---
_id: '16296'
abstract:
- lang: eng
  text: "Multiobjective optimization plays an increasingly important role in modern\r\napplications,
    where several objectives are often of equal importance. The task\r\nin multiobjective
    optimization and multiobjective optimal control is therefore\r\nto compute the
    set of optimal compromises (the Pareto set) between the\r\nconflicting objectives.
    Since the Pareto set generally consists of an infinite\r\nnumber of solutions,
    the computational effort can quickly become challenging\r\nwhich is particularly
    problematic when the objectives are costly to evaluate as\r\nis the case for models
    governed by partial differential equations (PDEs). To\r\ndecrease the numerical
    effort to an affordable amount, surrogate models can be\r\nused to replace the
    expensive PDE evaluations. Existing multiobjective\r\noptimization methods using
    model reduction are limited either to low parameter\r\ndimensions or to few (ideally
    two) objectives. In this article, we present a\r\ncombination of the reduced basis
    model reduction method with a continuation\r\napproach using inexact gradients.
    The resulting approach can handle an\r\narbitrary number of objectives while yielding
    a significant reduction in\r\ncomputing time."
author:
- first_name: Stefan
  full_name: Banholzer, Stefan
  last_name: Banholzer
- first_name: Bennet
  full_name: Gebken, Bennet
  id: '32643'
  last_name: Gebken
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
- first_name: Sebastian
  full_name: Peitz, Sebastian
  id: '47427'
  last_name: Peitz
  orcid: https://orcid.org/0000-0002-3389-793X
- first_name: Stefan
  full_name: Volkwein, Stefan
  last_name: Volkwein
citation:
  ama: 'Banholzer S, Gebken B, Dellnitz M, Peitz S, Volkwein S. ROM-Based Multiobjective
    Optimization of Elliptic PDEs via Numerical Continuation. In: Michael H, Roland
    H, Christian K, Michael U, Stefan U, eds. <i>Non-Smooth and Complementarity-Based
    Distributed Parameter Systems</i>. Springer; 2022:43-76. doi:<a href="https://doi.org/10.1007/978-3-030-79393-7_3">10.1007/978-3-030-79393-7_3</a>'
  apa: Banholzer, S., Gebken, B., Dellnitz, M., Peitz, S., &#38; Volkwein, S. (2022).
    ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation.
    In H. Michael, H. Roland, K. Christian, U. Michael, &#38; U. Stefan (Eds.), <i>Non-Smooth
    and Complementarity-Based Distributed Parameter Systems</i> (pp. 43–76). Springer.
    <a href="https://doi.org/10.1007/978-3-030-79393-7_3">https://doi.org/10.1007/978-3-030-79393-7_3</a>
  bibtex: '@inbook{Banholzer_Gebken_Dellnitz_Peitz_Volkwein_2022, place={Cham}, title={ROM-Based
    Multiobjective Optimization of Elliptic PDEs via Numerical Continuation}, DOI={<a
    href="https://doi.org/10.1007/978-3-030-79393-7_3">10.1007/978-3-030-79393-7_3</a>},
    booktitle={Non-Smooth and Complementarity-Based Distributed Parameter Systems},
    publisher={Springer}, author={Banholzer, Stefan and Gebken, Bennet and Dellnitz,
    Michael and Peitz, Sebastian and Volkwein, Stefan}, editor={Michael, Hintermüller
    and Roland, Herzog and Christian, Kanzow and Michael, Ulbrich and Stefan, Ulbrich},
    year={2022}, pages={43–76} }'
  chicago: 'Banholzer, Stefan, Bennet Gebken, Michael Dellnitz, Sebastian Peitz, and
    Stefan Volkwein. “ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical
    Continuation.” In <i>Non-Smooth and Complementarity-Based Distributed Parameter
    Systems</i>, edited by Hintermüller Michael, Herzog Roland, Kanzow Christian,
    Ulbrich Michael, and Ulbrich Stefan, 43–76. Cham: Springer, 2022. <a href="https://doi.org/10.1007/978-3-030-79393-7_3">https://doi.org/10.1007/978-3-030-79393-7_3</a>.'
  ieee: 'S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, and S. Volkwein, “ROM-Based
    Multiobjective Optimization of Elliptic PDEs via Numerical Continuation,” in <i>Non-Smooth
    and Complementarity-Based Distributed Parameter Systems</i>, H. Michael, H. Roland,
    K. Christian, U. Michael, and U. Stefan, Eds. Cham: Springer, 2022, pp. 43–76.'
  mla: Banholzer, Stefan, et al. “ROM-Based Multiobjective Optimization of Elliptic
    PDEs via Numerical Continuation.” <i>Non-Smooth and Complementarity-Based Distributed
    Parameter Systems</i>, edited by Hintermüller Michael et al., Springer, 2022,
    pp. 43–76, doi:<a href="https://doi.org/10.1007/978-3-030-79393-7_3">10.1007/978-3-030-79393-7_3</a>.
  short: 'S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein, in: H. Michael,
    H. Roland, K. Christian, U. Michael, U. Stefan (Eds.), Non-Smooth and Complementarity-Based
    Distributed Parameter Systems, Springer, Cham, 2022, pp. 43–76.'
date_created: 2020-03-13T12:45:31Z
date_updated: 2022-03-14T13:04:51Z
department:
- _id: '101'
- _id: '655'
doi: 10.1007/978-3-030-79393-7_3
editor:
- first_name: Hintermüller
  full_name: Michael, Hintermüller
  last_name: Michael
- first_name: Herzog
  full_name: Roland, Herzog
  last_name: Roland
- first_name: Kanzow
  full_name: Christian, Kanzow
  last_name: Christian
- first_name: Ulbrich
  full_name: Michael, Ulbrich
  last_name: Michael
- first_name: Ulbrich
  full_name: Stefan, Ulbrich
  last_name: Stefan
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/pdf/1906.09075.pdf
oa: '1'
page: 43-76
place: Cham
publication: Non-Smooth and Complementarity-Based Distributed Parameter Systems
publication_identifier:
  isbn:
  - 978-3-030-79392-0
publisher: Springer
status: public
title: ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation
type: book_chapter
user_id: '47427'
year: '2022'
...