---
_id: '10595'
abstract:
- lang: eng
  text: In this article we show that the boundary of the Pareto critical set of an
    unconstrained multiobjective optimization problem (MOP) consists of Pareto critical
    points of subproblems where only a subset of the set of objective functions is
    taken into account. If the Pareto critical set is completely described by its
    boundary (e.g., if we have more objective functions than dimensions in decision
    space), then this can be used to efficiently solve the MOP by solving a number
    of MOPs with fewer objective functions. If this is not the case, the results can
    still give insight into the structure of the Pareto critical set.
article_type: original
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. On the hierarchical structure of Pareto critical
    sets. <i>Journal of Global Optimization</i>. 2019;73(4):891-913. doi:<a href="https://doi.org/10.1007/s10898-019-00737-6">10.1007/s10898-019-00737-6</a>
  apa: Gebken, B., Peitz, S., &#38; Dellnitz, M. (2019). On the hierarchical structure
    of Pareto critical sets. <i>Journal of Global Optimization</i>, <i>73</i>(4),
    891–913. <a href="https://doi.org/10.1007/s10898-019-00737-6">https://doi.org/10.1007/s10898-019-00737-6</a>
  bibtex: '@article{Gebken_Peitz_Dellnitz_2019, title={On the hierarchical structure
    of Pareto critical sets}, volume={73}, DOI={<a href="https://doi.org/10.1007/s10898-019-00737-6">10.1007/s10898-019-00737-6</a>},
    number={4}, journal={Journal of Global Optimization}, author={Gebken, Bennet and
    Peitz, Sebastian and Dellnitz, Michael}, year={2019}, pages={891–913} }'
  chicago: 'Gebken, Bennet, Sebastian Peitz, and Michael Dellnitz. “On the Hierarchical
    Structure of Pareto Critical Sets.” <i>Journal of Global Optimization</i> 73,
    no. 4 (2019): 891–913. <a href="https://doi.org/10.1007/s10898-019-00737-6">https://doi.org/10.1007/s10898-019-00737-6</a>.'
  ieee: B. Gebken, S. Peitz, and M. Dellnitz, “On the hierarchical structure of Pareto
    critical sets,” <i>Journal of Global Optimization</i>, vol. 73, no. 4, pp. 891–913,
    2019.
  mla: Gebken, Bennet, et al. “On the Hierarchical Structure of Pareto Critical Sets.”
    <i>Journal of Global Optimization</i>, vol. 73, no. 4, 2019, pp. 891–913, doi:<a
    href="https://doi.org/10.1007/s10898-019-00737-6">10.1007/s10898-019-00737-6</a>.
  short: B. Gebken, S. Peitz, M. Dellnitz, Journal of Global Optimization 73 (2019)
    891–913.
date_created: 2019-07-10T08:13:31Z
date_updated: 2022-01-06T06:50:46Z
department:
- _id: '101'
doi: 10.1007/s10898-019-00737-6
intvolume: '        73'
issue: '4'
language:
- iso: eng
page: 891-913
publication: Journal of Global Optimization
publication_identifier:
  issn:
  - 0925-5001
  - 1573-2916
publication_status: published
status: public
title: On the hierarchical structure of Pareto critical sets
type: journal_article
user_id: '47427'
volume: 73
year: '2019'
...
---
_id: '16677'
author:
- 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: Witting K, Ober-Blöbaum S, Dellnitz M. A variational approach to define robustness
    for parametric multiobjective optimization problems. <i>Journal of Global Optimization</i>.
