Generalized Metric Subregularity and Regularity with Respect to an Admissible Function

Abstract

In this paper, adopting an admissible function $\varphi$, we consider a kind of generalized metric subregularity/regularity of a multifunction $F$ with respect to $\varphi$, which is a natural generalization of the Hölder metric regularity. In the special case when $F$ is the subdifferential mapping of a proper lower semicontinuous function $f$, it is known that such a generalized metric subregularity is very closely related to the well-posedness of $f$. Using the technique of variational analysis and in terms of the coderivative, we established some sufficient conditions for a multifunction to be metrically subregular/regular with respect to an admissible function $\varphi$. In particular, we extend some existing results on the metric regularity and Hölder metric regularity.