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I came across a concept called dual norm in my optimization course, which I am not familiar with. I am trying to understand what it is and how to compute it.

from Wiki, the definition of the dual norm is,

Let ${\displaystyle \|\cdot \|}$ be a norm on ${\mathbb {R} ^{n}.}$ The associated dual norm, denoted $\|\cdot\|_{*}$ is defined as $$\|z\|_{*}=\sup \left\{z^{\top} x \mid\|x\| \leq 1\right\}$$

My questions are:

  1. What is this dual norm used for, why do we need it?

  2. Is that the primal norm can be any like 1-norm, 2-norm, inf-norm, and the dual norm is still defined as above?

  3. How to compute this dual norm? Does Python or Matlab provide such a function? If not, I noticed that it is essentially an optimization problem, with an objective function: maximize:$z^{\top} x$, and an inequality constraint $\|x\| \leq 1$, can I compute it in this way?

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  • $\begingroup$ math.stackexchange.com/questions/265721/… $\endgroup$
    – NoNames
    Feb 13, 2021 at 15:57
  • $\begingroup$ Yes the definition of the dual is the same for any norm and actually the dual of the 1-norm is the inf-norm (and vice-versa). You have correctly identified that computing the dual norm can be done by solving a (convex) optimization problem. $\endgroup$
    – Surb
    Feb 13, 2021 at 18:22

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