Local minimum ($f(x)\ge f(x_{\ast })\text{near }{x}_{\ast }$)

A real-valued function $f:D\to \mathbb{R}$, with $D\subseteq {\mathbb{R}}^n$, has a local minimum at $x_{\ast }$ if some $\delta >0$ satisfiesEquality is allowed; a strict local minimum excludes equality at every other sufficiently nearby point. The comparison uses a neighbourhood relative to the domain of a function. On an open set, this is an ordinary small ball around $x_{\ast }$. At an endpoint, only points belonging to the domain of a function are tested. Some elementary texts reserve "local" for interior points; here endpoints are allowed under the relative convention, but the zero-gradient condition still requires an interior point.

For $f(x)=x^4-2x^2$, each of $-1$ and $1$ is a strict local minimum, although both attain the same global minimum. For $h(x)=|x|$, zero is a strict local minimum despite the missing derivative. Thus the definition compares values and does not require differentiability.

A local minimum describes a valley that cannot be improved by a sufficiently small move; mathematical optimization also asks whether a better value lies farther away. At a smooth interior local minimum, the gradient is zero. If the function is twice continuously differentiable nearby, its Hessian matrix is positive semidefinite: restrict to a line through the point and its second derivative is nonnegative. The converse fails for a degenerate Hessian matrix, as $-x^4$ at zero shows.

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.

A local minimum is strict when every other point of the domain of a function in some neighbourhood has a larger value. Strictness is local: two separated strict local minima can have the same value, as $x^4-2x^2$ at $\pm 1$ demonstrates. Negating the function gives a strict local maximum.