I am struggling to calculate the dual of the following conic problem:
$$\inf\{a\lambda_a+b\lambda_b\,\colon w_1,w_2,\lambda_a,\lambda_b\in\mathbb{R}, (1,\lambda_a,w_2-w_1)\in\mathbb{G}_1 \text{, and} (1,\lambda_b,w_2-w_1)\in\mathbb{G_1} \}.$$
Where $a$ and $b$ are fixed real scalars and $\mathbb{G}_1$ is defined as:
$$\mathbb{G}_=\Bigg\{\,(x\oplus\theta\oplus\beta) \in\mathbb{R}^1_+\oplus\mathbb{R}_+\oplus\mathbb{R}_+\,\colon \theta\sum_{i\in [n]}\exp\bigg(\frac{-x}{\theta}\bigg)\leq\beta \Bigg\}.$$ Where we consider $0\exp(\frac{\alpha}{0})=0$ for each $\alpha\in\mathbb{R}_+$.
I found a formula describing the dual cone of $\mathbb{G}_1$ in:
https://tel.archives-ouvertes.fr/tel-00006861/document
But don't know how to use it.
Any help is appreciated!