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.

In this paper, we propose a two-step iterative algorithm based on projection technique for solving system of monotone nonlinear equations with convex constraints. The proposed two-step algorithm uses two search directions which are defined using the well-known Barzilai and Borwein (BB) spectral parameters.The BB spectral parameters can be viewed as the approximations of Jacobians with scalar multiple of identity matrices. If the Jacobians are close to symmetric matrices with clustered eigenvalues then the BB parameters are expected to behave nicely. We present a new line search technique for generating the separating hyperplane projection step of Solodov and Svaiter (1998) that generalizes the one used in most of the existing literature. We establish the convergence result of the algorithm under some suitable assumptions. Preliminary numerical experiments demonstrate the efficiency and computational advantage of the algorithm over some existing algorithms designed for solving similar problems. Finally, we apply the proposed algorithm to solve image deblurring problem.

In this paper, we present a derivative-free algorithm for nonlinear monotone equations with convex constraints. The search direction is a product of a positive parameter and the negation of a residual vector. At each iteration step, the algorithm generates a descent direction independent from the line search used. Under appropriate assumptions, the global convergence of the algorithm is given. Numerical experiments show the algorithm has advantages over the recently proposed algorithms by Gao and He (Calcolo 55 (2018) 53) and Liu and Li (Comput. Math. App. 70 (2015) 2442–2453).

In this paper, we present a brief survey of methods for solving nonlinear least-squares problems. We pay specific attention to methods that take into account the special structure of the problems. Most of the methods discussed belong to the quasi-Newton family (i.e. the structured quasi-Newton methods (SQN)). Our survey comprises some of the traditional and modern developed methods for nonlinear least-squares problems. At the end, we suggest a few topics for further research.