# Penalty & Augmented Kaczmarz Methods For Linear Systems & Linear Feasibility Problems

arXiv: https://arxiv.org/abs/2205.08085 · HTML: https://mlsanity.com/paper/2205.08085/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2022.

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

- [Augmented Newton Method for Optimization: Global Linear Rate and Momentum Interpretation](https://mlsanity.com/paper/2205.11033/) (2022)
- [ALS: Augmented Lagrangian Sketching Methods for Linear Systems](https://mlsanity.com/paper/2208.06152/) (2022)

## Abstract (arXiv)

In this work, we shed light on the so-called Kaczmarz method for solving Linear System (LS) and Linear Feasibility (LF) problems from a optimization point of view. We introduce well-known optimization approaches such as Lagrangian penalty and Augmented Lagrangian in the Randomized Kaczmarz (RK) method. In doing so, we propose two variants of the RK method namely the Randomized Penalty Kacmarz (RPK) method and Randomized Augmented Kacmarz (RAK) method. We carry out convergence analysis of the proposed methods and obtain linear convergence results.