Stable local minimizers

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.

• I am waiting for the complete solution for my question. Thank you for all kind comments. – Blind May 10 '17 at 1:13

I think the function $f(x) = x^4$ is satisfying stable well posed local minimizer condition at $x=0$ but not Tilt stability. Am I right?
• You are right. The point $\bar{x}=0$ is the strict stable local minimizer of $f(x)=x^4$ and so it is also a well posed local minimizer of $f$ (see the above remark). It is easy to check that $\bar{x}$ is not a stable Lipschitz (strong) local minimizer of $f$. How about the remain counterexamples? Do you have any ideas about them? – Blind Apr 21 '17 at 18:47