<p>In this paper, we propose a hybrid acceleration Douglas-Rachford splitting method for solving large-scale absolute value equations of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Ax-|x|=b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>x</mi> <mo>-</mo> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation>. Under the condition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Vert A^{-1}\Vert \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msup> <mi>A</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">‖</mo> <mo>≤</mo> <mn>1</mn> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the global convergence and iteration complexity of the proposed algorithm are established. To our knowledge, the iteration-complexity result for finding an <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\epsilon$</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-approximation solution with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\epsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, in this paper, is new under such a condition, when compared with existing algorithms for solving absolute value equations. Numerical experiments demonstrate the effectiveness of the proposed method.</p>

A hybrid acceleration Douglas-Rachford splitting method for solving large-scale absolute value equations

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发表日期：

2025年6月7日

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Jinbao Jian,
Mingxia Wang,
Jianghua Yin

摘要

In this paper, we propose a hybrid acceleration Douglas-Rachford splitting method for solving large-scale absolute value equations of the form $Ax-|x|=b$ $A x - | x | = b$ . Under the condition $\Vert A^{-1}\Vert \le 1$ $‖ A^{- 1} ‖ \leq 1$ , the global convergence and iteration complexity of the proposed algorithm are established. To our knowledge, the iteration-complexity result for finding an $\epsilon$ $ϵ$ -approximation solution with $\epsilon >0$ $ϵ > 0$ , in this paper, is new under such a condition, when compared with existing algorithms for solving absolute value equations. Numerical experiments demonstrate the effectiveness of the proposed method.