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Relaxed method for optimization problems with cardinality constraints

Author

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  • Yan-Chao Liang

    (Henan Normal University)

  • Gui-Hua Lin

    (Shanghai University)

Abstract

In this paper, we review optimality conditions and constraint qualifications for the optimization problems with cardinality constraints (OPCC). OPCC is a class of optimization problems with important applications. In this paper, we provide a relaxed method for OPCC. We show that the Mangasarian-Fromovitz constraint qualification or constant positive linear dependence constraint qualification holds for the relaxed problem under some mild conditions. We provide that the local solution of the relaxed problem converges to the M-stationarity of OPCC under appropriate conditions. Furthermore, we obtain that the inexact stationary points of relaxed problem converges to the M-stationarity of OPCC under very weaker conditions. Numerical experiments show the effectiveness of the proposed method.

Suggested Citation

  • Yan-Chao Liang & Gui-Hua Lin, 2024. "Relaxed method for optimization problems with cardinality constraints," Journal of Global Optimization, Springer, vol. 88(2), pages 359-375, February.
    • Handle: RePEc:spr:jglopt:v:88:y:2024:i:2:d:10.1007_s10898-023-01317-5
    DOI: 10.1007/s10898-023-01317-5
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    References listed on IDEAS

    as
    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.
    2. Christian Kanzow & Andreas B. Raharja & Alexandra Schwartz, 2021. "Sequential optimality conditions for cardinality-constrained optimization problems with applications," Computational Optimization and Applications, Springer, vol. 80(1), pages 185-211, September.
    3. 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.
    4. Christian Kanzow & Andreas B. Raharja & Alexandra Schwartz, 2021. "An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 793-813, June.
    5. Dimitris Bertsimas & Romy Shioda, 2009. "Algorithm for cardinality-constrained quadratic optimization," Computational Optimization and Applications, Springer, vol. 43(1), pages 1-22, May.
    6. 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.
    7. Xiaojin Zheng & Xiaoling Sun & Duan Li & Jie Sun, 2014. "Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach," Computational Optimization and Applications, Springer, vol. 59(1), pages 379-397, October.
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