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.

This paper proposes an improved LS-RMIL-type conjugate gradient projection algorithm designed for solving systems of nonlinear equations with convex constraints. The algorithm introduces a search direction that maintains sufficient descent and trust-region properties independent of the line search approach. It operates under relatively mild conditions, requiring only continuity and monotonicity of nonlinear equations, thus avoiding the need for stronger assumptions such as Lipschitz continuity. The global convergence of the algorithm is established under these relaxed conditions. Furthermore, numerical experiments demonstrate that the algorithm exhibits superior efficiency and stability, particularly in solving large-scale nonlinear systems and in applications such as impulse noise image restoration, outperforming existing methods.