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.

翻译：我们阐述并给出矩阵数值半径及其对偶范数的半定规划刻画的完整自包含证明。我们证明，在给定数据及精度参数$|\log \varepsilon|$条件下，利用椭球法或短步原始内点法可在多项式时间内计算$\varepsilon$精度的数值半径及其对偶范数。我们应用所得结果，给出$2\times n\times m$阶实张量的谱范数和核范数关于数值半径及其对偶范数的简单表达式。