A modified fixed point iteration method for solving the system of absolute value equations. (4th March 2022)

• Record Type: Journal Article
• Authors: Yu, Dongmei; Chen, Cairong; Han, Deren
• Is Part Of: Optimization. Volume 71:Number 3(2022)
• Page Start: 449
• Page End: 461
• Publication Date: 2022-03-04
• Subjects: Absolute value equations -- fixed point iteration -- linear convergence; 65F10 -- 65H10; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2020.1804568
• Languages: English
• ISSNs: 0233-1934

Abstract

ABSTRACT: The fixed point iteration (FPI) method proposed by Ke [Appl Math Lett. 2020;99:105990] for solving the absolute value equations (AVE) with the form A x − | x | = b is interesting for its simplicity and efficiency. However, its convergence is only guaranteed for the case that 0 < ∥ A − 1 ∥ < 2 2, excluding the possible case that 2 2 ≤ ∥ A − 1 ∥ < 1 . To complete the gap, we develop a modified FPI (MFPI) method for solving the AVE with 0 < ∥ A − 1 ∥ < 1, which, besides keeping the simplicity of FPI, improves its efficiency by judiciously choosing the involving parameter. Under mild conditions, we prove its linear convergence. We present some preliminary numerical results for 0 < ∥ A − 1 ∥ < 1, demonstrating its convergence; and compare it with FPI when 0 < ∥ A − 1 ∥ < 2 2, illustrating its superiority.