On the variational behaviour of functions with positive steepest descent rate

TL;DR

This paper explores the variational properties of nonsmooth functions using the steepest descent rate, linking stability phenomena like sharp minimality and error bounds, with characterizations in metric and Banach spaces.

Contribution

It introduces a unified analysis of stability properties of nonsmooth functions via the steepest descent rate, extending known results in Banach spaces.

Findings

Positivity of the steepest descent rate characterizes stability properties.

Relationships between sharp minimality, superstability, and error bounds are established.

Characterizations in metric spaces generalize known Banach space results.

Abstract

This paper investigates some aspects of the variational behaviour of nonsmooth functions, with special emphasis on certain stability phenomena. Relationships linking such properties as sharp minimality, superstability, error bound and sufficiency of first-order optimality conditions are discussed. Their study is performed by employing the steepest descent rate, a rather general tool, which is adequate for a metric space analysis. The positivity of the steepest descent rate is then characterized in terms of $\Phi$-subdifferentials. If specialized to a Banach space setting, the resulting characterizations subsume known results on the stability of error bounds.