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