I'm studying the paper [1] and trying to understand the primal-dual problems introduced in the middle of page 4 of [1]. Below is the problem in question:
Let $A$ be a fixed $n \times m$ matrix, where each element is positive. Consider the following primal problem:
\begin{align*} \min \quad & -\sum_{i=1}^n \log(g_i) \\ \mathrm{subject~to} \quad & Af = g, \\ & 1_m^\top f = 1, \\ & f \succeq 0, \end{align*} where $\succeq$ denotes elementwise inequality.
The authors claim that the corresponding dual problem becomes \begin{align*} \max_{\nu \in \mathbb R^n} \quad & \sum_{i=1}^n \log(\nu_i) \\ \mathrm{subject~to} \quad & A^\top \nu \preceq n 1_m, \\ & \nu \succeq 0. \end{align*}
Unfortunately, I cannot derive the primal to dual. Could anyone provide ideas or hints for converting this primal problem to its dual form? Thank you for any hints or suggestions.
[1] Koenker, R., & Gu, J. (2017). REBayes: An R Package for Empirical Bayes Mixture Methods. Journal of Statistical Software, 82(8), 1–26. https://doi.org/10.18637/jss.v082.i08
