How to linearize the square of a positive continuous variable

Question

Suppose we have a positive continuous variables $0 \le x \le UB$ where $UB$ is a known upper bound.

How can we linearize the term $x^2$?

Answer 1

I'm not sure in which context you're working. To linearize the function $f(x) = x^2$ at $a \in [0, UB]$ (see here), use:

$f(a) + f'(a)(x-a) \le f(x), \forall x \in [0, UB]$

Since $f$ is convex, the linearization is a lower bound.