# dual norm optimization problem, how to dervie the second formula from the first one?

There is a post that someone answered it already, but I think his/her answer is how to validate the formula, not the way to derive the formula. A dual norm optimization problem

This formula comes from the paper : https://arxiv.org/pdf/2010.01412.pdf, it's about SAM optimizer.

$$\epsilon^*(w)= \underset{ ||\epsilon||_p<=\rho }{argmax}$$ $${\epsilon^T\nabla_wL_S(w)}$$ (1)

With classical dual norm prolem and the solution is:

$$\hat{\epsilon}(w)=\rho sign(\nabla_WL_S(w))|\nabla_WL_S(w)|^{q-1}/(||\nabla_wL_S(w)||_q^q)^{1/p}$$ (2)

when $$1/p + 1/q = 1$$ and $$|.|^{q-1}$$ denotes elementwise absolute value and power.

What I understand is I can use Holder's inequality to derive that

$$\epsilon^Tx<=||\epsilon||_p ||x||_q<=\rho||x||_q$$ when $$x = \nabla_wL_S(w)$$

So it seems to me the formula should be

$$\epsilon^*(w)=\underset{ ||\epsilon||_p<=\rho }{argmax}$$ $$\rho||x||_q$$

I don't know where sign and $${||\nabla_WL_S(w)||_q^q}^{1/p}$$ come from. Can anyone show me how I can dervie first formula to the second one?