An Inertial-Based Hybrid PRP-HS-Type CGP Algorithm for Nonlinear Equations With Convex Constraints and Its Applications

This paper presents an inertial-based hybrid conjugate gradient projection algorithm for solving nonlinear equations with convex constraints. The proposed algorithm integrates Polak–Ribière–Polyak and Hestenes–Stiefel methods within a conjugate gradient framework, incorporating an inertial-relaxed technique to accelerate iterative convergence. At each iteration, an inertial extrapolation step is employed to enhance the next iterate, followed by a projection step that ensures feasibility. The search direction satisfies both the sufficient descent and trust region properties without relying on line search approaches. Global convergence is established under weak and mild assumptions. Extensive numerical experiments demonstrate that the proposed algorithm is efficient and competitive relative to existing methods. Furthermore, the practical applicability of the proposed algorithm is illustrated through its successful implementation in image denoising problems.