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Computing Robustness to Sparse Diagonal Uncertainty

  • Christoffer Kjellson
  • Emma Tegling
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Abstract

A new robustness metric ν was recently proposed as a substitute for the structured singular value μ to better capture robustness to sparse diagonal uncertainty, but its computation has remained an open problem. In this paper, we show that computing ν is equivalent to maximizing the spectral radius of a nonnegative matrix product. This equivalence allows us to transfer existing results on spectral-radius maximization to ν, including a refined upper bound and conditions under which the bounds coincide. We then provide reformulations and structural results that enable an algorithm to solve the nonconvex optimization problem for nontrivial problems using global solvers. This also enables identification of the most fragile parts of the system. Our results do not yet provide a scalable solution for computing ν, but they are an important step toward computability and interpretability.

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Cite

BibTeXRIS
BibTeX
@misc{kjellson2026computing,
  title = {{Computing Robustness to Sparse Diagonal Uncertainty}},
  author = {Christoffer Kjellson and Emma Tegling},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14445},
  eprint = {2609.14445},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14445},
}
RIS
TY  - UNPB
TI  - Computing Robustness to Sparse Diagonal Uncertainty
AU  - Christoffer Kjellson
AU  - Emma Tegling
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14445
UR  - https://arxiv.org/abs/2609.14445
AB  - A new robustness metric ν was recently proposed as a substitute for the structured singular value μ to better capture robustness to sparse diagonal uncertainty, but its computation has remained an open problem. In this paper, we show that computing ν is equivalent to maximizing the spectral radius of a nonnegative matrix product. This equivalence allows us to transfer existing results on spectral-radius maximization to ν, including a refined upper bound and conditions under which the bounds coincide. We then provide reformulations and structural results that enable an algorithm to solve the nonconvex optimization problem for nontrivial problems using global solvers. This also enables identification of the most fragile parts of the system. Our results do not yet provide a scalable solution for computing ν, but they are an important step toward computability and interpretability.
ER  -