What is the meaning of $\inf$?

Question

I see it often in solution of some questions:

$$ \inf_x f(x,y) $$

as a next step of the solution usually I see something like this:

$$ \nabla_xf=0 $$

and then return $ x $ that satisfy it.

Answer 1

The infimum of a set of numbers is the greatest lower bound. It it similar to, but not always the same as, the minimum.

The problem with minimum is that it might not exist. A standard example from early analysis courses is $$ 1+\frac{1}{n} \quad \forall n \in {\mathbb N}$$ which yields the set $\{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \frac{6}{5}, \ldots \}$ What is the minimum of this set? There is none: take any value you like for $n$ and someone else can take the value $n+1$ to find a smaller number than yours that is also in the set.

The infimum though does exist: it's $1$. This is because no matter how large you take $n$ to be, $(1+1/n) > 1$, but you can get as close as you like to $1$. If you want to be no more than $10^{-17}$ away from $1$ just take $n=10^{17}$ or larger and you're there.

Infima (or infimums if you prefer modern English plurals) arise in minimisation problems as your example suggests: if you have a compact set and a continuous function defined on it, then the infimum will also be the minimum. Even if you don't, knowing what the infimum looks like bounds any possible solution to the minimisation problem and may allow you to describe a family of near-solutions to it.

Finally: in the real numbers infima don't need to exist: consider $1+n\cdot (-1)^n$ This has no infimum and will tend towards both positive and negative infinity. However, in some circumstances (convex analysis especially) we extend the real numbers by adding $\pm \infty$ and then the infimum can be $-\infty$. It can also be useful, when dealing with upper- and lower- semicontinuous functions to have them take $\pm \infty$ as values to indicate we are outside the range of interest.