.1007/978-3-319-96104-0_2
language:
- iso: eng
place: Cham
publication: Numerical and Evolutionary Optimization – NEO 2017
publication_identifier:
  isbn:
  - '9783319961033'
  - '9783319961040'
  issn:
  - 1860-949X
  - 1860-9503
publication_status: published
status: public
title: A Descent Method for Equality and Inequality Constrained Multiobjective Optimization
  Problems
type: conference
user_id: '47427'
year: '2018'
...
---
_id: '8752'
abstract:
- lang: eng
  text: In this article we develop a gradient-based algorithm for the solution of
    multiobjective optimization problems with uncertainties. To this end, an additional
    condition is derived for the descent direction in order to account for inaccuracies
    in the gradients and then incorporated into a subdivision algorithm for the computation
    of global solutions to multiobjective optimization problems. Convergence to a
    superset of the Pareto set is proved and an upper bound for the maximal distance
    to the set of substationary points is given. Besides the applicability to problems
    with uncertainties, the algorithm is developed with the intention to use it in
    combination with model order reduction techniques in order to efficiently solve
    PDE-constrained multiobjective optimization problems.
author:
- first_name: Sebastian
  full_name: Peitz, Sebastian
  id: '47427'
  last_name: Peitz
  orcid: https://orcid.org/0000-0002-3389-793X
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
citation:
  ama: 'Peitz S, Dellnitz M. Gradient-Based Multiobjective Optimization with Uncertainties.
    In: <i>NEO 2016</i>. Cham; 2017:159-182. doi:<a href="https://doi.org/10.1007/978-3-319-64063-1_7">10.1007/978-3-319-64063-1_7</a>'
  apa: Peitz, S., &#38; Dellnitz, M. (2017). Gradient-Based Multiobjective Optimization
    with Uncertainties. In <i>NEO 2016</i> (pp. 159–182). Cham. <a href="https://doi.org/10.1007/978-3-319-64063-1_7">https://doi.org/10.1007/978-3-319-64063-1_7</a>
  bibtex: '@inproceedings{Peitz_Dellnitz_2017, place={Cham}, title={Gradient-Based
    Multiobjective Optimization with Uncertainties}, DOI={<a href="https://doi.org/10.1007/978-3-319-64063-1_7">10.1007/978-3-319-64063-1_7</a>},
    booktitle={NEO 2016}, author={Peitz, Sebastian and Dellnitz, Michael}, year={2017},
    pages={159–182} }'
  chicago: Peitz, Sebastian, and Michael Dellnitz. “Gradient-Based Multiobjective
    Optimization with Uncertainties.” In <i>NEO 2016</i>, 159–82. Cham, 2017. <a href="https://doi.org/10.1007/978-3-319-64063-1_7">https://doi.org/10.1007/978-3-319-64063-1_7</a>.
  ieee: S. Peitz and M. Dellnitz, “Gradient-Based Multiobjective Optimization with
    Uncertainties,” in <i>NEO 2016</i>, 2017, pp. 159–182.
  mla: Peitz, Sebastian, and Michael Dellnitz. “Gradient-Based Multiobjective Optimization
    with Uncertainties.” <i>NEO 2016</i>, 2017, pp. 159–82, doi:<a href="https://doi.org/10.1007/978-3-319-64063-1_7">10.1007/978-3-319-64063-1_7</a>.
  short: 'S. Peitz, M. Dellnitz, in: NEO 2016, Cham, 2017, pp. 159–182.'
date_created: 2019-03-29T13:28:56Z
date_updated: 2022-01-06T07:04:00Z
department:
- _id: '101'
doi: 10.1007/978-3-319-64063-1_7
language:
- iso: eng
page: 159-182
place: Cham
project:
- _id: '52'
  name: Computing Resources Provided by the Paderborn Center for Parallel Computing
publication: NEO 2016
publication_identifier:
  isbn:
  - '9783319640624'
  - '9783319640631'
  issn:
  - 1860-949X
  - 1860-9503
publication_status: published
status: public
title: Gradient-Based Multiobjective Optimization with Uncertainties
type: conference
user_id: '47427'
year: '2017'
...
