Consider the following cone:
$$\mathbb{G}_n=\Bigg\{\,(x\oplus\theta\oplus\kappa) \in\mathbb{R}^n\oplus\mathbb{R}_+\oplus\mathbb{R}_+\,\colon \theta\sum_{i\in [n]}\exp\bigg(\frac{-x_i}{\theta}\bigg)\leq\kappa \Bigg\}.$$
Where $0\exp(\frac{\alpha}{0})=0$ for each $\alpha\in\mathbb{R}$.
I am trying to calculate the dual of $\mathbb{G}_n$. A similar result is presented in https://tel.archives-ouvertes.fr/tel-00006861/document Sec 6.3 and my attempt is to adapt it to my case. However, I found the following problems:
1- While considering the case where $\theta=0$, we still have that $(x,0,\kappa)\in\mathbb{G}_n$ for each $(x,\kappa)\in\mathbb{R}^n\oplus\mathbb{R}_+$. Because $x$ is not necessairly nonnegative, I couldn't obtain any corresponding sign constraints on $x^\ast$ and $\kappa^\ast$ (besides $x^\ast=0$ and $\kappa^\ast\geq0$).
2- While minimizing a single term of the summation, I would obtain that $t_i=−\log(\frac{|x_i^\ast|}{\kappa^\ast})$ because there is no $\log$ for negative numbers, is this right?
3- Why is the minimum equal to zero when $\kappa^\ast=0$?
4- The two final cases colapse and then we obtain that the minimum is $−x_i^\ast\log(\frac{x_i^\ast}{\kappa^\ast})$ either way?
5- Finally, we conclude that $x^\ast\in$?, $\kappa^\ast\in$? and $\theta^\ast\geq\sum_i x_i^\ast\log(\frac{x_i^\ast}{\kappa^\ast})−x_i^\ast$. Right?
Can anyone help me to fix these?