Combining the projection method of Solodov and Svaiter with the Liu-Storey and Fletcher Reeves conjugate gradient algorithm of Djordjević for unconstrained minimization problems, a hybrid conjugate gradient algorithm is proposed and extended to solve convex constrained nonlinear monotone equations. Under some suitable conditions, the global convergence result of the proposed method is established. Furthermore, the proposed method is applied to solve the ℓ 1 -norm regularized problems to restore sparse signal and image in compressive sensing. Numerical comparisons of the proposed algorithm versus some other conjugate gradient algorithms on a set of benchmark test problems, sparse signal reconstruction and image restoration in compressive sensing show that the proposed scheme is computationally more efficient and robust than the compared schemes.