---
_id: '16670'
author:
- first_name: Oliver
  full_name: Schütze, Oliver
  last_name: Schütze
- first_name: Katrin
  full_name: Witting, Katrin
  last_name: Witting
- first_name: Sina
  full_name: Ober-Blöbaum, Sina
  last_name: Ober-Blöbaum
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
citation:
  ama: 'Schütze O, Witting K, Ober-Blöbaum S, Dellnitz M. Set Oriented Methods for
    the Numerical Treatment of Multiobjective Optimization Problems. In: <i>EVOLVE-
    A Bridge between Probability, Set Oriented Numerics and Evolutionary Computation</i>.
    Berlin, Heidelberg; 2013. doi:<a href="https://doi.org/10.1007/978-3-642-32726-1_5">10.1007/978-3-642-32726-1_5</a>'
  apa: Schütze, O., Witting, K., Ober-Blöbaum, S., &#38; Dellnitz, M. (2013). Set
    Oriented Methods for the Numerical Treatment of Multiobjective Optimization Problems.
    In <i>EVOLVE- A Bridge between Probability, Set Oriented Numerics and Evolutionary
    Computation</i>. Berlin, Heidelberg. <a href="https://doi.org/10.1007/978-3-642-32726-1_5">https://doi.org/10.1007/978-3-642-32726-1_5</a>
  bibtex: '@inbook{Schütze_Witting_Ober-Blöbaum_Dellnitz_2013, place={Berlin, Heidelberg},
    title={Set Oriented Methods for the Numerical Treatment of Multiobjective Optimization
    Problems}, DOI={<a href="https://doi.org/10.1007/978-3-642-32726-1_5">10.1007/978-3-642-32726-1_5</a>},
    booktitle={EVOLVE- A Bridge between Probability, Set Oriented Numerics and Evolutionary
    Computation}, author={Schütze, Oliver and Witting, Katrin and Ober-Blöbaum, Sina
    and Dellnitz, Michael}, year={2013} }'
  chicago: Schütze, Oliver, Katrin Witting, Sina Ober-Blöbaum, and Michael Dellnitz.
    “Set Oriented Methods for the Numerical Treatment of Multiobjective Optimization
    Problems.” In <i>EVOLVE- A Bridge between Probability, Set Oriented Numerics and
    Evolutionary Computation</i>. Berlin, Heidelberg, 2013. <a href="https://doi.org/10.1007/978-3-642-32726-1_5">https://doi.org/10.1007/978-3-642-32726-1_5</a>.
  ieee: O. Schütze, K. Witting, S. Ober-Blöbaum, and M. Dellnitz, “Set Oriented Methods
    for the Numerical Treatment of Multiobjective Optimization Problems,” in <i>EVOLVE-
    A Bridge between Probability, Set Oriented Numerics and Evolutionary Computation</i>,
    Berlin, Heidelberg, 2013.
  mla: Schütze, Oliver, et al. “Set Oriented Methods for the Numerical Treatment of
    Multiobjective Optimization Problems.” <i>EVOLVE- A Bridge between Probability,
    Set Oriented Numerics and Evolutionary Computation</i>, 2013, doi:<a href="https://doi.org/10.1007/978-3-642-32726-1_5">10.1007/978-3-642-32726-1_5</a>.
  short: 'O. Schütze, K. Witting, S. Ober-Blöbaum, M. Dellnitz, in: EVOLVE- A Bridge
    between Probability, Set Oriented Numerics and Evolutionary Computation, Berlin,
    Heidelberg, 2013.'
date_created: 2020-04-16T10:05:59Z
date_updated: 2022-01-06T06:52:54Z
department:
- _id: '101'
doi: 10.1007/978-3-642-32726-1_5
language:
- iso: eng
place: Berlin, Heidelberg
publication: EVOLVE- A Bridge between Probability, Set Oriented Numerics and Evolutionary
  Computation
publication_identifier:
  isbn:
  - '9783642327254'
  - '9783642327261'
  issn:
  - 1860-949X
  - 1860-9503
publication_status: published
status: public
title: Set Oriented Methods for the Numerical Treatment of Multiobjective Optimization
  Problems
type: book_chapter
user_id: '15701'
year: '2013'
...