Local Minima

Definition (Local Minima)

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a real-valued function. The weak local minima is defined by the weak local minimizer

$$(\exists \epsilon >0)(y\in N_{\epsilon }(x)⟹f(x)\le f(y))$$

for some $\epsilon$ neighborhood¹ around $x$. See here for a more rigorous definition. The strong local minima is defined by the strict/strong local minimizer:

$$(\exists \epsilon >0)(y\in N_{\epsilon }(x)\wedge x\ne y⟹f(x)<f(y))$$

In a Level Set, the local minima is one point (or a set of single points).

In class, it is defined slightly differently but they are logically equivalent.

---

Footnotes

1. For the purposes of this class, a neighborhood is the set of points no more than $\epsilon$ distance from some chosen point $x$. Distance is measured by the Euclidean norm. Sometimes the notation $B(x,\epsilon )$ is used, but they mean the same thing. In ${\mathbb{R}}^2$, $B(x,\epsilon )$ is a filled circle. In ${\mathbb{R}}^3$, $B(x,\epsilon )$ is a solid sphere. ↩