---
res:
bibo_abstract:
- "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@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Sebastian
foaf_name: Peitz, Sebastian
foaf_surname: Peitz
foaf_workInfoHomepage: http://www.librecat.org/personId=47427
orcid: https://orcid.org/0000-0002-3389-793X
bibo_doi: 10.17619/UNIPB/1-176
dct_date: 2017^xs_gYear
dct_language: eng
dct_title: " \tExploiting structure in multiobjective optimization and optimal control@"
...