Newton-SOR Iterative Method with Lagrangian Function for Large-Scale Nonlinear Constrained Optimization Problems

Abstract

With the rapid development of computer technology and the wide application of nonlinear constrained optimization problems, many researchers are committed to solve large-scale constrained optimization problems. In this article, a new combinatorial iterative method is proposed on the basis of previous research, which can efficiently solve large-scale nonlinear constrained optimization problems. We first transform a large nonlinear constrained optimization problem into a corresponding unconstrained optimization problem by using the Lagrange multiplier method, and then the Newton iterative method is used to solve the transformed unconstrained optimization problem. To perform the iterative method, we need to compute its Newton direction, and the inverse matrix of Hessian matrix. To deal with the large-scale Hessian matrix, calculation of the inverse matrix for the Hessian matrix may not be easy to be determined. To overcome this issue, we propose the matrix iteration method to compute the Newton direction by solving the linear system as the internal iteration solution. Therefore, this paper investigates a Newton-SOR (NSOR) iterative method to solve this problem, in which the proposed NSOR iterative method combines the Newton method with Successive Over-Relaxation (SOR) iterative method. Based on the numerical experiments, the effectiveness of the proposed NSOR iterative method is more effective than the Newton-Gauss-Seidel (NGS) iterative method in terms of computing time and number of iterations.