I am trying to calculate the dual of some cones that I don't know 'a priori'.
For example, looking at MOSEK https://docs.mosek.com/MOSEKModelingCookbook-letter.pdf it seems that he already know the formula describing the dual cone and then proves that his guess is right. But and if I do not have any guess?
For example, how could I explictly calculate the dual of the following proper cone?
$$\mathbb{H}_n = \Bigg\{\,(x\oplus y\oplus \delta)\in\mathbb{R}^n_+\oplus\mathbb{R}_+^n\oplus\mathbb{R}^n\,\colon x_i\log\bigg(\frac{x_i}{y_i}\bigg)\leq\delta_i,\text{ for each } i\in [n]\Bigg\}.$$