A globally convergent inertial-relaxed hybrid CGPM for constrained nonlinear equations with its applications

Dandan Li; Songhua Wang; Jiaqi Wu; Dandan Li; Songhua Wang; Jiaqi Wu

doi:10.3934/math.20261042

AIMS Mathematics

2026, Volume 11, Issue 8: 26006-26025. doi: 10.3934/math.20261042

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A globally convergent inertial-relaxed hybrid CGPM for constrained nonlinear equations with its applications

1.
School of Artificial Intelligence, Guangzhou Huashang College, Guangzhou 511300, China
2.
School of Mathematics, Physics and Statistics, Baise University, Baise 533099, China
3.
School of Computer Science, Guangdong University of Finance, Guangzhou 510521, China

Received: 21 June 2026 Revised: 31 July 2026 Accepted: 17 August 2026 Published: 20 August 2026
MSC : 65K05, 90C30, 90C56, 90C90

This paper presents a globally convergent inertial-relaxed hybrid conjugate gradient projection (IRHCGP) method for solving constrained nonlinear equations. The proposed method integrates an inertial-relaxed extrapolation strategy into a hybrid conjugate gradient projection framework. A new hybrid conjugate parameter is constructed to generate an effective search direction, which is proved to satisfy both the sufficient descent condition and a trust-region-type bound. These properties provide the theoretical foundation for establishing the global convergence of the proposed method under mild assumptions. Extensive numerical experiments are conducted on large-scale constrained nonlinear equations with various dimensions and initial points, and the proposed method is further applied to impulse noise image restoration. The results demonstrate that the proposed method is competitive and superior to several existing methods in terms of running time, number of function evaluations, and number of iterations.
- nonlinear equations,
- inertial-relaxed technique,
- sufficient descent,
- global convergence,
- impulse noise image restoration
Citation: Dandan Li, Songhua Wang, Jiaqi Wu. A globally convergent inertial-relaxed hybrid CGPM for constrained nonlinear equations with its applications[J]. AIMS Mathematics, 2026, 11(8): 26006-26025. doi: 10.3934/math.20261042

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Abstract

This paper presents a globally convergent inertial-relaxed hybrid conjugate gradient projection (IRHCGP) method for solving constrained nonlinear equations. The proposed method integrates an inertial-relaxed extrapolation strategy into a hybrid conjugate gradient projection framework. A new hybrid conjugate parameter is constructed to generate an effective search direction, which is proved to satisfy both the sufficient descent condition and a trust-region-type bound. These properties provide the theoretical foundation for establishing the global convergence of the proposed method under mild assumptions. Extensive numerical experiments are conducted on large-scale constrained nonlinear equations with various dimensions and initial points, and the proposed method is further applied to impulse noise image restoration. The results demonstrate that the proposed method is competitive and superior to several existing methods in terms of running time, number of function evaluations, and number of iterations.

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