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Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods for Cardinality-Constrained Optimization Problems

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  • Max Bucher

    (Technische Universität Darmstadt)

  • Alexandra Schwartz

    (Technische Universität Darmstadt)

Abstract

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.

Suggested Citation

  • Max Bucher & Alexandra Schwartz, 2018. "Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods for Cardinality-Constrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 383-410, August.
    • Handle: RePEc:spr:joptap:v:178:y:2018:i:2:d:10.1007_s10957-018-1320-7
    DOI: 10.1007/s10957-018-1320-7
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    References listed on IDEAS

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    1. Walter Murray & Howard Shek, 2012. "A local relaxation method for the cardinality constrained portfolio optimization problem," Computational Optimization and Applications, Springer, vol. 53(3), pages 681-709, December.
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    6. Martin Branda & Max Bucher & Michal Červinka & Alexandra Schwartz, 2018. "Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization," Computational Optimization and Applications, Springer, vol. 70(2), pages 503-530, June.
    7. Lei Guo & Gui-Hua Lin & Jane J. Ye, 2013. "Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 33-64, July.
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    Cited by:

    1. S. Lämmel & V. Shikhman, 2022. "On nondegenerate M-stationary points for sparsity constrained nonlinear optimization," Journal of Global Optimization, Springer, vol. 82(2), pages 219-242, February.
    2. Ademir A. Ribeiro & Mael Sachine & Evelin H. M. Krulikovski, 2022. "A Comparative Study of Sequential Optimality Conditions for Mathematical Programs with Cardinality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 1067-1083, March.

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