A greedy randomized average block projection method for linear feasibility problems

TL;DR

This paper introduces a new greedy randomized average block projection (GRABP) method that accelerates solving large-scale linear feasibility problems, demonstrating superior convergence and performance over existing methods through theoretical analysis and numerical experiments.

Contribution

The paper develops a novel GRABP method combining greedy probability and block averaging, with proven linear convergence and improved efficiency over current algorithms.

Findings

GRABP converges linearly in expectation.

GRABP outperforms existing methods like RP, SKM, GSKM, and Nesterov-accelerated SKM.

Numerical results confirm the efficiency and effectiveness of GRABP.

Abstract

The randomized projection (RP) method is a simple iterative scheme for solving linear feasibility problems and has recently gained popularity due to its speed and low memory requirement. This paper develops an accelerated variant of the standard RP method by using two ingredients: the greedy probability criterion and the average block approach, and obtains a greedy randomized average block projection (GRABP) method for solving large-scale systems of linear inequalities. We prove that this method converges linearly in expectation under different choices of extrapolated stepsizes. Numerical experiments on both randomly generated and real-world data show the advantage of GRABP over several state-of-the-art solvers, such as the randomized projection (RP) method, the sampling Kaczmarz Motzkin (SKM) method, the generalized SKM (GSKM) method, and the Nesterov acceleration of SKM method.