Local minimizer
= Local minimizer
{title2=$f(x_*)\leq f(x)\quad(x\in D\cap U)$}
A <local minimizer> is a point $x_*$ of the <domain of a function> at which that <function> attains a <local minimum>: some <neighbourhood> $U$ of $x_*$ satisfies $f(x_*)\leq f(x)$ 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.