Q: Stable local minimizers of functions on a Banach space Let $X$ be a Banach space and $f:X\rightarrow (-\infty,\infty]$ be a lower semicontinuous function. We are interested in  some conceptions of local minimizer:

*

*We say that  $\bar{x}\in X$ is a stable strict local minimizer of $f$ if $f(\bar{x})$ is finite and there exists $\varepsilon>0$ such that the mapping
$$
 M:x^*\mapsto\text{argmin}_{\|x-\bar{x}\|\leq\varepsilon}\{f(x)-\langle x^*,x\rangle\}
 $$
from $X^*$ to $X$ is single-valued on some neighborhood of $0$ with $M(0)=\bar{x}$.


*We say that  $\bar{x}\in X$ is a stable well posed local minimizer of $f$ if $f(\bar{x})$ is finite and there exists $\varepsilon>0$ such that for every vector $x^*\in X^*$ near the origin, there is a point $x_{x^*}$ in $U:=\overline{B}(\bar{x},\varepsilon)$ with $x_0=\bar{x}$ so that in terms of the perturbed functions $f_{x^*}(\cdot):=f(\cdot)-\langle x^*,\cdot\rangle$ we have
$$
 f_{x^*}(x_{x^*})=\inf_{x\in U}f_{x^*}(x)
 $$
and every sequence $(x_n)\subset U$ along which $f_{x^*}(x_n)\rightarrow f_{x^*}(x_{x^*})$ obeys $\|x_n-x_{x^*}\|\rightarrow 0$.


*We say that  $\bar{x}\in X$ is a stable strong local minimizer of $f$ if $f(\bar{x})$ is finite and there exist $\varepsilon>0$ and $\kappa>0$ such that for every vector $x^*\in X^*$ near the origin, there is a point $x_{x^*}$ in $U:=\overline{B}(\bar{x},\varepsilon)$ with $x_0=\bar{x}$ so that in terms of the perturbed functions $f_{x^*}:=f(\cdot)-\langle x^*,\cdot\rangle$ the inequality
$$
 f_{x^*}(x)\geq f_{x^*}(x_{x^*})+\kappa\|x-x_{x^*}\|^2
 $$
holds for every $x\in U$.


*A point $\bar{x}\in X$ is called a stable Lipschitz local minimizer of the function $f$ if $f(\bar{x})$ is finite and there exists  $\varepsilon>0$ such that the mapping
$$
 M:x^*\mapsto\text{argmin}_{\|x-\bar{x}\|\leq\varepsilon}\{f(x)-\langle x^*,x\rangle\}
 $$
is single-valued and Lipschitz continuous on some neighborhood of $0$ with $M(0)=\bar{x}$.
We have observed that:

*

*stable strong local minimizer $\Rightarrow$ stable well posed local minimizer $\Rightarrow$ stable strict local minimizer.


*stable strong local minimizer $\Rightarrow$ stable Lipschitz local minimizer $\Rightarrow$ stable strict local minimizer.


*When $X$ is finite dimensional,  stable strict local minimizer $\Leftrightarrow$ stable well posed local minimizer.


*When $X$ is finite dimensional,  stable strong local minimizer $\Leftrightarrow$ stable Lipschitz local minimizer.
We would like to construct:

*

*a stable well posed local minimizer of $f$ which is not a stable strong local minimizer of $f$


*a stable strict local minimizer of $f$ which is not a stable well posed local minimizer of $f$


*a stable Lipschitz local minimizer of $f$ which is not a stable strong local minimizer of $f$


*a stable Lipschitz local minimizer of $f$ which is not a stable well posed local minimizer of $f$


*a stable well posed local minimizer of $f$ which is not a stable Lipschitz local minimizer of $f$
Thank you for all kind help and comments.
