Research On Augmented Lagrangian Function Based Trust Region Method For Optimization Problems With Inequality Constraints

Constrained optimization problems are widely used in many fields such as finance,network and transportation,digital integrated design,image processing and so on.Which have important theoretical research significance and practical value.This thesis proposes an exponential augmented Lagrangian function based trust region method for optimization problems with inequality constraints.The main works are summarized as follows:1.In order to overcome the disadvantage that it costs too high to exactly solve the subproblem in the second step of the traditional augmented Lagrangian method,a trust region method based on an exponential augmented Lagrangian function is proposed,and an update strategy that is different from the traditional penalty parameter is designed.In the traditional augmented Lagrangian method,the corresponding augmented Lagrangian function needs to be accurately minimized in each iteration.However,obtaining such an exact solution of subproblem is too expensive,and when the nonlinearity of functions in the original problem is high,the corresponding subproblem is not easy to solve.Therefore,this thesis transforms the minimization of the subproblem into minimization of the quadratic approximation of augmented Lagrangian function,and combines the trust region technique to ensure the rationality of the approximation.At the same time,because penalty parameter has a great influence on reducing constraint violation and solving trust region subproblem,this thesis considers the relationship among the predicted descent,constraint violation and trust region radius,and designs a new penalty parameter updating strategy.And then this thesis establishes a detailed trust region algorithm based on exponential augmented Lagrangian function.2.The thesis demonstrates that the proposed trust region algorithm based on the augmented Lagrangian function is globally convergence,which is different from the local convergence of the traditional the traditional augmented Lagrangian method.In this thesis,under the conditions that the objective function and the constraint function are second-order continuous differentiable,the iteration points generated by the algorithm are uniformly bounded,the other assumptions,it is proved that the iteration sequence points generated by the algorithm are feasible,and the iteration sequence points converges globally to the KKT point of the original optimization problems with inequality constraints.3.According to the proposed algorithm,the classical examples of optimization problems with inequality constraints are numerically tested,and the obtained numerical results are compared with those obtained by the traditional augmented Lagrangian algorithm and a related algorithm in the literature.The numerical results show that the proposed method is feasible and effective.