An inertial hybrid CGP-based algorithm with restart strategy for constrained nonlinear equations and impulse noise image restoration

Yan Xia; Dandan Li; Yan Xia; Dandan Li

doi:10.3934/math.20251037

AIMS Mathematics

2025, Volume 10, Issue 10: 23360-23379. doi: 10.3934/math.20251037

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Research article Topical Sections

An inertial hybrid CGP-based algorithm with restart strategy for constrained nonlinear equations and impulse noise image restoration

Yan Xia ,
Dandan Li ^,

School of Artificial Intelligence, Guangzhou Huashang College, Guangzhou 511300, China

Received: 26 August 2025 Revised: 24 September 2025 Accepted: 10 October 2025 Published: 15 October 2025
MSC : 65K05, 90C56

The conjugate gradient method is widely recognized as one of the most efficient approaches for solving large-scale optimization problems. In this paper, we have propose a novel hybrid conjugate gradient projection (CGP)-based algorithm that integrates an improved conjugate coefficient derived from the Hestenes-Stiefel (HS) and Polak-Ribière-Polak (PRP) formulas. The proposed algorithm exhibits several key characteristics: (ⅰ) The hybrid coefficient with a single parameter was employed to construct a search direction that ensures both the sufficient descent condition and trust-region feature, enhanced via a restart strategy; (ⅱ) we incorporated an inertial-relaxed scheme alongside a projection technique in a hybrid CGP-based framework for further improving performance; (ⅲ) we established the global convergence of the proposed algorithm under relaxed assumptions, providing a solid theoretical foundation; and (iv) extensive numerical experiments demonstrated the superior numerical performance of the proposed algorithm compared to existing algorithms on large-scale constrained nonlinear equations and impulse noise image restoration problems.
- nonlinear equations,
- conjugate gradient projection method,
- inertial-related scheme,
- theoretical analysis,
- impulse noise image restoration
Citation: Yan Xia, Dandan Li. An inertial hybrid CGP-based algorithm with restart strategy for constrained nonlinear equations and impulse noise image restoration[J]. AIMS Mathematics, 2025, 10(10): 23360-23379. doi: 10.3934/math.20251037

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Abstract

The conjugate gradient method is widely recognized as one of the most efficient approaches for solving large-scale optimization problems. In this paper, we have propose a novel hybrid conjugate gradient projection (CGP)-based algorithm that integrates an improved conjugate coefficient derived from the Hestenes-Stiefel (HS) and Polak-Ribière-Polak (PRP) formulas. The proposed algorithm exhibits several key characteristics: (ⅰ) The hybrid coefficient with a single parameter was employed to construct a search direction that ensures both the sufficient descent condition and trust-region feature, enhanced via a restart strategy; (ⅱ) we incorporated an inertial-relaxed scheme alongside a projection technique in a hybrid CGP-based framework for further improving performance; (ⅲ) we established the global convergence of the proposed algorithm under relaxed assumptions, providing a solid theoretical foundation; and (iv) extensive numerical experiments demonstrated the superior numerical performance of the proposed algorithm compared to existing algorithms on large-scale constrained nonlinear equations and impulse noise image restoration problems.

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