A New Conjugate Gradient Projection Method for Convex Constrained Nonlinear Equations

The conjugate gradient projection method is one of the most effective methods for solving large-scale monotone nonlinear equations with convex constraints. In this paper, a new conjugate parameter is designed to generate the search direction, and an adaptive line search strategy is improved to yield the step size, and then, a new conjugate gradient projection method is proposed for large-scale monotone nonlinear equations with convex constraints. Under mild conditions, the proposed method is proved to be globally convergent. A large number of numerical experiments for the presented method and its comparisons are executed, which indicates that the presented method is very promising. Finally, the proposed method is applied to deal with the recovery of sparse signals.

A Modified Fletcher–Reeves Conjugate Gradient Method for Monotone Nonlinear Equations with Some Applications

One of the fastest growing and efficient methods for solving the unconstrained minimization problem is the conjugate gradient method (CG). Recently, considerable efforts have been made to extend the CG method for solving monotone nonlinear equations. In this research article, we present a modification of the Fletcher–Reeves (FR) conjugate gradient projection method for constrained monotone nonlinear equations. The method possesses sufficient descent property and its global convergence was proved using some appropriate assumptions. Two sets of numerical experiments were carried out to show the good performance of the proposed method compared with some existing ones. The first experiment was for solving monotone constrained nonlinear equations using some benchmark test problem while the second experiment was applying the method in signal and image recovery problems arising from compressive sensing.

A Nonmonotone Projection Method for Constrained System of Nonlinear Equations

This paper deals with the nonmonotone projection algorithm for constrained nonlinear equations. For some starting points, the previous projection algorithms for the problem may encounter slow convergence which is related to the monotone behavior of the iterative sequence as well as the iterative direction. To circumvent this situation, we adopt the nonmonotone technique introduced by Dang to develop a nonmonotone projection algorithm. After constructing the nonmonotone projection algorithm, we show its convergence under some suitable condition. Preliminary numerical experiment is reported at the end of this paper, from which we can see that the algorithm we propose converges more quickly than that of the usual projection algorithm for some starting points.