We study the computability of the operator norm of a matrix with respect to norms induced by linear operators. Our findings reveal that this problem can be solved exactly in polynomial time in certain situations, and we discuss how it can be approximated in other cases. Along the way, we investigate the concept of push-forward and pull-back of seminorms, which leads us to uncover novel duality principles that come into play when optimizing over the unit ball of norms.

翻译：我们研究了由线性算子诱导的范数下矩阵算子范数的可计算性。我们的研究结果表明，在某些情况下该问题可在多项式时间内精确求解，并讨论了其他情况下的近似方法。在此过程中，我们探讨了半范数的前推与拉回概念，这引导我们发现了在范数单位球上优化时发挥作用的新对偶原理。