The Improved Euler Method For Trust Region Subproblem Based On Differential Equation Model

With the development and application of computer,the nonlinear optimization theory and method has become an important branch of operational research,and is widely used in the natural science,system engineering,economic management,optimization design and other fields.In nonlinear optimization,the trust region method is widely used,and has become a research focus because of its good properties such as strong well-posedness and convergence.At present,the trust region method and the line search method are two kinds of important method to solve unconstrained optimization problems.So far,in terms of how to construct trust region subproblem model has a lot of discussion.For solving trust region subproblems with the initially proposed quadratic model,the dogleg method has attracted much attention in recent years.With the in-depth study of dogleg method,the trust region subproblem optimal curve of the differential equation model is put forward by director Wang Xi-yun and senior Li Liang.A class of Euler’s tangents are built according to the model,which have achieved good numerical results and opened up a new calculation way for solving trust region subproblems.Based on the study about differential equation model which is put forward by Wang Xi-yun and Li Liang,this paper mainly improves the Euler’s tangent for solving trust region subproblems,then an improved Euler’s tangent algorithm for solving trust region subproblems is proposed.The following is the concrete research content of this paper:In the first chapter,the background and research status of the trust region algorithm and trust region subproblems are introduced,and the breakthroughs of this paper are briefly introduced.In the second chapter,an improved Euler’s tangent algorithm for solving trust region subproblems is proposed.Based on Euler’s tangent algorithm,we improves the assumed condition of the theorem,and the step of the algorithm is simplified.In this chapter,the properties of the improved Euler’s tangent for solving trust region subproblems is analyzed,and the numerical experiments demonstrate that the advantages of the new algorithm compared with the original algorithm are less number of iterations,short computing time,and so on.and the new algorithm is effective and feasible.In the third chapter,an improved implicit Euler’s tangent algorithm for solving trust region subproblems is proposed.Based on implicit Euler’s tangent algorithm,we improves the assumed condition of the theorem,and the step of the algorithm is simplified.In this chapter,the properties of the improved implicit Euler’s tangent for solving trust region subproblems is analyzed,and the numerical experiments demonstrate that the advantages of the new algorithm compared with the original algorithm are less number of iterations,short computing time,and so on.and the new algorithm is effective and practical.In the fourth chapter,an improved mean Euler’s tangent algorithm for solving trust region subproblems is proposed.Based on mean Euler’s tangent algorithm,we improves the assumed condition of the theorem,and the step of the algorithm is simplified.In this chapter,the properties of the improved mean Euler’s tangent for solving trust region subproblems is analyzed,and the numerical experiments demonstrate that the advantages of the new algorithm compared with the original algorithm are less number of iterations,short computing time,and so on.And the new algorithm is effective and feasible.