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.

翻译：本文提出了一个统一的框架来研究双鞍点问题解的部分条件数，并给出了其闭式表达式。该统一框架将著名的部分范数条件数、部分混合条件数和部分分量条件数作为特例包含其中。进一步地，我们推导了部分范数条件数、部分混合条件数和部分分量条件数的尖锐上界，这些上界计算高效且无需昂贵的Kronecker积运算。通过施加保持双鞍点问题分块矩阵结构的扰动，我们分析了当分块矩阵呈现线性结构时的结构化部分范数条件数、部分混合条件数和部分分量条件数。利用双鞍点问题与等式约束不定最小二乘问题之间的关系，我们恢复了等式约束不定最小二乘问题的部分条件数。数值结果验证了所推导上界的尖锐性，并证明了其在估计部分条件数方面的有效性。