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Inexact penalty decomposition methods for optimization problems with geometric constraints

Author

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  • Christian Kanzow

    (University of Würzburg)

  • Matteo Lapucci

    (University of Florence)

Abstract

This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints are nonconvex and complicated, like cardinality constraints, disjunctive programs, or matrix problems involving rank constraints. By a variable duplication and decomposition strategy, the method presented here explicitly handles these difficult constraints, thus generating iterates which are feasible with respect to them, while the remaining (standard and supposingly simple) constraints are tackled by sequential penalization. Inexact optimization steps are proven sufficient for the resulting algorithm to work, so that it is employable even with difficult objective functions. The current work is therefore a significant generalization of existing papers on penalty decomposition methods. On the other hand, it is related to some recent publications which use an augmented Lagrangian idea to solve optimization problems with geometric constraints. Compared to these methods, the decomposition idea is shown to be numerically superior since it allows much more freedom in the choice of the subproblem solver, and since the number of certain (possibly expensive) projection steps is significantly less. Extensive numerical results on several highly complicated classes of optimization problems in vector and matrix spaces indicate that the current method is indeed very efficient to solve these problems.

Suggested Citation

  • Christian Kanzow & Matteo Lapucci, 2023. "Inexact penalty decomposition methods for optimization problems with geometric constraints," Computational Optimization and Applications, Springer, vol. 85(3), pages 937-971, July.
  • Handle: RePEc:spr:coopap:v:85:y:2023:i:3:d:10.1007_s10589-023-00475-2
    DOI: 10.1007/s10589-023-00475-2
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    References listed on IDEAS

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    1. Matteo Lapucci & Tommaso Levato & Marco Sciandrone, 2021. "Convergent Inexact Penalty Decomposition Methods for Cardinality-Constrained Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(2), pages 473-496, February.
    2. 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.
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    Cited by:

    1. Matteo Lapucci & Alessio Sortino, 2024. "On the Convergence of Inexact Alternate Minimization in Problems with $$\ell _0$$ ℓ 0 Penalties," SN Operations Research Forum, Springer, vol. 5(2), pages 1-11, June.

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    1. Matteo Lapucci & Alessio Sortino, 2024. "On the Convergence of Inexact Alternate Minimization in Problems with $$\ell _0$$ ℓ 0 Penalties," SN Operations Research Forum, Springer, vol. 5(2), pages 1-11, June.

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