Posted on Categories:Optimization Theory, 优化理论, 数学代写

# 数学代写|优化理论代写Optimization Theory代考|Closeness of Functions

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## 数学代写|优化理论代写Optimization Theory代考|Closeness of Functions

If two points are said to be close to one another, a geometric interpretation springs immediately to mind. But what do we mean when we say two functions are close to one another? To give a precise meaning to the term “close” we next introduce the concept of a norm.
DEFINITION 4-5
The norm in n-dimensional Euclidean space is a rule of correspondence that assigns to each point $\mathbf{q}$ a real number. The norm of $\mathbf{q}$, denoted by $|\mathbf{q}|$, satisfies the following properties:

$|\mathbf{q}| \geq 0$ and $|\mathbf{q}|=0$ if and only if $\mathbf{q}=\mathbf{0}$.

$|\alpha q|=|\alpha| \cdot|q \mid|$ for all real numbers $\alpha$.

$\left|q^{(1)}+q^{(2)}\right| \leq\left|q^{(1)}\right|+\left|q^{(2)}\right|$.
When we say that two points $\mathbf{q}^{(1)}$ and $\mathbf{q}^{(2)}$ are close together, we mean that
$\left|\mathbf{q}^{(1)}-\mathbf{q}^{(2)}\right|$ is small.
Example 4.1-5. What is a suitable norm for two-dimensional Euclidean space? It is easily verified that
$$|\mathbf{q}|_2 \triangleq \sqrt{q_1^2+q_2^2}, \text { or }|\mathbf{q}|_1 \triangleq\left|q_1\right|+\left|q_2\right|$$
satisfies properties (4.1-14). Now suppose that a point $\mathbf{q}^{(1)}$ is specified and it is required that $\left|\mathbf{q}^{(2)}-\mathbf{q}^{(1)}\right|<\delta$. What are the acceptable locations for $q^{(2)}$ ? If $|\mathbf{q}|_2$ is used as the norm, $\mathbf{q}^{(2)}$ must lie within the circle centered at $q^{(1)}$ having radius $\delta$ as shown in Fig. 4-2(a). On the other hand, if $|\mathbf{q}|_1$ is used as the norm, the acceptable locations for $q^{(2)}$ are as shown in Fig, 4-2(b).

## 数学代写|优化理论代写Optimization Theory代考|The Increment of a Functional

In order to consider extreme values of a function, we now define the concept of an increment.
DEFINITION 4-7
If $\mathbf{q}$ and $\mathbf{q}+\Delta \mathbf{q}$ are elements for which the function $f$ is defined, then the increment of $f$, denoted by $\Delta f$, is
$$\Delta f \triangleq f(\mathbf{q}+\Delta \mathbf{q})-f(\mathbf{q})$$
Notice that $\Delta f$ depends on both $\mathbf{q}$ and $\Delta \mathbf{q}$, in general, so to be more explicit we would write $\Delta f(\mathbf{q}, \Delta \mathbf{q})$.
Example 4.1-7. Consider the function
$$f(\mathbf{q})=q_1^2+2 q_1 q_2 \text { for all real } q_1, q_2 .$$
The increment of $f$ is
\begin{aligned} \Delta f= & f(\mathbf{q}+\Delta \mathbf{q})-f(\mathbf{q})=\left[q_1+\Delta q_1\right]^2 \ & +2\left[q_1+\Delta q_1\right]\left[q_2+\Delta q_2\right]-\left[q_1^2+2 q_1 q_2\right] \ = & 2 q_1 \Delta q_1+\left[\Delta q_1\right]^2+2 \Delta q_1 q_2+2 \Delta q_2 q_1+2 \Delta q_1 \Delta q_2 \end{aligned}
In an analogous manner, we next define the increment of a functional. DEFINITION 4-8

If $\mathbf{x}$ and $\mathbf{x}+\delta \mathbf{x}$ are functions for which the functional $J$ is defined, then the increment of $J$, denoted by $\Delta J$, is
$$\Delta J \triangleq J(\mathbf{x}+\delta \mathbf{x})-J(\mathbf{x})$$
Again, to be more explicit, we would write $\Delta J(\mathbf{x}, \delta \mathbf{x})$ to emphasize that the increment depends on the functions $\mathbf{x}$ and $\delta \mathbf{x} . \delta \mathbf{x}$ is called the variation of the function $\mathbf{x}$.

## 数学代写|优化理论代写Optimization Theory代考|Closeness of Functions

n维欧几里得空间中的范数是一个对应的规则，它赋予每个点$\mathbf{q}$一个实数。$\mathbf{q}$的范数用$|\mathbf{q}|$表示，满足以下性质:

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。