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.

翻译：在这项工作中,我们从最优化的角度来说明所谓的卡茨马尔兹解决线性系统和线性可行性(LS)问题的方法,我们采用了众所周知的优化方法,例如拉格朗加惩罚和加固拉格朗加因随机卡茨马尔兹(RK)方法,为此,我们提出了RK方法的两种变体,即随机惩罚卡茨马尔兹(RPK)方法和随机加增卡克马尔兹(RAK)方法,我们对拟议方法进行趋同分析,并获得线性趋同结果。