Exact and inexact Douglas–Rachford splitting methods for solving large-scale sparse absolute value equations. (11th February 2022)

• Record Type: Journal Article
• Authors: Chen, Cairong; Yu, Dongmei; Han, Deren
• Is Part Of: IMA journal of numerical analysis. Volume 43:Number 2(2023)
• Page Start: 1036
• Page End: 1060
• Publication Date: 2022-02-11
• Subjects: absolute value equation -- Douglas–Rachford splitting method -- exact and inexact -- global and linear convergence -- LSQR; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drab105
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: Exact and inexact Douglas–Rachford splitting methods are developed to solve the large-scale sparse absolute value equation (AVE) $Ax - |x| =b$, where $A\in \mathbb {R}^{n\times n}$ and $b\in \mathbb {R}^n$ . The inexact method adopts a relative error tolerance and, therefore, in the inner iterative processes, the LSQR method is employed to find a qualified approximate solution of each subproblem, resulting in a lower cost for each iteration. When $\|A^{-1}\|\le 1$ and the solution set of the AVE is nonempty, the algorithms are globally and linearly convergent. When $\|A^{-1}\|= 1$ and the solution set of the AVE is empty, the sequence generated by the exact algorithm diverges to infinity on a trivial example. Numerical examples are presented to demonstrate the viability and robustness of the proposed methods.