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Convergence rate estimates for penalty methods revisited

Author

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  • A. F. Izmailov

    (Lomonosov Moscow State University, MSU)

  • M. V. Solodov

    (IMPA – Instituto de Matemática Pura e Aplicada)

Abstract

For the classical quadratic penalty, it is known that the distance from the solution of the penalty subproblem to the solution of the original problem is at worst inversely proportional to the value of the penalty parameter under the linear independence constraint qualification, strict complementarity, and the second-order sufficient optimality conditions. Moreover, using solutions of the penalty subproblem, one can obtain certain useful Lagrange multipliers estimates whose distance to the optimal ones is also at least inversely proportional to the value of the parameter. We show that the same properties hold more generally, namely, under the (weaker) strict Mangasarian–Fromovitz constraint qualification and second-order sufficiency (and without strict complementarity). Moreover, under the linear independence constraint qualification and strong second-order sufficiency (also without strict complementarity), we demonstrate local uniqueness and Lipschitz continuity of stationary points of penalty subproblems. In addition, those results follow from the analysis of general power penalty functions, of which quadratic penalty is a special case.

Suggested Citation

  • A. F. Izmailov & M. V. Solodov, 2023. "Convergence rate estimates for penalty methods revisited," Computational Optimization and Applications, Springer, vol. 85(3), pages 973-992, July.
    • Handle: RePEc:spr:coopap:v:85:y:2023:i:3:d:10.1007_s10589-023-00476-1
    DOI: 10.1007/s10589-023-00476-1
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    References listed on IDEAS

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    1. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
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    Cited by:

    1. Marcelo Gallardo & Manuel Loaiza & Jorge Ch'avez, 2024. "Congestion and Penalization in Optimal Transport," Papers 2410.07363, arXiv.org, revised Oct 2024.

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