This work studies how a generalized error bound (GEB) property in the ambient Euclidean space transfers to the active manifold under \(\mathcal{C}^1\) partial smoothness. The GEB generalizes the standard error bound, which upper bounds the distance to a subset of critical points using first-order stationarity residuals, by allowing the distance to be composed with a … Read more
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, 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