The structured distance to ill-posedness for conic systems

TL;DR

The paper addresses conditioning of conic linear systems by defining the distance to ill-posedness as the smallest structured perturbation $F+\sum_i P_i T_i Q_i$ that makes the system nonsurjective. It develops a rank-one reduction within a sublinear set-valued mapping framework and proves a duality-based series of equivalences, culminating in four equal characterizations of the structured distance to nonsurjectivity, with attainment results when finite. By extending Eckart-Young and connecting to structured singular values and $\mu$-analysis, the work provides a unified, duality-driven theory for assessing how small structured perturbations can destroy well-posedness in conic systems. The results hold for general norms and do not require lifting to higher dimensions, making the theory broadly applicable to conditioning analysis in numerical linear algebra and control theory.

Abstract

An important measure of conditioning of a conic linear system is the size of the smallest structured perturbation making the system ill-posed. We show that this measure is unchanged if we restrict to perturbations of low rank. We thereby derive a broad generalization of the classical Eckart-Young result characterizing the distance to ill-posedness for a linear map.

Theorems & Definitions (7)

• Theorem 2.1: rank-one reduction
• Corollary 2.2: rank-one reduction for sums
• Theorem 3.2: open mapping
• Theorem 3.5: rank reduction and surjectivity
• Theorem 4.1: theorem of the alternative
• Theorem 4.7: duality
• Theorem 5.1: distance to nonsurjectivity