Stable Well-posedness and Tilt stability with respect to an admissible function
Stable Well-posedness and Tilt stability with respect to an admissible function
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相对于容许函数的稳定适定性和倾斜稳定性
DOI：
10.1051/cocv/2016067
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
Jiangxing Zhu
中科院分区：
文献类型：
--
作者：
Xi Yin Zheng;Jiangxing Zhu
In this paper, adopting an admissible function $\varphi$, we introduce and study the stable well-posedness and $\varphi$-tilt-stable local minimum for a proper lower semicontinuous extended real-valued function $f$ on a Banach space when $f$ undergoes tilt perturbations. We prove that $f$ has stable well-posedness at $\bar x$ with respect to an admissible function $\varphi$ if and only if $f$ gives a $(\varphi')^{-1}$-tilt-stable local minimum at $\bar x$. In terms of an admissible function $\psi$, we also consider the metric $\psi$-regularity of the subdifferential mapping $\partial f$. We prove that metric $\varphi'$-regularity of $\partial f$ at $(\bar x,0)$ is a sufficient condition for $f$ to stable well-posedness at $\bar x$ with respect to $\varphi$ and that metric $\varphi'$-regularity of $\partial\overline{\rm co}(f+\delta_{B[\bar x,r]})$ at $(\bar x,0)$ for some $r>0$ is a necessary condition for $f$ to have stable well-posedness at $\bar x$ with respect to $\varphi$. Moreover, we consider the corresponding issues for the non-isolated minimizer case. In the special case when $\varphi(t)=t^2$, our results reduce to some existing main results on the tilt stability in Poliquin and Rockafellar's sense.
