Research On Trust-region Filter-SQP Algorithms For Nonlinear Constrained Optimization Problems

Sequential quadratic programming method is the most commonly used to solve nonlinear constrained optimization problems. It can achieve the superlinear convergence rate which is similar to Newton method. In 2002, Leyffer and Fletcher firstly introduced the filter method. This method is easy to calculate and has good convergence properties. Soon after that the filter-SQP method became a hot research topic in optimization theory. However, both the classic SQP method and the filter-SQP method could generate the Maratos effect, which will do harm to convergence properties. In this paper, we deeply study various algorithms to overcome the Maratos effect. Then we make some modifications on the trust-region filter-SQP method introduced by Leyffer, Fletcher and Toint. Firstly, we use the Lagrangian function value (not objective function value) and a modified constraint violation in the filter. Secondly, we set a switching criterion for the calculation of Lagrangian multipliers to the nonlinear problems. When current iteration point gets closer to the optimum point, the formula of computation of the Lagrangian multipliers is established by solving a Least squares problem so as to improve the convergent rate. Lastly, we make full use of the gradients and the Hessian matrix of current iteration point, and update the trust region radius adaptively. To enhance the efficiency of the iterative algorithm, the sufficient descent conditions are also modified in this paper. Through the convergence analysis, the new algorithm is proved to have global convergence properties. Moreover, the results of numerical experiments show that the new algorithm is effective.