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  - 
TY  - THES
A1  - Schwartz, Alexandra
T1  - Mathematical Programs with Complementarity Constraints: Theory, Methods and Applications
T1  - Mathematische Programme mit Komplementaritätsrestriktionen: Theorie, Verfahren und Anwendungen
N2  - 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.
N2  - Das Thema dieser Dissertation sind mathematische Programme mit Komplementaritätsrestriktionen (MPCC). Zunächst wurde eine ökonomische Anwendung dieses Problemklasse betrachtet, das sogenannte Wettbewerbsdesignproblem. Während es für dieses spezielle Problem möglich war eine analytische Lösung herzuleiten, ist dies im Allgemeinen nicht möglich. Daher wurden anschließend Optimalitätsbedingungen, die für eine numerische Lösung verwendet werden können, betrachtet. Genauer wurde ein stärkeres Fritz-John Resultat als die bisher bekannten zusammen mit neuen Constraint Qualifications hergeleitet und anschließend zum Beweis eines exakten Penaltyresultates benutzt. Schließlich wurden zur numerischen Lösung von MPCCs sogenannte Relaxationsverfahren betrachtet. Zusätzlich zur Verbesserung der Resultate für bekannte Verfahren wurde eine neue Relaxierung mit starken Konvergenzeigenschaften vorgeschlagen und ein numerischer Vergleich aller Verfahren auf Basis der MacMPEC Testsammlung durchgeführt.
KW  - Zwei-Ebenen-Optimierung
KW  - Nash-Gleichgewicht
KW  - Constraint-Programmierung
KW  - Wettbewerbsdesign
KW  - Nichtlineare Optimierung
KW  - Nichtkonvexe Optimierung
KW  - Kombinatorische Optimierung
KW  - Numerik
KW  - MPEC
Y1  - 2011
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-64891
ER  -