---
_id: '10594'
abstract:
- lang: eng
text: "Multiobjective optimization plays an increasingly important role in modern
applications, where several criteria are often of equal importance. The task in
multiobjective optimization and multiobjective optimal control is therefore to
compute\r\nthe set of optimal compromises (the Pareto set) between the conflicting
objectives.\r\n\r\nSince – in contrast to the solution of a single objective optimization
problem – the\r\nPareto set generally consists of an infinite number of solutions,
the computational\r\neffort can quickly become challenging. This is even more
the case when many problems have to be solved, when the number of objectives is
high, or when the objectives\r\nare costly to evaluate. Consequently, this thesis
is devoted to the identification and\r\nexploitation of structure both in the
Pareto set and the dynamics of the underlying\r\nmodel as well as to the development
of efficient algorithms for solving problems with\r\nadditional parameters, with
a high number of objectives or with PDE-constraints.\r\nThese three challenges
are addressed in three respective parts.\r\n\r\nIn the first part, predictor-corrector
methods are extended to entire Pareto sets.\r\nWhen certain smoothness assumptions
are satisfied, then the set of parameter dependent Pareto sets possesses additional
structure, i.e. it is a manifold. The tangent\r\nspace can be approximated numerically
which yields a direction for the predictor\r\nstep. In the corrector step, the
predicted set converges to the Pareto set at a new\r\nparameter value. The resulting
algorithm is applied to an example from autonomous\r\ndriving.\r\n\r\nIn the second
part, the hierarchical structure of Pareto sets is investigated. When\r\nconsidering
a subset of the objectives, the resulting solution is a subset of the Pareto\r\nset
of the original problem. Under additional smoothness assumptions, the respective
subsets are located on the boundary of the Pareto set of the full problem. This\r\nway,
the “skeleton” of a Pareto set can be computed and due to the exponential\r\nincrease
in computing time with the number of objectives, the computations of\r\nthese
subsets are significantly faster which is demonstrated using an example from\r\nindustrial
laundries.\r\n\r\nIn the third part, PDE-constrained multiobjective optimal control
problems are\r\naddressed by reduced order modeling methods. Reduced order models
exploit the\r\nstructure in the system dynamics, for example by describing the
dynamics of only the\r\nmost energetic modes. The model reduction introduces an
error in both the function values and their gradients, which has to be taken into
account in the development of\r\nalgorithms. Both scalarization and set-oriented
approaches are coupled with reduced\r\norder modeling. Convergence results are
presented and the numerical benefit is\r\ninvestigated. The algorithms are applied
to semi-linear heat flow problems as well\r\nas to the Navier-Stokes equations.\r\n"
author:
- first_name: Sebastian
full_name: Peitz, Sebastian
id: '47427'
last_name: Peitz
orcid: https://orcid.org/0000-0002-3389-793X
citation:
ama: Peitz S. * Exploiting Structure in Multiobjective Optimization and Optimal
Control*.; 2017. doi:10.17619/UNIPB/1-176
apa: Peitz, S. (2017). * Exploiting structure in multiobjective optimization
and optimal control*. https://doi.org/10.17619/UNIPB/1-176
bibtex: '@book{Peitz_2017, title={ Exploiting structure in multiobjective optimization
and optimal control}, DOI={10.17619/UNIPB/1-176},
author={Peitz, Sebastian}, year={2017} }'
chicago: Peitz, Sebastian. * Exploiting Structure in Multiobjective Optimization
and Optimal Control*, 2017. https://doi.org/10.17619/UNIPB/1-176.
ieee: S. Peitz, * Exploiting structure in multiobjective optimization and optimal
control*. 2017.
mla: Peitz, Sebastian. * Exploiting Structure in Multiobjective Optimization
and Optimal Control*. 2017, doi:10.17619/UNIPB/1-176.
short: S. Peitz, Exploiting Structure in Multiobjective Optimization and Optimal
Control, 2017.
date_created: 2019-07-10T08:12:22Z
date_updated: 2022-01-06T06:50:46Z
ddc:
- '510'
department:
- _id: '101'
doi: 10.17619/UNIPB/1-176
file:
- access_level: closed
content_type: application/pdf
creator: speitz
date_created: 2020-03-13T12:52:50Z
date_updated: 2020-03-13T12:52:50Z
file_id: '16298'
file_name: Dissertation_Peitz.pdf
file_size: 16636801
relation: main_file
success: 1
file_date_updated: 2020-03-13T12:52:50Z
has_accepted_license: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://d-nb.info/1139356542/34
oa: '1'
project:
- _id: '52'
name: Computing Resources Provided by the Paderborn Center for Parallel Computing
publication_status: published
status: public
title: " \tExploiting structure in multiobjective optimization and optimal control"
type: dissertation
user_id: '47427'
year: '2017'
...