# A convergence framework for inexact nonconvex and nonsmooth algorithms and its applications to several iterations

arXiv: https://arxiv.org/abs/1709.04072 · HTML: https://mlsanity.com/paper/1709.04072/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Sep 2017.

## Built on by (most cited first, 1 of 1)

- [Plug-and-Play Methods Provably Converge Even with Improperly Trained Denoisers: Convergence by Architectural Design](https://mlsanity.com/paper/2609.33087/) (2026)

## Abstract (arXiv)

In this paper, we consider the convergence of an abstract inexact nonconvex and nonsmooth algorithm. We promise a pseudo sufficient descent condition and a pseudo relative error condition, which are both related to an auxiliary sequence, for the algorithm; and a continuity condition is assumed to hold. In fact, a lot of classical inexact nonconvex and nonsmooth algorithms allow these three conditions. Under a special kind of summable assumption on the auxiliary sequence, we prove the sequence generated by the general algorithm converges to a critical point of the objective function if being assumed Kurdyka- Lojasiewicz property. The core of the proofs lies in building a new Lyapunov function, whose successive difference provides a bound for the successive difference of the points generated by the algorithm. And then, we apply our findings to several classical nonconvex iterative algorithms and derive the corresponding convergence results