ABSTRACT In this paper, we propose a novel conjugate parameter in the search direction by enhancing the modified Polak‐Ribiére‐Polayk and Hestenes‐Stiefel methods through the hybrid technique. The constructed search direction inherently satisfies both the sufficient descent and trust region properties without the need for any line search strategies. By integrating this direction with the inertial‐relaxed technique, we develop an inertial conjugate gradient projection (CGP)‐based method for solving constrained nonlinear equations. Theoretical analysis confirms that the proposed method guarantees global convergence without requiring a Lipschitz continuity assumption. Experimental results further demonstrate the effectiveness of the proposed method in solving large‐scale constrained nonlinear equations and impulse noise image recovery applications.
Li et al. (2026) studied this question.