Randomized Kaczmarz, Motzkin Method and Sampling Kaczmarz Motzkin (SKM) algorithms are commonly used iterative techniques for solving a system of linear inequalities (i.e., \(Ax \le b\)). 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 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 and Active Set Method 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.

随机Kaczmarz，Motzkin方法和抽样Kaczmarz Motzkin（SKM）算法是求解线性不等式（即\（Ax \ le b \））的常用迭代技术。由于方程的线性系统代表了解决许多优化问题的建模范例，因此这些随机和迭代技术在不同领域的研究人员中越来越受欢迎。在这项工作中，我们提出了一种通用采样Kaczmarz Motzkin（GSKM）方法，该方法将迭代方法统一为一个框架。除了通用框架外，我们还以SKM方法提出了Nesterov型加速方案，称为“可能加速采样Kaczmarz Motzkin”（PASKM）。对于所提出的采样分布，我们从\（L_2 \）范数角度证明了GSKM和PASKM算法的收敛性定理。此外，对于拟议的GSKM和PASKM算法，我们证明了迭代的Cesaro平均的亚线性收敛。从GSKM算法的收敛定理中，我们发现了几种著名算法的收敛结果，例如Kaczmarz方法，Motzkin方法和SKM算法。我们使用随机生成的和真实的（使用支持向量机和Netlib LP进行分类）测试实例进行全面的数值实验，以证明所提出方法的效率。我们将所提出的算法与SKM，内部点方法和活动集方法进行了比较在计算时间和解决方案质量方面。在大多数问题实例中，所提出的广义算法和加速算法明显优于最新方法。