    2013:331-345. doi:<a href="https://doi.org/10.1007/s10898-012-9972-6">10.1007/s10898-012-9972-6</a>
  apa: Witting, K., Ober-Blöbaum, S., &#38; Dellnitz, M. (2013). A variational approach
    to define robustness for parametric multiobjective optimization problems. <i>Journal
    of Global Optimization</i>, 331–345. <a href="https://doi.org/10.1007/s10898-012-9972-6">https://doi.org/10.1007/s10898-012-9972-6</a>
  bibtex: '@article{Witting_Ober-Blöbaum_Dellnitz_2013, title={A variational approach
    to define robustness for parametric multiobjective optimization problems}, DOI={<a
    href="https://doi.org/10.1007/s10898-012-9972-6">10.1007/s10898-012-9972-6</a>},
    journal={Journal of Global Optimization}, author={Witting, Katrin and Ober-Blöbaum,
    Sina and Dellnitz, Michael}, year={2013}, pages={331–345} }'
  chicago: Witting, Katrin, Sina Ober-Blöbaum, and Michael Dellnitz. “A Variational
    Approach to Define Robustness for Parametric Multiobjective Optimization Problems.”
    <i>Journal of Global Optimization</i>, 2013, 331–45. <a href="https://doi.org/10.1007/s10898-012-9972-6">https://doi.org/10.1007/s10898-012-9972-6</a>.
  ieee: K. Witting, S. Ober-Blöbaum, and M. Dellnitz, “A variational approach to define
    robustness for parametric multiobjective optimization problems,” <i>Journal of
    Global Optimization</i>, pp. 331–345, 2013.
  mla: Witting, Katrin, et al. “A Variational Approach to Define Robustness for Parametric
    Multiobjective Optimization Problems.” <i>Journal of Global Optimization</i>,
    2013, pp. 331–45, doi:<a href="https://doi.org/10.1007/s10898-012-9972-6">10.1007/s10898-012-9972-6</a>.
  short: K. Witting, S. Ober-Blöbaum, M. Dellnitz, Journal of Global Optimization
    (2013) 331–345.
date_created: 2020-04-16T10:14:04Z
date_updated: 2022-01-06T06:52:54Z
department:
- _id: '101'
doi: 10.1007/s10898-012-9972-6
language:
- iso: eng
page: 331-345
publication: Journal of Global Optimization
publication_identifier:
  issn:
  - 0925-5001
  - 1573-2916
publication_status: published
status: public
title: A variational approach to define robustness for parametric multiobjective optimization
  problems
type: journal_article
user_id: '15701'
year: '2013'
...
---
_id: '16668'
author:
- first_name: Oliver
  full_name: Schütze, Oliver
  last_name: Schütze
- first_name: Marco
  full_name: Laumanns, Marco
  last_name: Laumanns
- first_name: Carlos A.
  full_name: Coello Coello, Carlos A.
  last_name: Coello Coello
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
- first_name: El-Ghazali
  full_name: Talbi, El-Ghazali
  last_name: Talbi
citation:
  ama: Schütze O, Laumanns M, Coello Coello CA, Dellnitz M, Talbi E-G. Convergence
    of stochastic search algorithms to finite size pareto set approximations. <i>Journal
    of Global Optimization</i>. 2008:559-577. doi:<a href="https://doi.org/10.1007/s10898-007-9265-7">10.1007/s10898-007-9265-7</a>
  apa: Schütze, O., Laumanns, M., Coello Coello, C. A., Dellnitz, M., &#38; Talbi,
    E.-G. (2008). Convergence of stochastic search algorithms to finite size pareto
    set approximations. <i>Journal of Global Optimization</i>, 559–577. <a href="https://doi.org/10.1007/s10898-007-9265-7">https://doi.org/10.1007/s10898-007-9265-7</a>
  bibtex: '@article{Schütze_Laumanns_Coello Coello_Dellnitz_Talbi_2008, title={Convergence
    of stochastic search algorithms to finite size pareto set approximations}, DOI={<a
    href="https://doi.org/10.1007/s10898-007-9265-7">10.1007/s10898-007-9265-7</a>},
    journal={Journal of Global Optimization}, author={Schütze, Oliver and Laumanns,
    Marco and Coello Coello, Carlos A. and Dellnitz, Michael and Talbi, El-Ghazali},
    year={2008}, pages={559–577} }'
  chicago: Schütze, Oliver, Marco Laumanns, Carlos A. Coello Coello, Michael Dellnitz,
    and El-Ghazali Talbi. “Convergence of Stochastic Search Algorithms to Finite Size
    Pareto Set Approximations.” <i>Journal of Global Optimization</i>, 2008, 559–77.
