Local minimizer ($f(x_{\ast })\le f(x)(x\in D\cap U)$)

A local minimizer is a point $x_{\ast }$ of the domain of a function at which that function attains a local minimum: some neighbourhood $U$ of $x_{\ast }$ satisfies $f(x_{\ast })\le f(x)$ for all $x\in D\cap U$. The point and the attained value are different objects. Every global minimizer is a local minimizer, but the converse need not hold.