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Perturbation Analysis of Singular Semidefinite Programs and Its Applications to Control Problems

Author

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  • Yoshiyuki Sekiguchi

    (Tokyo University of Marine Science and Technology)

  • Hayato Waki

    (Kyushu University)

Abstract

We consider sensitivity of a semidefinite program under perturbations in the case that the primal problem is strictly feasible and the dual problem is weakly feasible. When the coefficient matrices are perturbed, the optimal values can change discontinuously as explained in concrete examples. We show that the optimal value of such a semidefinite program changes continuously under conditions involving the behavior of the minimal faces of the perturbed dual problems. In addition, we determine what kinds of perturbations keep the minimal faces invariant, by using the reducing certificates, which are produced in facial reduction. Our results allow us to classify the behavior of the minimal face of a semidefinite program obtained from a control problem.

Suggested Citation

  • Yoshiyuki Sekiguchi & Hayato Waki, 2021. "Perturbation Analysis of Singular Semidefinite Programs and Its Applications to Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 52-72, January.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:1:d:10.1007_s10957-020-01780-0
    DOI: 10.1007/s10957-020-01780-0
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    References listed on IDEAS

    as
    1. J.F. Sturm & S. Zhang, 1998. "On Sensitivity of Central Solutions in Semidefinite Programming," Tinbergen Institute Discussion Papers 98-040/4, Tinbergen Institute.
    2. Hayato Waki & Maho Nakata & Masakazu Muramatsu, 2012. "Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization," Computational Optimization and Applications, Springer, vol. 53(3), pages 823-844, December.
    3. Hayato Waki & Masakazu Muramatsu, 2013. "Facial Reduction Algorithms for Conic Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 188-215, July.
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    Cited by:

    1. Nguyen Ngoc Luan & Do Sang Kim & Nguyen Dong Yen, 2022. "Two Optimal Value Functions in Parametric Conic Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 574-597, June.

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