@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}
}
@phdthesis{Schwartz2011,
  author    = {Schwartz, Alexandra},
  title     = {Mathematical Programs with Complementarity Constraints: Theory, Methods and Applications},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-64891},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2011},
  abstract  = {The subject of this thesis are mathematical programs with complementarity conditions (MPCC). At first, an economic example of this problem class is analyzed, the problem of effort maximization in asymmetric n-person contest games. While an analytical solution for this special problem could be derived, this is not possible in general for MPCCs. Therefore, optimality conditions which might be used for numerical approaches where considered next. More precisely, a Fritz-John result for MPCCs with stronger properties than those known so far was derived together with some new constraint qualifications and subsequently used to prove an exact penalty result. Finally, to solve MPCCs numerically, the so called relaxation approach was used. Besides improving the results for existing relaxation methods, a new relaxation with strong convergence properties was suggested and a numerical comparison of all methods based on the MacMPEC collection conducted.},
  subject      = {Zwei-Ebenen-Optimierung},
  language  = {en}
}