Application of Newton-SOR Iteration with Linear Weighted Lagrange Approach for Solving Multi-objective Constrained Optimization Problems

In recent years, multi-objective constrained optimization problems have received extensive attention from optimization enthusiasts due to their wide range of applications, which have many conflicting objective functions and complex constraints. In view of this situation, this paper firstly transforms the original multi-objective constrained optimization problem into a single-objective unconstrained optimization problem using the linear weighting method and the Lagrange multiplier approach subject to the characteristics of the objective functions and constraints. Secondly, based on the respective advantages of Newton's method for solving unconstrained optimization problems and the successive over relaxation method for solving linear systems of equations, we propose the Newton successive over relaxation (NSOR) iterative method. Having the advantages of Newton's method of fast convergence speed and the successive over relaxation iterative method of small storage, the proposed NSOR iteration improved the efficiency of the convergence rate in solving the proposed problem. Finally, numerical experiments demonstrate that the proposed iteration can still find pareto solutions without the need of downscaling and interactive iterations. In addition, the NSOR has higher computational efficiency than the Newton-Gauss–Seidel (NGS) iterative method.