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A Practical Second-Order Optimality Condition for Cardinality-Constrained Problems with Application to an Augmented Lagrangian Method

Author

Listed:
  • Jean C. A. Medeiros

    (State University of Campinas)

  • Ademir A. Ribeiro

    (Federal University of Paraná)

  • Mael Sachine

    (Federal University of Paraná)

  • Leonardo D. Secchin

    (Federal University of Espírito Santo)

Abstract

This paper addresses the mathematical programs with cardinality constraints (MPCaC). We first define two new tailored (strong and weak) second-order necessary conditions, MPCaC-SSONC and MPCaC-WSONC. We then propose a constraint qualification (CQ), namely, MPCaC-relaxed constant rank constraint qualification (MPCaC-RCRCQ), and establish the validity of MPCaC-SSONC at minimizers under this new CQ. All the concepts proposed here are based on the so-called M-stationarity, which is a suitable first-order stationarity for MPCaC. Furthermore, they are defined using only original variables, without the help of auxiliary variables of an augmented problem commonly considered in this context. This makes the proposed second-order stationarity concepts suitable for MPCaC. We illustrate the applicability of MPCaC-WSONC to derive global convergence for a second-order augmented Lagrangian algorithm on MPCaCs under MPCaC-RCRCQ. The relationship between the tailored MPCaC-WSONC and the standard WSONC (applied to a reformulated problem) are treated, showing that MPCaC-WSONC is a strong condition to study global convergence of algorithms in the MPCaC context.

Suggested Citation

  • Jean C. A. Medeiros & Ademir A. Ribeiro & Mael Sachine & Leonardo D. Secchin, 2025. "A Practical Second-Order Optimality Condition for Cardinality-Constrained Problems with Application to an Augmented Lagrangian Method," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-25, August.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:2:d:10.1007_s10957-025-02705-5
    DOI: 10.1007/s10957-025-02705-5
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    References listed on IDEAS

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    1. R. Andreani & C. E. Echagüe & M. L. Schuverdt, 2010. "Constant-Rank Condition and Second-Order Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 255-266, August.
    2. 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.
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    4. 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.
    5. M.L. Flegel & C. Kanzow, 2005. "Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 595-614, March.
    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. 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.
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