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<p>The generalized absolute value equation (GAVE) finds applications across multiple disciplines, including operations research, economic modeling, engineering, scientific computing, and management science. However, effectively solving this equation remains a challenging problem due to its nonlinear nature. This paper introduces a novel process by utilizing the identity <Equation ID="Equa"> <EquationSource Format="TEX">\(|y| = 2\max(y,0) - y,\)</EquationSource> </Equation></p><p>in the context of GAVEs. With this formulation as a foundation, we derive a matrix splitting of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( A + B \)</EquationSource> </InlineEquation>, enabling the proposal of two new fixed-point iterative methods for the GAVE <Equation ID="Equb"> <EquationSource Format="TEX">\(\hspace{3cm} Ay - B|y| = b. \)</EquationSource> </Equation></p><p>To establish the theoretical validity of the proposed methods, a rigorous convergence analysis is conducted under suitable assumptions. Moreover, we demonstrate their efficiency and reliability in solving GAVEs under a variety of test cases through extensive numerical experiments. These findings are promising and may encourage additional exploration in this field.</p>

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Two fixed-point iteration methods for generalized absolute value equations: formulation and analysis

  • 期刊论文

  • 发表日期：

    2025年11月10日

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  • Rashid Ali

摘要

The generalized absolute value equation (GAVE) finds applications across multiple disciplines, including operations research, economic modeling, engineering, scientific computing, and management science. However, effectively solving this equation remains a challenging problem due to its nonlinear nature. This paper introduces a novel process by utilizing the identity \(|y| = 2\max(y,0) - y,\)

in the context of GAVEs. With this formulation as a foundation, we derive a matrix splitting of \( A + B \) , enabling the proposal of two new fixed-point iterative methods for the GAVE \(\hspace{3cm} Ay - B|y| = b. \)

To establish the theoretical validity of the proposed methods, a rigorous convergence analysis is conducted under suitable assumptions. Moreover, we demonstrate their efficiency and reliability in solving GAVEs under a variety of test cases through extensive numerical experiments. These findings are promising and may encourage additional exploration in this field.