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Local Convergence Analysis of a Primal–Dual Method for Bound-Constrained Optimization Without SOSC

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  • Paul Armand

    (Université de Limoges - Laboratoire XLIM)

  • Ngoc Nguyen Tran

    (Quy Nhon University)

Abstract

We propose a local convergence analysis of a primal–dual interior point algorithm for the solution of a bound-constrained optimization problem. The algorithm includes a regularization technique to prevent singularity of the matrix of the linear system at each iteration, when the second-order sufficient conditions do not hold at the solution. These conditions are replaced by a milder assumption related to a local error-bound condition. This new condition is a generalization of the one used in unconstrained optimization. We show that by an appropriate updating strategy of the barrier parameter and of the regularization parameter, the proposed algorithm owns a superlinear rate of convergence. The analysis is made thanks to a boundedness property of the inverse of the Jacobian matrix arising in interior point algorithms. An illustrative example is given to show the behavior and the gain obtained by this regularization strategy.

Suggested Citation

  • Paul Armand & Ngoc Nguyen Tran, 2021. "Local Convergence Analysis of a Primal–Dual Method for Bound-Constrained Optimization Without SOSC," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 96-116, April.
    • Handle: RePEc:spr:joptap:v:189:y:2021:i:1:d:10.1007_s10957-021-01822-1
    DOI: 10.1007/s10957-021-01822-1
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    References listed on IDEAS

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    1. ,, 2004. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 20(2), pages 427-429, April.
    2. Paul Armand & Joël Benoist & Jean-Pierre Dussault, 2012. "Local path-following property of inexact interior methods in nonlinear programming," Computational Optimization and Applications, Springer, vol. 52(1), pages 209-238, May.
    3. Hamid Esmaeili & Morteza Kimiaei, 2016. "A trust-region method with improved adaptive radius for systems of nonlinear equations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 109-125, February.
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    5. Ying-Jie Li & Dong-Hui Li, 2009. "Truncated regularized Newton method for convex minimizations," Computational Optimization and Applications, Springer, vol. 43(1), pages 119-131, May.
    6. Hamid Esmaeili & Morteza Kimiaei, 2016. "A trust-region method with improved adaptive radius for systems of nonlinear equations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 109-125, February.
    7. ,, 2004. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 20(1), pages 223-229, February.
    8. Paul Armand & Isaï Lankoandé, 2017. "An inexact proximal regularization method for unconstrained optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(1), pages 43-59, February.
    9. Paul Armand & Riadh Omheni, 2017. "A Mixed Logarithmic Barrier-Augmented Lagrangian Method for Nonlinear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 523-547, May.
    10. Paul Tseng & Sangwoon Yun, 2014. "Incrementally Updated Gradient Methods for Constrained and Regularized Optimization," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 832-853, March.
    11. Kenji Ueda & Nobuo Yamashita, 2014. "A regularized Newton method without line search for unconstrained optimization," Computational Optimization and Applications, Springer, vol. 59(1), pages 321-351, October.
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