@misc{Kanzow2022,
  author    = {Kanzow, Christian},
  title     = {Y. Cui, J.-S. Pang: "Modern Nonconvex Nondifferentiable Optimization"},
  series = {Jahresbericht der Deutschen Mathematiker-Vereinigung},
  volume    = {124},
  journal   = {Jahresbericht der Deutschen Mathematiker-Vereinigung},
  number    = {2},
  issn      = {0012-0456},
  doi       = {10.1365/s13291-022-00250-y},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324346},
  pages     = {137-143},
  year      = {2022},
  abstract  = {No abstract available.},
  language  = {en}
}
@article{KanzowMehlitz2022,
  author    = {Kanzow, Christian and Mehlitz, Patrick},
  title     = {Convergence properties of monotone and nonmonotone proximal gradient methods revisited},
  series = {Journal of Optimization Theory and Applications},
  volume    = {195},
  journal   = {Journal of Optimization Theory and Applications},
  number    = {2},
  issn      = {0022-3239},
  doi       = {10.1007/s10957-022-02101-3},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324351},
  pages     = {624-646},
  year      = {2022},
  abstract  = {Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the objective function is of simple enough structure. The available convergence theory associated with these methods (mostly) requires the derivative of the smooth part of the objective function to be (globally) Lipschitz continuous, and this might be a restrictive assumption in some practically relevant scenarios. In this paper, we readdress this classical topic and provide convergence results for the classical (monotone) proximal gradient method and one of its nonmonotone extensions which are applicable in the absence of (strong) Lipschitz assumptions. This is possible since, for the price of forgoing convergence rates, we omit the use of descent-type lemmas in our analysis.},
  language  = {en}
}
@article{KanzowRaharjaSchwartz2021,
  author    = {Kanzow, Christian and Raharja, Andreas B. and Schwartz, Alexandra},
  title     = {An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems},
  series = {Journal of Optimization Theory and Applications},
  volume    = {189},
  journal   = {Journal of Optimization Theory and Applications},
  number    = {3},
  issn      = {1573-2878},
  doi       = {10.1007/s10957-021-01854-7},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-269166},
  pages     = {793-813},
  year      = {2021},
  abstract  = {A reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.},
  language  = {en}
}
@article{KanzowRaharjaSchwartz2021,
  author    = {Kanzow, Christian and Raharja, Andreas B. and Schwartz, Alexandra},
  title     = {Sequential optimality conditions for cardinality-constrained optimization problems with applications},
  series = {Computational Optimization and Applications},
  volume    = {80},
  journal   = {Computational Optimization and Applications},
  number    = {1},
  issn      = {1573-2894},
  doi       = {10.1007/s10589-021-00298-z},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-269052},
  pages     = {185-211},
  year      = {2021},
  abstract  = {Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which is satisfied at every local minimizer without requiring any constraint qualification. We relate this condition to an existing M-type stationary concept by introducing a weak sequential constraint qualification based on a cone-continuity property. Finally, we present two algorithmic applications: We improve existing results for a known regularization method by proving that it generates limit points satisfying the aforementioned optimality conditions even if the subproblems are only solved inexactly. And we show that, under a suitable Kurdyka-Łojasiewicz-type assumption, any limit point of a standard (safeguarded) multiplier penalty method applied directly to the reformulated problem also satisfies the optimality condition. These results are stronger than corresponding ones known for the related class of mathematical programs with complementarity constraints.},
  language  = {en}
}