Heavy Ball Momentum Induced Sampling Kaczmarz Motzkin Methods for Linear Feasibility Problems

Abstract

The recently proposed Sampling Kaczmarz Motzkin (SKM) algorithm performs well in comparison with the state-of-the-art methods in solving large-scale Linear Feasibility (LF) problems. To explore the concept of momentum in the context of solving LF problems, in this work, we propose a momentum induced algorithm called Momentum Sampling Kaczmarz Motzkin (MSKM). The MSKM algorithm is developed by integrating the heavy ball momentum to the SKM algorithm. We provide a rigorous convergence analysis of the proposed MSKM algorithm from which we obtain convergence results of several Kaczmarz type methods for solving LF problems. Moreover, under somewhat weaker conditions, we establish a sub-linear convergence rate for the so-called Cesaro average of the sequence generated by the MSKM algorithm. We then back up the theoretical results via thorough numerical experiments on artificial and real datasets. For a fair comparison, we test our proposed method in comparison with the SKM method on a wide variety of test instances$:$ 1) randomly generated instances, 2) Netlib LPs and 3) linear classification test instances. We also compare the proposed method with the traditional Interior Point Method (IPM) and Active Set Method (ASM) on Netlib LPs. The proposed momentum induced algorithm significantly outperforms the basic SKM method (with no momentum) on all of the considered test instances. Furthermore, the proposed algorithm also performs well in comparison with IPM and ASM algorithms. Finally, we propose a stochastic version of the MSKM algorithm called Stochastic-Momentum Sampling Kaczmarz Motzkin (SSKM) to better handle large-scale real-world data. We conclude our work with a rigorous theoretical convergence analysis of the proposed SSKM algorithm.