    <a href="https://doi.org/10.1007/s10898-007-9265-7">https://doi.org/10.1007/s10898-007-9265-7</a>.
  ieee: O. Schütze, M. Laumanns, C. A. Coello Coello, M. Dellnitz, and E.-G. Talbi,
    “Convergence of stochastic search algorithms to finite size pareto set approximations,”
    <i>Journal of Global Optimization</i>, pp. 559–577, 2008.
  mla: Schütze, Oliver, et al. “Convergence of Stochastic Search Algorithms to Finite
    Size Pareto Set Approximations.” <i>Journal of Global Optimization</i>, 2008,
    pp. 559–77, doi:<a href="https://doi.org/10.1007/s10898-007-9265-7">10.1007/s10898-007-9265-7</a>.
  short: O. Schütze, M. Laumanns, C.A. Coello Coello, M. Dellnitz, E.-G. Talbi, Journal
    of Global Optimization (2008) 559–577.
date_created: 2020-04-16T10:02:17Z
date_updated: 2022-01-06T06:52:54Z
department:
- _id: '101'
doi: 10.1007/s10898-007-9265-7
language:
- iso: eng
page: 559-577
publication: Journal of Global Optimization
publication_identifier:
  issn:
  - 0925-5001
  - 1573-2916
publication_status: published
status: public
title: Convergence of stochastic search algorithms to finite size pareto set approximations
type: journal_article
user_id: '15701'
year: '2008'
...
---
_id: '16671'
author:
- first_name: Stefan
  full_name: Sertl, Stefan
  last_name: Sertl
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
citation:
  ama: Sertl S, Dellnitz M. Global Optimization using a Dynamical Systems Approach.
    <i>Journal of Global Optimization</i>. 2006:569-587. doi:<a href="https://doi.org/10.1007/s10898-005-4384-5">10.1007/s10898-005-4384-5</a>
  apa: Sertl, S., &#38; Dellnitz, M. (2006). Global Optimization using a Dynamical
    Systems Approach. <i>Journal of Global Optimization</i>, 569–587. <a href="https://doi.org/10.1007/s10898-005-4384-5">https://doi.org/10.1007/s10898-005-4384-5</a>
  bibtex: '@article{Sertl_Dellnitz_2006, title={Global Optimization using a Dynamical
    Systems Approach}, DOI={<a href="https://doi.org/10.1007/s10898-005-4384-5">10.1007/s10898-005-4384-5</a>},
    journal={Journal of Global Optimization}, author={Sertl, Stefan and Dellnitz,
    Michael}, year={2006}, pages={569–587} }'
  chicago: Sertl, Stefan, and Michael Dellnitz. “Global Optimization Using a Dynamical
    Systems Approach.” <i>Journal of Global Optimization</i>, 2006, 569–87. <a href="https://doi.org/10.1007/s10898-005-4384-5">https://doi.org/10.1007/s10898-005-4384-5</a>.
  ieee: S. Sertl and M. Dellnitz, “Global Optimization using a Dynamical Systems Approach,”
    <i>Journal of Global Optimization</i>, pp. 569–587, 2006.
  mla: Sertl, Stefan, and Michael Dellnitz. “Global Optimization Using a Dynamical
    Systems Approach.” <i>Journal of Global Optimization</i>, 2006, pp. 569–87, doi:<a
    href="https://doi.org/10.1007/s10898-005-4384-5">10.1007/s10898-005-4384-5</a>.
  short: S. Sertl, M. Dellnitz, Journal of Global Optimization (2006) 569–587.
date_created: 2020-04-16T10:07:46Z
date_updated: 2022-01-06T06:52:54Z
department:
- _id: '101'
doi: 10.1007/s10898-005-4384-5
language:
- iso: eng
page: 569-587
publication: Journal of Global Optimization
publication_identifier:
  issn:
  - 0925-5001
  - 1573-2916
publication_status: published
status: public
title: Global Optimization using a Dynamical Systems Approach
type: journal_article
user_id: '15701'
year: '2006'
...