Structured backward errors for KKT systems (Q1300867)

Karush-Kuhn-Tucker (KKT) systems are linear systems with coefficient matrices of the form \[ \begin{pmatrix} H & A\\ A^T & 0\end{pmatrix}, \] where \(H\) is symmetric. A normwise structured backward error for KKT systems is defined, and a computable formula of the structured backward error is obtained. Simple examples show that the structured backward error may be arbitrarily larger than the unstructured ones in the worst case, and a stable algorithm for solving KKT systems is not necessarily strongly stable. Consequently, the computable formula of the structured backward error may be useful for testing the strong stability of practical algorithms for solving KKT systems.