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

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