Automatic Differentiation, Extended System And Optimization Method

With the development of science and technology,nonlinear optimization method plays a more and more significant role in scientific computation and engineering analysis fields.As far as nonlinear optimization is concerned,the computation mostly depends on the first and high order derivatives as well as other items(e.g.the multiplication between Jacobian matrix and vector) of the objective function and constraint function.Automatic differentiation,which is characteristic of low cost and high accuracy compared with the traditional differentiation methods,is an efficient tool to calculate the derivatives.In this paper,from different standpoints of the basic and extended system,some research was carried on to compare and analyze the automatic differentiation method for the first order derivative,including the reverse mode and forward mode and so on.At the same time,these modes were applied to the augmented Lagrange multiplier.When attempting to solve the high order derivatives,this paper discussed the Chebyshev method which belongs to the third order derivative Halley class method,and improved the original algorithm. At last,some numerical experiments were done to realize the algorithm and demonstrate its high efficiency.