TY - JOUR
T1 - Perturbation Analysis of Singular Semidefinite Programs and Its Applications to Control Problems
AU - Sekiguchi, Yoshiyuki
AU - Waki, Hayato
N1 - Funding Information:
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.
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2021/1
Y1 - 2021/1
N2 - 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.
AB - 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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U2 - 10.1007/s10957-020-01780-0
DO - 10.1007/s10957-020-01780-0
M3 - Article
AN - SCOPUS:85096340976
SN - 0022-3239
VL - 188
SP - 52
EP - 72
JO - Journal of Optimization Theory and Applications
JF - Journal of Optimization Theory and Applications
IS - 1
ER -