This paper proposes a novel hybrid PRP-HS-LS-type conjugate gradient algorithm for solving constrained nonlinear systems of equations. The proposed algorithm presents several significant advancements and key features: (i) the conjugate parameter is constructed by utilizing the hybrid technique; (ii) the search direction, designed with the conjugate parameter, possesses sufficient descent and trust region properties without the need for a line search mechanism; (iii) the global convergence is rigorously established under general assumptions, notably without the requirement of the Lipschitz continuity condition; (vi) numerical experiments demonstrate the algorithm's efficiency, particularly in solving large-scale constrained nonlinear systems of equations and addressing the sparse signal restoration problem.

With a focus on image denoising applications, this paper proposed a three-term extended Rivaie-Mustafa-Ismail-Leong (RMIL) conjugate gradient projection (CGP)-based algorithm for solving constrained nonlinear equations. Unlike traditional methods, the proposed algorithm relied only on the continuity and monotonicity properties of the nonlinear equations, and did not require the more restrictive Lipschitz continuity condition. A rigorous convergence analysis was established under these relaxed assumptions. At the algorithmic level, a novel three-term search direction was constructed by extending previous two-term schemes through the introduction of a carefully designed scale factor, which effectively eliminates the need for a line search procedure. Comprehensive numerical experiments on standard benchmark problems demonstrated the algorithm's efficiency and competitiveness, consistently outperforming comparable three-term algorithms in terms of running time in seconds, number of iterations, and function evaluations. Furthermore, the proposed algorithm has been successfully applied to image denosing problems.

This paper proposes a modified Rivaie-Mohd-Ismail-Leong (RMIL)-type conjugate gradient algorithm for solving nonlinear systems of equations with convex constraints. The proposed algorithm offers several key characteristics: (1) The modified conjugate parameter is non-negative, thereby enhancing the proposed algorithm's stability. (2) The search direction satisfies sufficient descent and trust region properties without relying on any line search technique. (3) The global convergence of the proposed algorithm is established under general assumptions without requiring the Lipschitz continuity condition for nonlinear systems of equations. (4) Numerical experiments indicated that the proposed algorithm surpasses existing similar algorithms in both efficiency and stability, particularly when applied to large scale nonlinear systems of equations and signal recovery problems in compressed sensing.