Explanation of "unbounded below"

Question

In Boyd & Vandenberghe, I found this part:

It would be really helpful if someone could explain clearly the above terms $\nu^T Ax$ and $\nu^T y$ with examples. Thanks.

Answer 1

An function $f(x)$ is unbounded below if we can make it as negative as we like by choosing appropriate values of $x$ (i.e. there is no constant $K$ so that $f(x) \geq K$ for all $x$).

For example, $f(x)=\nu^TAx$ is a linear function. If we have an $x_0$ for which it has a nonzero value, $f(-x_0)=-f(x_0)$ is smaller than zero, and then if we take $a$ positive, $f(-ax_0)=-af(x_0)$ can be made as negative as we like by choosing $a$ large enough.

For the other term, $-\log{y_i}$ is unbounded below by itself (take $y_i \to \infty$). But $\log{y_i}$ tends to $\infty$ slower than any positive power of $y_i$, and so in particular $-\log{y_i}+\nu_i y_i$ tends to $\infty$ for both $y \to 0$ and $y \to \infty$, if and only if $\nu_i>0$. Since it is continuous, the infimum exists and is attained. One then finds the minimum by differentiation: $-1/y_i+\nu_i=0$.