Sparse signal recovery is a concept that is not only central to compressed sensing problems, but also apparent in magnetic resonance imaging (MRI) problems, machine learning, as well as statistical inference. In each of these fields, the target is finding sparse solutions to linear systems of equations that are underdetermined or ill-conditioned. In this paper, an efficient modified Dai-Yuan conjugate gradient method that is globally convergent irrespective of the line search procedure employed was developed to reconstruct sparse signals in compressed sensing. Results of the experiments conducted show that the method is promising.

The conjugate gradient (CG) method is an optimization technique known for its rapid convergence; it has blossomed into significant developments and applications. Numerous variations of CG methods have emerged to enhance computational efficiency and address real-world challenges. This work presents a new conjugate gradient method for solving nonlinear unconstrained optimization problems by introducing a new conjugate gradient parameter. To improve the convergence properties, we have proposed a new inexact line search technique that fits in with the suggested approach and can also be useful for other gradient descent methods. The existence of a steplength that meets the new line search conditions is established. The generated descent direction and the convergence properties of the suggested approach are studied under the new line search conditions, where the global convergence is proven under mild assumptions. The proposed approach is evaluated on various test functions, and a comparison with recent similar algorithms is carried out. Furthermore, the proposed algorithm is applied for restoring images with different noise levels.

In the last decades, conjugate gradient methods have gained important applications in various scientific areas due to their low memory requirements and ability to solve problems of high dimensions. When analyzing a conjugate gradient method, the descent property of the search directions is always required, as it ensures that the search for the minimizer is in the correct direction. In this paper, we proposed a conjugate gradient method that always generates descent search directions under all line searches techniques. Moreover, we established the global convergence of the proposed method when it is applied under Wolfe or strong Wolfe line search. At the same time, to show the performance of the proposed method in practical computation, we compared it with other well-known methods and then applied it to train two-layer neural network models. The numerical results show that the proposed method is efficient.