Research On Some Optimization Algorithms And Applications For Solving Nonlinear Equations

Optimization theory and methods is an applied discipline,and it studies how to find the optimal solution from many feasible solutions to a practical problem.Solving nonlinear equations is an important part of optimization theory,and it has been widely applied in many fields of national economy such as finance,trade,management,scientific research,and so on.This thesis mainly studies some optimization algorithms and applications for solving nonlinear equations.The trust region and new secant methods based on fractional model for solving nonlinear equations are proposed.Methods for solving nonlinear equations are applied to find Z-eigenvalues of symmetric tensors for the first time,the quasi-Newton and conjugate gradient methods for computing the Z-eigenpairs of symmetric tensors are proposed.Some convergences of these methods are proved,and the numerical analysis are given.This thesis also studies some applications of the proposed methods in financial investment problems.The whole thesis is divided into eight chapters.Chapter I is introductory.Research background and significance,preliminary knowledge,research status and main contents are discussed.In Chapter II,main methods for solving nonlinear equations are introduced.We mainly study some algorithms for solving nonlinear equations in Chapters III and IV.In Chapter III,a trust region method based on fractional model is proposed for solving nonlinear equations.We demonstrate the property of the algorithm,prove the global and quadratic convergence of the algorithm,and report the comparative test results and analysis of our method and the trust region Newton method.The preliminary numerical results illustrate that our method is more efficient than the trust region Newton method for the functions whose curvatures change dramatically.In Chapter IV,a new secant method based on fractional model is established for solving nonlinear equations.An attractive feature of this method is the superlinear convergence under the local error bound condition without computation of Jacobian.We report the comparative experiment results and analysis of our method and the quasi-Newton method,the preliminary numerical results show that our method is more efficient than the quasi-Newton method.We mainly study the applications of some algorithms for solving nonlinear equations in Chapters V and VI.In Chapter V,we convert the problem for computing the Z-eigenvalues of symmetric tensors into solving nonlinear equations,and prove the symmetric property of Jacobian.Therefore,the quasi-Newton method for computing the Z-eigenpairs of symmetric tensors is proposed,the global and superlinear convergence of the proposed method is estab-lished.We give the comparative test results and analyses,the numerical results show that our method is more efficient than SS-HOPM in some sense.In Chapter VI,we improve the line search technique of the modified FR conjugate gradient method for symmetric nonlinear equa-tions,and propose a modified FR conjugate gradient method for computing the Z-eigenpairs of symmetric tensors.We prove the global convergence of our method,and report the numerical comparative experiment results and analysis of our method.The numerical results show the promising performance of our method.In Chapter VII,some applications of the proposed methods for solving nonlinear equations in financial investment problems are given.In the last Chapter,all the algorithms developed in the thesis are summarized and some problems that are worthwhile for further research are proposed.