Partial Condition Numbers for Double Saddle Point Problems

TL;DR

This paper develops a comprehensive framework for calculating and bounding partial condition numbers of double saddle point problems, including structured cases, with efficient formulas and numerical validation.

Contribution

It introduces a unified approach to partial condition numbers for DSPPs, deriving closed-form expressions, sharp bounds, and extending to structured problems and related EILS problems.

Findings

Derived closed-form expressions for partial CNs.

Established sharp, computationally efficient upper bounds.

Numerical results confirm the bounds' accuracy and effectiveness.

Abstract

This paper presents a unified framework for investigating the partial condition number (CN) of the solution of double saddle point problems (DSPPs) and provides closed-form expressions for it. This unified framework encompasses the well-known partial normwise CN (NCN), partial mixed CN (MCN) and partial componentwise CN (CCN) as special cases. Furthermore, we derive sharp upper bounds for the partial NCN, MCN and CCN, which are computationally efficient and free of expensive Kronecker products. By applying perturbations that preserve the structure of the block matrices of the DSPPs, we analyze the structured partial NCN, MCN and CCN when the block matrices exhibit linear structures. By leveraging the relationship between DSPP and equality constrained indefinite least squares (EILS) problems, we recover the partial CNs for the EILS problem. Numerical results confirm the sharpness of the derived upper bounds and demonstrate their effectiveness in estimating the partial CNs.