Quadratic growth and critical point stability of semi-algebraic functions

TL;DR

This paper establishes an equivalence between quadratic growth and strong metric subregularity of the subdifferential for semi-algebraic functions, providing new conditions for optimality and highlighting the importance of semi-algebraic structure.

Contribution

It demonstrates the equivalence between quadratic growth and subdifferential stability specifically for semi-algebraic functions, which was previously not well-understood.

Findings

Quadratic growth is equivalent to strong metric subregularity of the subdifferential for semi-algebraic functions.

Outside semi-algebraic functions, this equivalence may not hold.

Provides necessary and sufficient subdifferential-based conditions for optimality.

Abstract

We show that quadratic growth of a semi-algebraic function is equivalent to strong metric subregularity of the subdifferential --- a kind of stability of generalized critical points. In contrast, this equivalence can easily fail outside of the semi-algebraic setting. As a consequence, we derive necessary conditions and sufficient conditions for optimality in subdifferential terms.