Why the definition for optimal value is the $\inf{f_0(x)}$ rather than $\min{f_0(x)}$?

Question

Suppose an optimization problem

\begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & & f_0(x) \\ & \text{subject to} & & f_i(x) \leq b_i, \; i = 1, \ldots, m. \end{aligned} \end{equation*}

Then, the optimal value is defined as

$$p^{\star} = \inf\{f_{0}(x) \: | x \in \mathcal{A} \}$$

where $A$ is the feasible set. My question is that why we use $\inf$ rather than $\min$ for representing the optimal value?

Answer 1

Not every set has a minimum value, but every set that’s bounded below has an infimum value. If you used min instead of inf, you would lose the ability to talk about many functions

Answer 2

As mentioned, the minimum isn't always attainable in the feasible set. In this case you need to look for the infimum.

However, in the context of the problem statement, if $f$ is also a continuous function and $A$ (the obtainable set) is compact, then the minimum and infimum are the same and you can attain the infimum in the attainable set.