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

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.