# 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.

• 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? Apr 21 '17 at 16:48
• I am waiting for the complete solution for my question. Thank you for all kind comments. May 10 '17 at 1:13