Converse Inertial Step Approach and Its Applications in Solving Nonexpansive Mapping

Gangxing Yan and Tao Zhang ()
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Gangxing Yan: Faculty of Data Science, City University of Macau, Macao 999078, China
Tao Zhang: School of Mathematics and Statistics, Linyi University, Linyi 276000, China

Mathematics, 2025, vol. 13, issue 22, 1-26

Abstract: In spite of great successes of the inertial step approach (ISA) in various fields, we are investigating the converse inertial step approach (CISA) for the first time. First, the classical Picard iteration for solving nonexpansive mappings converges weakly with CISA integration. Its analysis is based on the newly developed weak quasi-Fejér monotonicity under mild assumptions. We also establish O ( 1 / k γ ) ( γ ∈ ( 0 , 1 ) ) and linear convergence rate under different assumptions. This extends the O ( 1 / k ) convergence rate of the Krasnosel’skiĭ–Mann iteration. A generalized version of CISA is then studied. Second, combining CISA with over-relaxed step approach for solving nonexpansive mappings leads to a new algorithm, which not only converges without restrictive assumptions but also allows an inexact calculation in each iteration. Third, with CISA integration, a Backward–Forward splitting algorithm succeeds in accepting a larger step-size, and a Peaceman–Rachford splitting algorithm is guaranteed to converge.

Keywords: converse inertial step approach; nonexpansive mappings; over-relaxed step approach (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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