# Subregularity of subdifferential mappings relative to the critical set and KL property of exponent 1/2

arXiv: https://arxiv.org/abs/1812.00558 · HTML: https://mlsanity.com/paper/1812.00558/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2018.

## Built on by (most cited first, 2 of 2)

- [Distributed Sparse Regression via Penalization](https://mlsanity.com/paper/2111.06530/) (2021)
- [Enhancing Convergence of Decentralized Gradient Tracking under the KL Property](https://mlsanity.com/paper/2412.09556/) (2024)

## Abstract (arXiv)

For a proper extended real-valued function, this work focuses on the relationship between the subregularity of its subdifferential mapping relative to the critical set and its KL property of exponent 1/2. When the function is lsc convex, we establish the equivalence between them under the continuous assumption on the critical set. Then, for the uniformly prox-regular function, under its continuity on the local minimum set, the KL property of exponent 1/2 on the local minimum set is shown to be equivalent to the subregularity of its subdifferential relative to this set. Moreover, for this class of nonconvex functions, under a separation assumption of stationary values, we show that the subregularity of its subdifferential relative to the critical set also implies its KL property of exponent 1/2. These results provide a bridge for the two kinds of regularity, and their application is illustrated by examples.