| This thesis mainly deals with optimization iteration method for non-linear equations. It is well known that the research of nonlinear problem became popular from 1950s and 1960s, and searching approximate solutions problem for nonlinear differential equations is an important branch of the nonlinear science. There are many useful methods for solving nonlinear dif-ferential equations, such as perturbation method, approximation method,Multi-scale method,Harmonic balance method,Homotopy method,Vari-ational iteration method,Adomain decomposition methods, etc. However, there is not a universal method in this field.In Chapter 2, we obtain a new iteration method for searching approxi-mate solutions problem for nonlinear differential equations. Our idea comes from Constant variation formula. We mainly consider the following equation where a(·),b(·) and f(·,u,u') is T-periodic in t. FromConstant variation formula, we obtain the following modified interation procedure Where Lagrange multiplier is For variational interation method, one has to obtain Lagrange multiplier from stability condition by modifying the above functional, while Lagrange multiplier can be determined by Constant variation formula, directly using our method. Thus the actual calculation greatly enhances its usefulness and to reduce unnecessary variation caused by the complex process of computing. In addition, variational iteration method will often lead to the emergence of a secular terms, like the need to select the optimal Lagrange multiplier and the initial conditions to guarantee the convergence of the iterative process, and in order to eliminate the secular terms options that would greatly increase the computational complexity. take advantage of our approach there will be no secular terms, which greatly saves computational time.In Chapter 3, we give a new iteration algorithm (SFIA), using this kind of iteration algorithm and Constant variation formula, we can obtain the iteration method for approximate solutions to nonlinear systems. this kind of iteration algorithm is irrelated to secular terms. Consider the 2n-dimensional equationWhere a(t) and b(t) are continuous, f(t, u, u), the derivative of f is bounded in D. We can obtain the followingThere are a lot of methods for searching approximate solutions problem, such as, Formal group methods, multi-scale method. All of these methods for solving the approximate solution in the process can not avoid secular terms, resulting in additional costs to eliminate secular terms, leading to a very low computation efficiency. Take advantage of our iterative method can be contains a secular terms, Using this kind of iteration method we can obtain the approximate solutions of nonlinear differential equations. Through a simple iterative procedure can be simplified with high precision Approximate solution, in addition to the secular terms because they are not elimination, thereby reducing a huge amount of computing.In Chapter 4, for programming problem with constraining conditions, we give a new kind of iteration method for solving the programming problem. Consider the following programming problem: The corresponding Lagrangian is Where g(x)= (g1(x), g2(x),=(g1(x),g2(x)…,gm(x))T, h(x)=(h1(x), h2(x),…, hl(x))T, y∈Rm,z∈Rl, and fi,gi,hiis a C2 function of x∈Rn in order to obtain the op-timal solution for the above programming problem, we provide the following iteration formulawhere the style of the Lagrange multiplier y, z, can be done on both sides of the previous type of variational get an algebraic equation on the multiplier, which can be obtained. And then back to the iterative formula. Variational method to study the use of such programming problem with constraints, as compared with other optimization methods, do not directly calculate the KKT conditions, but to use Lagrange function is directly iteration, Thereby greatly simplifying the operation steps, saving computation time.
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