# What is the most intuitive explanation for the concepts of weak local minimizer, strong local minimizer and isolated local minimizer?

Reading Nocedal's book on optimization, I came into the concept of local minimizer, which is a well-known concept in numerical optimization. However, I think I am having a rough time trying to come up with an intuitive understanding of the different types of local minimizer. Let us recall the following three definitions:

Def. A point $x^*$ is a weak local minimizer of $f$ if there is a neighborhood $\mathcal{N}$ of $x^*$ such that $f(x^*) \leq f(x)$ for all $x \in \mathcal{N}$.

Def. A point $x^*$ is a strong local minimizer of $f$ if there is a neighborhood $\mathcal{N}$ of $x^*$ such that $f(x^*) < f(x)$ for all $x \in \mathcal{N}$ with $x \neq x^*$.

Def. A point $x^*$ is an isolated local minimizer of $f$ if there is a neighborhood $\mathcal{N}$ of $x^*$ such that $x^*$ is the only local minimizer in $\mathcal{N}$.

Intuitively, a weak local minimizer is a point that achieves the smallest value of $f$ in its neighborhood. A strong local minimizer can be thought off as (to quote Nocedal et al.) "the outright winner in its neighborhood".

But I am failing to see an intuition behind the isolated local minimizer. How is a strong local minimizer not an isolated one, if in the neighborhood, it is the only minimizer?

I'll quote Remark 1.1 from this book:

"Isolated" implies "strict", but the inverse is not true. For example, let $f(x)$ be $$f(x) = \begin{cases} 0, & x = 0, \\ x^4 (\cos (1/x) + 2), & x \ne 0. \end{cases}$$ Here, $x^\ast = 0$ is "strict" but not "isolated".

The "strict minimizer" used in the book corresponds to the strong minimizer according to your definition. The point $x^\ast$ is the global minimum, but there are other local minima in every of its neighborhood.