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Combining stabilized SQP with the augmented Lagrangian algorithm

Author

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  • A. Izmailov
  • M. Solodov
  • E. Uskov

Abstract

For an optimization problem with general equality and inequality constraints, we propose an algorithm which uses subproblems of the stabilized SQP (sSQP) type for approximately solving subproblems of the augmented Lagrangian method. The motivation is to take advantage of the well-known robust behavior of the augmented Lagrangian algorithm, including on problems with degenerate constraints, and at the same time try to reduce the overall algorithm locally to sSQP (which gives fast local convergence rate under weak assumptions). Specifically, the algorithm first verifies whether the primal-dual sSQP step (with unit stepsize) makes good progress towards decreasing the violation of optimality conditions for the original problem, and if so, makes this step. Otherwise, the primal part of the sSQP direction is used for linesearch that decreases the augmented Lagrangian, keeping the multiplier estimate fixed for the time being. The overall algorithm has reasonable global convergence guarantees, and inherits strong local convergence rate properties of sSQP under the same weak assumptions. Numerical results on degenerate problems and comparisons with some alternatives are reported. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • A. Izmailov & M. Solodov & E. Uskov, 2015. "Combining stabilized SQP with the augmented Lagrangian algorithm," Computational Optimization and Applications, Springer, vol. 62(2), pages 405-429, November.
    • Handle: RePEc:spr:coopap:v:62:y:2015:i:2:p:405-429
    DOI: 10.1007/s10589-015-9744-6
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    References listed on IDEAS

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    1. D. Fernández & E. Pilotta & G. Torres, 2013. "An inexact restoration strategy for the globalization of the sSQP method," Computational Optimization and Applications, Springer, vol. 54(3), pages 595-617, April.
    2. A. F. Izmailov & M. V. Solodov, 2015. "Newton-Type Methods: A Broader View," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 577-620, February.
    3. A. Izmailov & M. Solodov, 2009. "Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints," Computational Optimization and Applications, Springer, vol. 42(2), pages 231-264, March.
    4. Philip Gill & Daniel Robinson, 2012. "A primal-dual augmented Lagrangian," Computational Optimization and Applications, Springer, vol. 51(1), pages 1-25, January.
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    1. A. F. Izmailov & M. V. Solodov & E. I. Uskov, 2019. "A globally convergent Levenberg–Marquardt method for equality-constrained optimization," Computational Optimization and Applications, Springer, vol. 72(1), pages 215-239, January.
    2. E. G. Birgin & R. D. Lobato & J. M. Martínez, 2016. "Packing ellipsoids by nonlinear optimization," Journal of Global Optimization, Springer, vol. 65(4), pages 709-743, August.
    3. Dominique Orban & Abel Soares Siqueira, 2020. "A regularization method for constrained nonlinear least squares," Computational Optimization and Applications, Springer, vol. 76(3), pages 961-989, July.
    4. A. F. Izmailov & E. I. Uskov, 2017. "Subspace-stabilized sequential quadratic programming," Computational Optimization and Applications, Springer, vol. 67(1), pages 129-154, May.
    5. A. F. Izmailov & M. V. Solodov & E. I. Uskov, 2016. "Globalizing Stabilized Sequential Quadratic Programming Method by Smooth Primal-Dual Exact Penalty Function," Journal of Optimization Theory and Applications, Springer, vol. 169(1), pages 148-178, April.
    6. Yuya Yamakawa & Takayuki Okuno, 2022. "A stabilized sequential quadratic semidefinite programming method for degenerate nonlinear semidefinite programs," Computational Optimization and Applications, Springer, vol. 83(3), pages 1027-1064, December.
    7. E. G. Birgin & R. D. Lobato & J. M. Martínez, 2017. "A nonlinear programming model with implicit variables for packing ellipsoids," Journal of Global Optimization, Springer, vol. 68(3), pages 467-499, July.
    8. A. F. Izmailov, 2021. "Accelerating convergence of a globalized sequential quadratic programming method to critical Lagrange multipliers," Computational Optimization and Applications, Springer, vol. 80(3), pages 943-978, December.

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