Let $x$ and $y$ be variables in $\mathbb{R}^{+}$, and assume I have a system of inequalities, e.g.:
$$ \begin{align} x \geq 1 \tag{1}\\ x \leq 5 \tag{2}\\ y \geq 1 \tag{3}\\ \log x + \log y \leq \log 6 \tag{4}\\ \end{align} $$
where some of the inequalities (here: 4) are nonlinear but convex and monotone, such as the sum of log variables above. I have plotted this example below. This would be a pretty straightforward problem if I only had the linear inequalities (1) to (3). In this instance, I could for example use the simplex algorithm. However, the presence of the nonlinear inequality (4) complicates matters.
My question: I am interested in a method or an algorithm that finds the vertices of the feasible region. Is this possible? If it is possible, can you recommend a method or some keywords for further study? If it is not possible, why?
(Mind that I am not interested in solving an optimization problem on the polytope - I only want the vertices of the feasible region.)
You already have two vertices: (1,1) and (5,1). You can find the other two by plugging in x=1 and x=5 into (4).
Let’s try x=1. $\log(1)=0$ so (4) becomes $\log(y) \leq \log(6)$, which is true iff $y\leq 6$. So the vertex is (1,6). Do the same with x=5 and you’ve found the four vertices.