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.

中文翻译：

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奇异半定程序的摄动分析及其在控制问题中的应用

在原始问题严格可行和对偶问题弱可行的情况下，我们考虑了扰动下半定程序的敏感性。当系数矩阵受到扰动时，最佳值可能会不连续地变化，如具体示例中所述。我们表明，在涉及扰动对偶问题的最小面的行为的条件下，这种半定程序的最佳值连续变化。此外，我们通过使用在面部缩减中产生的缩减证书来确定什么样的扰动保持最小面部不变。我们的结果使我们能够对从控制问题获得的半定程序的最小面的行为进行分类。