New Efficient Iterative Methods for Solving Systems of Absolute Value Equations

Abstract

This paper introduces new efficient iterative methods for solving systems of absolute value equations (AVEs) of the form Ax − |x| = b. The proposed methods address the inherent challenges posed by the NP-hard and nondifferentiable nature of AVEs, which arise in various applications, including linear complementarity problems, optimization, and engineering. We develop novel iteration schemes that combine matrix-splitting techniques with accelerated-convergence strategies, offering significant improvements over classical approaches such as Picard and Gauss–Seidel iterations. The convergence properties of the proposed methods are rigorously analyzed under appropriate conditions. Extensive numerical experiments demonstrate that the new methods achieve superior performance in terms of iteration counts, computational time, and solution accuracy compared to existing state-of-the-art algorithms. The results confirm the robustness and practical efficiency of our approaches for solving large-scale AVE systems.