Adaptive Cubic Regularization Methods For Nonlinear Constrained Optimization

New algorithms for solving nonlinear constrained optimization problems have e-merged in an endless stream since the1950s. There are more in-depth study of the methods to consider these problems with the rapid development of computer technolo-gy. Since one way of solving nonlinear constrained optimization problems is to transform them into unconstrained optimization problems by solving one or a series of their sub-problems, scholars always extend the approaches for solving unconstrained optimization problems to the nonlinear constrained optimization problems. An adaptive cubic regular-ization algorithm is a new method, which has been recently proposed for unconstrained optimization. This paper tentatively applies this new approach in solving constrained optimization problems, and obtains some achievements.The main content of this paper is organized as follows:In the second chapter, considering the adaptive cubic regularization approach for solving the nonlinear equation constrained optimization. Firstly, we construct approxi-mate function of the augmented Lagrangian function. Secondly, we give the subproblem of the original problem through combining the approximate function with the adaptive cubic regularization approach, and we update the Lagrange multiplier and the penalty parameter at the same time. In our method, we also make use of a filter technique and a merit function to decide whether the trail step should be accepted, which increase the opportunity of the trail step to be accepted. Finally, the adaptive cubic regularization algorithm for nonlinear equation constrained optimization and the convergence properties are given.The third chapter In this chapter, we add the inequality constraint condition into the nonlinear equation constrained optimization so that the nonlinear constrained optimiza-tion model is more general. By introducing slack variables and using active set method, we transform inequality constrained optimization to equation constrained optimization, which has been discussed in the second chapter. Finally, we use the same approach to get the unconstrained optimization problems and give the adaptive cubic regularization algorithm for nonlinear constrained optimization.