Solving Systems of Nonlinear Monotone Equations by Using a New Projection Approach

Abstract

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.