Quadratic Growth and Strong Metric Subregularity of the Subdifferential for a Class of Non-prox-regular Functions

TL;DR

This paper explores the relationship between quadratic growth and strong metric subregularity of the subdifferential for a class of non-prox-regular functions, providing new characterizations and properties.

Contribution

It establishes the equivalence of quadratic growth, strong metric subregularity, and positive definiteness of the subgradient graphical derivative for a broad class of functions.

Findings

Quadratic growth and strong metric subregularity are equivalent under certain conditions.

Characterizations of quadratic growth and subregularity are provided.

Properties of extended twice differentiable functions are examined.

Abstract

This paper mainly studies the quadratic growth and the strong metric subregularity of the subdifferential of a function that can be represented as the sum of a function twice differentiable in the extended sense and a subdifferentially continuous, prox-regular, twice epi-differentiable function. For such a function, which is not necessarily prox-regular, it is shown that the quadratic growth, the strong metric subregularity of the subdifferential at a local minimizer, and the positive definiteness of the subgradient graphical derivative at a stationary point are equivalent. In addition, other characterizations of the quadratic growth and the strong metric subregularity of the subdifferential are also given. Besides, properties of functions twice differentiable in the extended sense are examined.