Research On Direct Search Methods For Solving Unconstrained Optimization And Nonlinear Equations

Nonlinear optimization is a subject with the strong applications,and it have widely used in defense,economy,finance,engineering,trade and many other fields.In addition,the non-linear optimization problem is closely related to the nonlinear systems of equations,and many nonlinear optimization problems can be transformed into nonlinear systems of equations.This thesis mainly studies the direct search method for solving unconstrained optimization problems in nonlinear optimization and the direct search method for solving the nonlinear systems of equations.The thesis has four aspects of research content.The first is that we propose a direct search frame-based adaptive Barzilai-Borwein method for unconstrained optimization problems.The method is based on the framework of frame-based direct search algorithms which is proposed by Coope and Price.In each iteration,The method uses the minimum positive basis to construct the frame,and obtains the search direction by using the frame,then obtains the step size directly with the adaptive Barzilai-Borwein method,and finally rotates the minimum positive base according to the local property of the objective function.Under mild assumptions,we prove the convergence of this method.Numerical experimentations show that the proposed method is promising.This is the main content of Chapter three.The second is that we propose a new hybrid direct search method,which combines the frame-based direct search algorithm with radial basis function interpolation trust region model.In each iteration,the method uses a minimal positive basis to construct the frame and estab-lishes the radial basis function interpolation trust region model.When the objective function value of the trial point obtained by the radial basis function interpolation trust region model does not satisfy the decrease condition,the method employs PRP formula to get the search direction.Further more,the method rotates the minimal positive basis according to the lo-cal topography of objective function,which make the method more effective in practice.The convergence is established under some mild conditions.Some numerical results show that the proposed method is promising.This is the main content of Chapter fourth.The third is that we propose a new quasi-Newton equation based on the general quasi-Newton equation.The new quasi-Newton equation exploits additional information by assuming a quadratic relationship between the information from the last three iterates.Further more,we use the new quasi-Newton equation to construct a modified quasi-Newton method for solving nonlinear systems of equations.The modified quasi-Newton method possesses local superlinear convergence properties.Numerical experiments show that the modified quasi-Newton method is effective for solving small and medium scale nonlinear systems of equations.This is the main content of Chapter fifth.The fourth is that we extend RMIL conjugate gradient method in unconstrained optimiza-tion problem to solve nonlinear systems of equations under the direct search method of spectral residuals,and then presents a RMIL conjugate gradient direct method for solving nonlinear systems of equations.In each iteration,the method uses the RMIL conjugate gradient method to obtain the search direction,and employs nonmonotone line search conditions,then finally obtains the step length by the backtracking method.We prove the convergence of the method.Numerical experiments show that the method is effective for solving medium and large scale nonlinear systems of equations.This is the main content of Chapter sixth.