Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods for Cardinality-Constrained Optimization Problems
We consider nonlinear optimization problems with cardinality constraints. Based on a continuous reformulation, we introduce second-order necessary and sufficient optimality conditions. Under such a second-order condition, we can guarantee local uniqueness of Mordukhovich stationary points. Finally, we use this observation to provide extended local convergence theory for a Scholtes-type regularization method, which guarantees the existence and convergence of iterates under suitable assumptions. This convergence theory can also be applied to other regularization schemes.
KeywordsCardinality constraints Strong stationarity Mordukhovich stationarity Second-order optimality conditions Regularization method Scholtes regularization
Mathematics Subject Classification90C27 90C30 90C33 90C46 65K05
The work of Alexandra Schwartz and Max Bucher is supported by the ‘Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt. The authors would like to thank two anonymous referees and the editors for carefully reading the manuscript and asking the right questions, which helped to improve the paper.
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