The trust region method (TRM) is a very important technique to solve both of linear and nonlinear systems of equations. In this work, a new modified algorithm of a TRM with adaptive radius is introduced in purpose of solving systems of nonlinear equations. At each iteration, the new algorithm changes the trust region radius (TRR) automatically to reduce the subproblems resolving number when the current radius is rejected. The global convergence results of the new procedure under some appropriate conditions is established. The numerical effects indicate that the suggested algorithm is interesting and robustness.

The projection technique is one of the famous method and highly useful to solve the optimization problems and nonlinear systems of equations. In this work, a new projection approach for solving systems of nonlinear monotone equation is proposed combining with the conjugate gradient direction because of their low storage. The new algorithm can be used to solve the large-scale nonlinear systems of equations and satisfy the sufficient descent condition. The new algorithm generates appropriate direction then employs a good line search along this direction to reach a new point. If this point solves the problem then the algorithm stops, otherwise, it constructs a suitable hyperplane that strictly separate the current point from the solution set. The next iteration is obtained by projection the new point onto the separating hyperplane. We proved that the line search of the new projection algorithm is well defined. Furthermore, we established the global convergence under some mild conditions. The numerical experiment indicates that the new method is effective and very well.

Conjugate gradient methods more used in the field of unconstrained optimization, particularly large scale problems, Armijo condition one of the simple rule are commonly used to analyses and applications of CG methods. In this paper we exhibit a new modified for the Armijo condition with established converges globally of the conjugate gradient method and best numerical results.

There are many methods derived from the conjugate gradient method, the most famous of which is the FR method (Fletcher–Reeves). Most of the methods are found to solve large unconstrained optimization problems. In this paper, we made a modified to the FR method, so that it achieves better numerical results as well as the conditions of global convergence. The numerical experiment showed the efficiency and robustness of the new method.

The Conjugate Gradient (CG) method of unconstrained optimization algorithms possesses good properties like less requirement memory and global convergence properties. Many modified algorithms have been made to this method, as well as new suggestions for its work to obtain the best results. In this article, a modified of conjugate gradient method is introduced, so that global convergence was smoothly proven, and the numerical results of the proposed method were compared with other methods, and the results were better in its general form, this confirms the strength, durability, and effectiveness of the proposed new method.