SIAM Journal on Optimization, Volume 31, Issue 1, Page 812-836, January 2021.
In semidefinite programming a proposed optimal solution may be quite poor in spite of having sufficiently small residual in the optimality conditions. This issue may be framed in terms of the discrepancy between forward error (the unmeasurable “true error'') and backward error (the measurable violation of optimality conditions). In [SIAM J. Optim., 10 (2000), pp. 1228--1248], Sturm provided an upper bound on forward error in terms of backward error and singularity degree. In this work we provide a method to bound the maximum rank over all optimal solutions and use this result to obtain a lower bound on forward error for a class of convergent sequences. This lower bound complements the upper bound of Sturm. The results of Sturm imply that semidefinite programs with slow convergence necessarily have large singularity degree. Here we show that large singularity degree is, in some sense, also a sufficient condition for slow convergence for a family of external-type “central” paths. Our results are supported by numerical observations.

中文翻译：

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半定规划中的误差界和奇异度

SIAM优化杂志，第31卷，第1期，第812-836页，2021年1月。
在半定编程中，尽管在最优条件下残差足够小，但所提出的最优解可能仍然很差。可以根据前向误差（不可测量的“真实误差”）和后向误差（可测量的违反最优性条件）之间的差异来构造此问题。在[SIAM J. Optim。，10（2000），第1228--1248页]中，Sturm就后向误差和奇异程度提供了前向误差的上限。在这项工作中，我们提供了一种在所有最优解上限制最大秩的方法，并使用此结果为一类收敛序列获得前向误差的下限。此下限与Sturm的上限互补。Sturm的结果表明，收敛速度较慢的半定程序必然具有较大的奇异度。在这里，我们表明，从某种意义上讲，大的奇异度也是一系列外部类型的“中心”路径缓慢收敛的充分条件。我们的结果得到数值观察的支持。