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Extending zero-duality-gap non-Euclidean S-Lemma results to ℓ∞-norm contractivity

Establish necessary and sufficient zero-duality-gap conditions for a non-Euclidean S-Lemma that certify contractivity with respect to the ℓ∞ norm, extending the existing results known for the ℓ1 norm with Metzler matrices.

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Background

The paper discusses recent advances on non-Euclidean S-Lemma formulations that yield necessary and sufficient conditions for zero duality gap in the ℓ1 norm with Metzler matrices.

A key unresolved issue is whether analogous zero-duality-gap results can be obtained to certify contractivity with respect to the ℓ∞ norm, which would broaden the computational toolkit for non-Euclidean contraction analysis.

References

For contraction with respect to non-Euclidean norms, a non-Euclidean S-Lemma was recently proposed in and necessary and sufficient conditions for zero duality gap were shown in the case of the $\ell_1$ norm with Metzler matrices. It remains an open problem whether such results can be extended to other important cases such as contractivity with respect to the $\ell_\infty$ norm.

— Perspectives on Contractivity in Control, Optimization, and Learning (2404.11707 - Davydov et al., 17 Apr 2024) in Section 6 (Conjectures and Future Directions), Theory