Sampling Kaczmarz-Motzkin method for linear feasibility problems: generalization and acceleration

Randomized Kaczmarz, Motzkin Method and Sampling Kaczmarz (SKM) algorithms are commonly used iterative techniques for solving a system of linear inequalities (i.e., \(Ax \le b\)). As systems equations represent modeling paradigm many optimization problems, these randomized gaining popularity among researchers in different domains. In this work, we propose Generalized (GSKM) method that unifies the methods into single framework. addition to general framework, Nesterov-type acceleration scheme SKM called Probably Accelerated (PASKM). We prove convergence theorems both GSKM PASKM \(L_2\) norm perspective with respect proposed sampling distribution. Furthermore, sub-linear Cesaro average iterates algorithms. From theorem algorithm, find results several well-known like method, algorithm. perform thorough numerical experiments using randomly generated real-world (classification support vector machine Netlib LP) test instances demonstrate efficiency methods. compare SKM, Interior Point Active Set terms computation time solution quality. majority problem instances, generalized accelerated significantly outperform state-of-the-art