Convergence analysis of a three-term extended RMIL CGP-based algorithm for constrained nonlinear equations and image denoising applications

Dandan Li; Songhua Wang; Yan Xia; Xuejie Ma; Dandan Li; Songhua Wang; Yan Xia; Xuejie Ma

doi:10.3934/era.2025160

Electronic Research Archive

2025, Volume 33, Issue 6: 3584-3612. doi: 10.3934/era.2025160

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Research article

Convergence analysis of a three-term extended RMIL CGP-based algorithm for constrained nonlinear equations and image denoising applications

1.
School of Artificial Intelligence, Guangzhou Huashang College, Guangzhou 511300, China
2.
School of Mathematics, Physics and Statistics, Baise University, Baise 533099, China

Received: 27 February 2025 Revised: 20 May 2025 Accepted: 04 June 2025 Published: 11 June 2025

With a focus on image denoising applications, this paper proposed a three-term extended Rivaie-Mustafa-Ismail-Leong (RMIL) conjugate gradient projection (CGP)-based algorithm for solving constrained nonlinear equations. Unlike traditional methods, the proposed algorithm relied only on the continuity and monotonicity properties of the nonlinear equations, and did not require the more restrictive Lipschitz continuity condition. A rigorous convergence analysis was established under these relaxed assumptions. At the algorithmic level, a novel three-term search direction was constructed by extending previous two-term schemes through the introduction of a carefully designed scale factor, which effectively eliminates the need for a line search procedure. Comprehensive numerical experiments on standard benchmark problems demonstrated the algorithm's efficiency and competitiveness, consistently outperforming comparable three-term algorithms in terms of running time in seconds, number of iterations, and function evaluations. Furthermore, the proposed algorithm has been successfully applied to image denosing problems.
- constrained nonlinear equations,
- conjugate gradient projection method,
- Lipschitz continuity,
- convergence analysis,
- image denoising
Citation: Dandan Li, Songhua Wang, Yan Xia, Xuejie Ma. Convergence analysis of a three-term extended RMIL CGP-based algorithm for constrained nonlinear equations and image denoising applications[J]. Electronic Research Archive, 2025, 33(6): 3584-3612. doi: 10.3934/era.2025160

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Abstract

With a focus on image denoising applications, this paper proposed a three-term extended Rivaie-Mustafa-Ismail-Leong (RMIL) conjugate gradient projection (CGP)-based algorithm for solving constrained nonlinear equations. Unlike traditional methods, the proposed algorithm relied only on the continuity and monotonicity properties of the nonlinear equations, and did not require the more restrictive Lipschitz continuity condition. A rigorous convergence analysis was established under these relaxed assumptions. At the algorithmic level, a novel three-term search direction was constructed by extending previous two-term schemes through the introduction of a carefully designed scale factor, which effectively eliminates the need for a line search procedure. Comprehensive numerical experiments on standard benchmark problems demonstrated the algorithm's efficiency and competitiveness, consistently outperforming comparable three-term algorithms in terms of running time in seconds, number of iterations, and function evaluations. Furthermore, the proposed algorithm has been successfully applied to image denosing problems.

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