On Generalization and Acceleration of Randomized Projection Methods for Linear Feasibility Problems

Randomized Kaczmarz (RK), Motzkin Method (MM) and Sampling Kaczmarz Motzkin (SKM) algorithms are commonly used iterative techniques for solving linear system of inequalities (i.e., $Ax \leq b$). As linear systems of equations represents a modeling paradigm for solving many optimization problems, these randomized and iterative techniques are gaining popularity among researchers in different domains. In this work, we propose a Generalized Sampling Kaczamrz Motzkin (GSKM) method that unifies the iterative methods into a single framework. In addition to the general framework, we propose a Nesterov type acceleration scheme in the SKM method called as Probably Accelerated Sampling Kaczamrz Motzkin (PASKM). We prove the convergence theorems for both GSKM and PASKM algorithms in the L2 norm perspective with respect to the proposed sampling distribution. Furthermore, from the convergence theorem of GSKM algorithm, we find the convergence results of several well known algorithms like Kaczmarz method, Motzkin method and SKM algorithm. We perform thorough numerical experiments using both randomly generated and real life (classification with support vector machine and Netlib LP) test instances to demonstrate the efficiency of the proposed methods. We compare the proposed algorithms with SKM, Interior Point Method (IPM) and Active Set Method (ASM) in terms of computation time and solution quality. In majority of the problem instances, the proposed generalized and accelerated algorithms significantly outperform the state-of-the-art methods.