@phdthesis{Lechner2022,
  author    = {Lechner, Theresa},
  title     = {Proximal Methods for Nonconvex Composite Optimization Problems},
  doi       = {10.25972/OPUS-28907},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-289073},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2022},
  abstract  = {Optimization problems with composite functions deal with the minimization of the sum of a smooth function and a convex nonsmooth function. In this thesis several numerical methods for solving such problems in finite-dimensional spaces are discussed, which are based on proximity operators. After some basic results from convex and nonsmooth analysis are summarized, a first-order method, the proximal gradient method, is presented and its convergence properties are discussed in detail. Known results from the literature are summarized and supplemented by additional ones. Subsequently, the main part of the thesis is the derivation of two methods which, in addition, make use of second-order information and are based on proximal Newton and proximal quasi-Newton methods, respectively. The difference between the two methods is that the first one uses a classical line search, while the second one uses a regularization parameter instead. Both techniques lead to the advantage that, in contrast to many similar methods, in the respective detailed convergence analysis global convergence to stationary points can be proved without any restricting precondition. Furthermore, comprehensive results show the local convergence properties as well as convergence rates of these algorithms, which are based on rather weak assumptions. Also a method for the solution of the arising proximal subproblems is investigated. In addition, the thesis contains an extensive collection of application examples and a detailed discussion of the related numerical results.},
  subject      = {Optimierung},
  language  = {en}
}
@article{KanzowLechner2021,
  author    = {Kanzow, Christian and Lechner, Theresa},
  title     = {Globalized inexact proximal Newton-type methods for nonconvex composite functions},
  series = {Computational Optimization and Applications},
  volume    = {78},
  journal   = {Computational Optimization and Applications},
  number    = {2},
  doi       = {10.1007/s10589-020-00243-6},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-283715},
  pages     = {377-410},
  year      = {2021},
  abstract  = {Optimization problems with composite functions consist of an objective function which is the sum of a smooth and a (convex) nonsmooth term. This particular structure is exploited by the class of proximal gradient methods and some of their generalizations like proximal Newton and quasi-Newton methods. The current literature on these classes of methods almost exclusively considers the case where also the smooth term is convex. Here we present a globalized proximal Newton-type method which allows the smooth term to be nonconvex. The method is shown to have nice global and local convergence properties, and some numerical results indicate that this method is very promising also from a practical point of view.},
  language  = {en}
}
@misc{KanzowLechner2021,
  author    = {Kanzow, Christian and Lechner, Theresa},
  title     = {Correction to: Globalized inexact proximal Newton-type methods for nonconvex composite functions},
  series = {Computational Optimization and Applications},
  volume    = {80},
  journal   = {Computational Optimization and Applications},
  number    = {2},
  doi       = {10.1007/s10589-021-00302-6},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-348858},
  pages     = {679-680},
  year      = {2021},
  abstract  = {No abstract available.},
  language  = {en}
}