Trust Region Subproblems Based On Differential Equation Model Research On The Prediction And Correction Algorithms

Trust region algorithm is widely used in solving nonlinear optimization problems because of its strong global convergence.In recent years,the research on trust region algorithm focuses on the construction of trust region model and the solution of trust region subproblem.Quadratic model is the most widely used trust region model because of its simple calculation form.In the process of solving the subproblem of quadratic model,the traditional broken line method and the broken line method based on differential equation model show the advantages of high efficiency and rapidity,and open up an efficient path for the research of trust region algorithm.In this paper,based on the differential equation model of the optimal curve,a variety of new algorithms are proposed to solve the trust region subproblem of the quadratic model based on the predictor corrector scheme in the positive and uncertain Hessian matrix respectively.Based on the differential equation model of the optimal curve,when Hessian matrix is positive,the first chapter puts forward an algorithm to solve the positive definite trust region subproblem based on the Milne Hamming prediction correction scheme,analyzes the properties of the algorithm and obtains that the Milne Hamming prediction correction broken line is closer to the optimal curve through many numerical experiments,and the numerical effect of the optimal solution is better than that of approaching the optimal curve The average Euler tangent algorithm is better;The second chapter puts forward an algorithm based on the Adams fourth-order predictor corrector scheme is proposed to solve the positive definite trust region subproblem.The properties of the algorithm are analyzed and several numerical experiments show that the Adams fourth-order predictor corrector polyline is closer to the optimal curve,and the numerical effect of the optimal solution is better than the average Eulerian tangent algorithm;When Hessian matrix is not fixed,based on the new algorithm in the first two chapters,in the third chapter,B-P decomposition method is used to modify the Milne Hamming prediction and correction algorithm,and the Milne Hamming prediction and correction algorithm for solving the non stator problem is proposed.After a number of numerical experiments,the numerical effect of the new algorithm is better than that of the modified segmented secant method.In the fourth chapter,the modified Cholesky decomposition method is used to modify the Adams fourth-order prediction In this paper,the Adams fourth-order predictor corrector is proposed to solve the non stator problem,and the numerical results of the new algorithm are better than that of the modified secant method.