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

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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.

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Acknowledgements

The first author was supported by JSPS KAKENHI Grant Number JP15K04993 and JP19K03631. A part of his work was done when he stayed in University of Konstanz with the financial support from Tokyo University of Marine Science and Technology. The second author was supported by JSPS KAKENHI Grant Numbers JP22740056, JP26400203, JP17H01700, JP20K11696, and ERATO HASUO Metamathematics for Systems Design Project (No.JPMJER1603), JST. We would like to thank Noboru Sebe in Kyushu Institute of Technology for fruitful discussions and anonymous referees for their careful reading of the paper and for their comments that helped us to improve the presentation of the paper.

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Authors and Affiliations

  1. Tokyo University of Marine Science and Technology, 2-1-6, Etchujima, Koto, Tokyo, 135-8533, Japan

    Yoshiyuki Sekiguchi

  2. Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Hayato Waki

Authors
  1. Yoshiyuki Sekiguchi
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  2. Hayato Waki
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Correspondence to Yoshiyuki Sekiguchi.

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Communicated by Levent Tunçel.

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Sekiguchi, Y., Waki, H. Perturbation Analysis of Singular Semidefinite Programs and Its Applications to Control Problems. J Optim Theory Appl 188, 52–72 (2021). https://doi.org/10.1007/s10957-020-01780-0

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