Computational complexity of real structured singular value in ℓp setting

This paper studies the structured singular value (μ) problem with real parameters bounded by an ℓ_p norm. Our main result shows that this generalized μ problem is NP-hard for any given rational number p ϵ [1, ∞], whenever k, the size of the smallest repeated block, exceeds 1. This result generalizes the known result that the conventional μ problem (with p - ∞) is NP-hard. However, our proof technique is different from the known proofs for the p - ∞ case as these proofs do not generalize to p ≠ ∞. For k - 1 and p - ∞, the μ problem is known to be NP-hard. We provide an alternative proof of this result. For k = 1 and p finite the issue of NP-hardness remains unresolved. When every block has size 1, and p - 2 we outline some potential difficulties in computing μ.