KL property – Optimization Online

The complexity of first-order optimization methods from a metric perspective

Published: 2023/05/04

Nonsmooth Optimization, Other Topics complexity, first order algorithms, KL property, nonsmooth optimization, semi-algebraic, slope

A central tool for understanding first-order optimization algorithms is the Kurdyka-Lojasiewicz inequality. Standard approaches to such methods rely crucially on this inequality to leverage sufficient decrease conditions involving gradients or subgradients. However, the KL property fundamentally concerns not subgradients but rather “slope”, a purely metric notion. By highlighting this view, and avoiding any use of … Read more

Identifiability, the KL property in metric spaces, and subgradient curves

Published: 2022/05/05

Adrian Lewis

Tonghua Tian

Nonsmooth Optimization active manifold, identification, KL property, partly smooth, subgradient descent, variational analysis

Identifiability, and the closely related idea of partial smoothness, unify classical active set methods and more general notions of solution structure. Diverse optimization algorithms generate iterates in discrete time that are eventually confined to identifiable sets. We present two fresh perspectives on identifiability. The first distills the notion to a simple metric property, applicable not … Read more