Let $x_{1},...,x_{N}$ and $y_{1},...,y_{N}$ be scalar-values variables. Now assume that I have two sets of linear inequalities inequalities:
$$c_{i,1}x_{1} + \dots + c_{i,N}x_{N} \leq a_{i} \quad i=1,\dots,M \tag{1} $$ $$c_{j,1}y_{1} + \dots + c_{j,N}y_{N} \leq b_{j} \quad j=1,\dots,K \tag{2}$$
where $N,M,K\in\mathbb{N}^{+}$, and all the $a$, $b$, and $c$ are known.
Goal: I want to find the vertices of the resulting feasible region.
Catch: The $x_n$ and $y_n$ for $n=1,\dots,N$ are not independent, but in fact coupled nonlinearly as $y_n = \log x_n$.
Does this problem have a solution? Do you know of a method that may be able to find the vertices of the resulting feasible region?