Towards an optimal condition number of certain augmented Lagrangian‐type saddle‐point matrices

We present an analysis for minimizing the condition number of nonsingular parameter‐dependent 2 × 2 block‐structured saddle‐point matrices with a maximally rank‐deficient (1,1) block. The matrices arise from an augmented Lagrangian approach. Using quasidirect sums, we show that a decomposition akin to simultaneous diagonalization leads to an optimization based on the extremal nonzero eigenvalues and singular values of the associated block matrices. Bounds on the condition number of the parameter‐dependent matrix are obtained, and we demonstrate their tightness on some numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.