# Inertial Proximal Alternating Linearized Minimization (iPALM) for Nonconvex and Nonsmooth Problems

## TL;DR

This paper introduces an inertial version of the PALM algorithm for nonconvex, nonsmooth optimization problems with semi-algebraic data, proving its global convergence and demonstrating its effectiveness through numerical experiments.

## Contribution

It develops and analyzes an inertial PALM algorithm with proven global convergence for complex nonconvex, nonsmooth problems, extending existing methods.

## Key findings

- Proves global convergence of inertial PALM to critical points.
- Demonstrates effectiveness in blind image deconvolution.
- Shows improved performance in sparse matrix factorization.

## Abstract

In this paper we study nonconvex and nonsmooth optimization problems with semi-algebraic data, where the variables vector is split into several blocks of variables. The problem consists of one smooth function of the entire variables vector and the sum of nonsmooth functions for each block separately. We analyze an inertial version of the Proximal Alternating Linearized Minimization (PALM) algorithm and prove its global convergence to a critical point of the objective function at hand. We illustrate our theoretical findings by presenting numerical experiments on blind image deconvolution, on sparse non-negative matrix factorization and on dictionary learning, which demonstrate the viability and effectiveness of the proposed method.