Study Of The Adaptive Trust Region Method And Its Application In Engineering

The establishment of a mathematical model to optimize the design problem and choosing the appropriate optimization methods are two important contents of mechanical optimization design. In order to grasp the optimal design method, it needs to update knowledge in optimization theory, modeling and computer applications; as the development of CAD/CAM and CIMS (Computer Integrated Manufacturing System), it becomes an indispensable technology and contemporary aspects. Because mechanical optimization design applies mathematical methods to seek the best solution for mechanical design, it firstly establishes a corresponding mathematical model based on the actual mechanical design problems that should have mathematical form to describe the actual design problems. In creating a mathematical model, it is necessary to apply the expertise to determine design constraints and the pursuit of the goal to build the connection between these variables. Once established, the mechanical optimization problem becomes a mathematical problem. We can use the theory of mathematical programming to choose the appropriate optimization methods according to the characteristics of the mathematical model. For solving optimization problems in practical application, trust region method is an important method, which need not require the Hessian matrix to be positive definite at each iteration and also can be applied to solve some ill-conditioned problems. Moreover, it has strong convergence and robustness. Because of these advantages, trust region method becomes popular in nonlinear optimization. Firstly, we study a class of nonmonotone self-adaptive trust region methods, under the reasonable conditions, we establish the appropriate models and discuss the properties, for example, the global and the superlinear convergence. It can be used in the field of machinery. The main contents are as follows:1. A new class of trust region method is given. Firstly, with the help of the nonmonotone technique, we propose an adaptive trust region method. At every iteration point, during the iterative process, the algorithm does not require the function value decline. The non-monotonic property accelerates the convergence rate, especially in the presence of the narrow curved valley. It makes full use of the information of the current and the previous iterative points to define the trust region radius at each iteration which can make the quadratic model approach the objective function better in the given trust region. This adaptive technique not only overcomes the blindness of selection for the initial trust region radius, but also reduces the complexity of the problem and so that the speed of the given algorithm can be improved. Finally, under some certain assumptions, we prove the convergence properties of above method.2. On the basis of the quadratic model trust region subproblem, a nonmonotone trust region method with line search is proposed. At the current iteration, we take advantage of the trust region techniques to find the next iteration point, if the approximate solution of the subproblem can not be unaccepted, the method finds the next iterative with the help of the nonmonotone line search technique. Therefore, this method can always find a successful iterative point by solving the subproblem only once at each iteration. Thus a large quantity of computation is avoidable and the convergence rate of the new method is accelerated. Under some assumptions, the global convergence property is proved.3. Nonmonotone adaptive trust region methods in the design of mechanical optimization, and the methods and the optimization method were compared to show that non-monotonic adaptive trust region methods are effective and feasible.