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An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

Author

Listed:
  • Christian Kanzow

    (University of Würzburg)

  • Andreas B. Raharja

    (University of Würzburg)

  • Alexandra Schwartz

    (Technische Universität Dresden)

Abstract

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.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:189:y:2021:i:3:d:10.1007_s10957-021-01854-7
    DOI: 10.1007/s10957-021-01854-7
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    References listed on IDEAS

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    1. Dimitris Bertsimas & Romy Shioda, 2009. "Algorithm for cardinality-constrained quadratic optimization," Computational Optimization and Applications, Springer, vol. 43(1), pages 1-22, May.
    2. R. Andreani & J. M. Martinez & M. L. Schuverdt, 2005. "On the Relation between Constant Positive Linear Dependence Condition and Quasinormality Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 473-483, May.
    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. D.P. Bertsekas & A.E. Ozdaglar, 2002. "Pseudonormality and a Lagrange Multiplier Theory for Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 114(2), pages 287-343, August.
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    Cited by:

    1. 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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