  Randomized Kaczmarz (RK), Motzkin Method (MM) and Sampling Kaczmarz Motzkin
(SKM) algorithms are commonly used iterative techniques for solving a system of
linear inequalities (i.e., $Ax \leq b$). As linear systems of equations
represent 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 Kaczmarz
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
Kaczmarz Motzkin (PASKM). We prove the convergence theorems for both GSKM and
PASKM algorithms in the $L_2$ norm perspective with respect to the proposed
sampling distribution. Furthermore, we prove sub-linear convergence for the
Cesaro average of iterates for the proposed GSKM and PASKM algorithms.From the
convergence theorem of the GSKM algorithm, we find the convergence results of
several well-known algorithms like the Kaczmarz method, Motzkin method and SKM
algorithm. We perform thorough numerical experiments using both randomly
generated and real-world (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
the majority of the problem instances, the proposed generalized and accelerated
algorithms significantly outperform the state-of-the-art methods.
