Friends, I am reading the book Convex Optimization by Boyd and I have encountered the inequality
$$f(\theta x+(1-\theta) y) \leq \theta f(x)+(1-\theta) f(y)$$
for $0 \leq \theta \leq 1$ and any $x$ and $y$.
When $x \notin \text{dom} f$, $f(x) = \infty$. So when both sides of the inequality are $\infty$, how are we able to compare?
In the text book, there is a sentence to explain; "Of course here we must interpret the inequality using extended arithmetic and ordering." What is "extended arithmetic and ordering"?
The extended reals is the real number set $ \mathbb{R} $ with the addition of two new symbols $ \infty $ and $ - \infty $. With axioms for these two symbols so it works in a consistent way.