Hölder Stable Minimizers, Tilt Stability, and Hölder metric Regularity of Subdifferentials
Xi Yin Zheng, Kung Fu Ng · SIAM Journal on Optimization · 2015
Using techniques of variational analysis and dual techniques for smooth conjugate functions, for a local minimizer of a proper lower semicontinuous function $f$ on a Banach space, $p\in (0,\;+\infty)$ and $q=\frac{1+p}{p}$, we prove that the following two properties are always equivalent: (i) $\bar x$ is a stable $q$-order minimizer of $f$ and (ii) $\bar x$ is a tilt-stable $p$-order minimizer of $f$. We also consider their relationships in conjunction with the $p$-order strong metric regularity of the subdifferential mapping $\partial f$.