On semidefinite programming characterizations of the numerical radius and its dual norm

TL;DR

This paper provides semidefinite programming characterizations for the numerical radius and its dual norm, demonstrating polynomial-time computability and applying these results to tensor spectral and nuclear norms.

Contribution

It introduces semidefinite programming characterizations for the numerical radius and its dual norm, with proofs and polynomial-time algorithms, and applies these to tensor norm computations.

Findings

Numerical radius and dual norm are polynomially computable within any precision.

Semidefinite programming characterizations are self-contained and rigorously proved.

Application to tensor norms provides new formulas for spectral and nuclear norms.

Abstract

We state and give self contained proofs of semidefinite programming characterizations of the numerical radius and its dual norm for matrices. We show that the computation of the numerical radius and its dual norm within $\varepsilon$ precision are polynomially time computable in the data and $|\log \varepsilon |$ using either the ellipsoid method or the short step, primal interior point method. We apply our results to give a simple formula for the spectral and nuclear norm of $2\times n\times m$ real tensor in terms of the numerical radius and its dual norm.