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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.

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  • $\begingroup$ I am waiting for the complete solution for my question. Thank you for all kind comments. $\endgroup$ – Blind May 10 '17 at 1:13
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Just for your information here stable Lipschitz local minimizer is known as tilt stability in literatures.

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?

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    $\begingroup$ this is a comment, not an answer $\endgroup$ – Solomonoff's Secret Apr 21 '17 at 16:55
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    $\begingroup$ How do you define an answer? He/She asked for four counter examples , and I did !! That is a counter example for cases 1 and 4 . !!!!!!!!!!!!!!!!!!!!!!!!!!! $\endgroup$ – Red shoes Apr 21 '17 at 17:00
  • $\begingroup$ 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? $\endgroup$ – Blind Apr 21 '17 at 18:47
  • $\begingroup$ @Solomonoff'sSecret Do you still think my post is a comment :)) $\endgroup$ – Red shoes Apr 21 '17 at 23:29
  • $\begingroup$ @nonlinearthought Note that I did not downvote you. $\endgroup$ – Solomonoff's Secret Apr 22 '17 at 2:30

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