摘要
本文主要研究求解非线性方程近似解析解的若干优化迭代方法.全书分为四章：第一章是绪论,第二章建立了一种与长期项无关的优化迭代算法(SFIA),考虑非线性微分方程,
其中a(t)和b(t)都是连续的,f(t,u,υ’)在区域D内的一次偏导数有界,且是关于υ和u’的一般的非线性函数.根据常数变易公式,并结合变分迭代法的基本思想,建立了下面的优化迭代公式：
通过上面的简单迭代程序,可以方便的得到精度很高的近似解。并且迭代程序可以自动消除高阶近似中的长期项,极大地减少了运算量,具有高的运算效率.
第三章考虑下面的微分方程：的近似周期解问题。其中a(.),b(.)和f(.,u,u’)关于时间t都是T-周期的.
根据常数变易公式,可以得到下面的迭代公式：其中Lagrange乘子
应用这个迭代公式可以方面地求出近似的解析周期解。方法直接利用常数变易公式来确定其中的Lagrange乘子,而不需要对校正泛函实施变分,不需要应用稳定性条件确定Lagrange乘子。此外,建立的迭代公式使得高阶近似中不会出现长期项,从而极大的节约了运算时间.
第四章,针对带有约束的规划问题,我们利用变分法给出一种求解约束问题新的迭代方法.考虑下面带约束的规划问题：相应的lagrange函数为：其中g(x)=(g1(x),g2(x),…,gm(x))T,h(x)=(h_1(x),h_2(x),…,h_ι(x))T,y∈Rm,z∈R_l,并且f_i,g_i,h_i关于x∈R_n为二次连续可微函数.为了得到上述优化问题的最优解,我们构造如下迭代公式：其中上式中的Lagrange乘子y,z,可以通过对上式两边做变分,得到一个关于乘子的代数方程,从而可以求得.然后带回上式中进行迭代.利用变分法来研究这类带有约束的规划问题,与其它的优化方法相比,不用直接计算KKT条件,而是利用Lagrange函数直接进行迭代,从而极大的简化了运算步骤,节约了运算时间.
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 equation
Where a(t) and b(t) are continuous, f(t, u, u), the derivative of f is bounded in D. We can obtain the following
There 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 formula
where 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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