TY  - JOUR
A1  - Kanzow, Christian
A1  - Raharja, Andreas B.
A1  - Schwartz, Alexandra
T1  - An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems
JF  - Journal of Optimization Theory and Applications
N2  - 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.
KW  - quasinormality constraint qualification
KW  - cardinality constraints
KW  - augmented Lagrangian
KW  - global convergence
KW  - stationarity
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-269166
SN  - 1573-2878
VL  - 189
IS  - 3
ER  - 
TY  - JOUR
A1  - Kanzow, Christian
A1  - Raharja, Andreas B.
A1  - Schwartz, Alexandra
T1  - Sequential optimality conditions for cardinality-constrained optimization problems with applications
JF  - Computational Optimization and Applications
N2  - 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.
KW  - augmented Lagrangian method
KW  - cardinality constraints
KW  - sequential optimality condition
KW  - conecontinuity type constraint qualification
KW  - relaxation method
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-269052
SN  - 1573-2894
VL  - 80
IS  - 1
ER  -