@phdthesis{Steck2018,
  author    = {Steck, Daniel},
  title     = {Lagrange Multiplier Methods for Constrained Optimization and Variational Problems in Banach Spaces},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-174444},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2018},
  abstract  = {This thesis is concerned with a class of general-purpose algorithms for constrained minimization problems, variational inequalities, and quasi-variational inequalities in Banach spaces. A substantial amount of background material from Banach space theory, convex analysis, variational analysis, and optimization theory is presented, including some results which are refinements of those existing in the literature. This basis is used to formulate an augmented Lagrangian algorithm with multiplier safeguarding for the solution of constrained optimization problems in Banach spaces. The method is analyzed in terms of local and global convergence, and many popular problem classes such as nonlinear programming, semidefinite programming, and function space optimization are shown to be included as special cases of the general setting. The algorithmic framework is then extended to variational and quasi-variational inequalities, which include, by extension, Nash and generalized Nash equilibrium problems. For these problem classes, the convergence is analyzed in detail. The thesis then presents a rich collection of application examples for all problem classes, including implementation details and numerical results.},
  subject      = {Optimierung},
  language  = {en}
}
@phdthesis{Boergens2020,
  author    = {B{\"o}rgens, Eike Alexander Lars Guido},
  title     = {ADMM-Type Methods for Optimization and Generalized Nash Equilibrium Problems in Hilbert Spaces},
  doi       = {10.25972/OPUS-21877},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-218777},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2020},
  abstract  = {This thesis is concerned with a certain class of algorithms for the solution of constrained optimization problems and generalized Nash equilibrium problems in Hilbert spaces. This class of algorithms is inspired by the alternating direction method of multipliers (ADMM) and eliminates the constraints using an augmented Lagrangian approach. The alternating direction method consists of splitting the augmented Lagrangian subproblem into smaller and more easily manageable parts. Before the algorithms are discussed, a substantial amount of background material, including the theory of Banach and Hilbert spaces, fixed-point iterations as well as convex and monotone set-valued analysis, is presented. Thereafter, certain optimization problems and generalized Nash equilibrium problems are reformulated and analyzed using variational inequalities and set-valued mappings. The analysis of the algorithms developed in the course of this thesis is rooted in these reformulations as variational inequalities and set-valued mappings. The first algorithms discussed and analyzed are one weakly and one strongly convergent ADMM-type algorithm for convex, linearly constrained optimization. By equipping the associated Hilbert space with the correct weighted scalar product, the analysis of these two methods is accomplished using the proximal point method and the Halpern method. The rest of the thesis is concerned with the development and analysis of ADMM-type algorithms for generalized Nash equilibrium problems that jointly share a linear equality constraint. The first class of these algorithms is completely parallelizable and uses a forward-backward idea for the analysis, whereas the second class of algorithms can be interpreted as a direct extension of the classical ADMM-method to generalized Nash equilibrium problems. At the end of this thesis, the numerical behavior of the discussed algorithms is demonstrated on a collection of examples.},
  subject      = {Constrained optimization},
  language  = {en}
}