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# 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 $\mathbf{q}=\mathbf{0}$.

$|\alpha q|=|\alpha| \cdot|\mathbf{q}|$ 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 $|q|_2$ is used as the norm, $\mathbf{q}^{(2)}$ must lie within the circle centered at $\mathbf{q}^{(1)}$ 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 $\mathbf{q}^{(2)}$ are as shown in Fig, 4-2(b).

Next, let us define the norm of a function.
DEFINITION 4-6
The norm of a function is a rule of correspondence that assigns to each function $\mathbf{x} \in \Omega$, defined for $t \in\left[t_0, t_f\right]$, a real number. The norm of $\mathbf{x}$, denoted by $|\mathbf{x}|$, satisfies the following properties:

1. $|\mathbf{x}| \geq 0$ and $|\mathbf{x}|=0$ if and only if $\mathbf{x}(t)=\mathbf{0}$ for all $t \in\left[t_0, t_f\right]$
$(4.1-15 a)$
2. $|\alpha \mathbf{x}|=|\alpha| \cdot|\mathbf{x}|$ for all real numbers $\alpha$.
3. $\left|\mathbf{x}^{(1)}+\mathbf{x}^{(2)}\right| \leq\left|\mathbf{x}^{(1)}\right|+\left|\mathbf{x}^{(2)}\right|$.
$(4.1-15 c)$

## 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 \ & \left.+2\left[q_1+\Delta q_1\right] 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 $\mathbf{x}$.

## 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:

$|\mathbf{q}| \geq 0$$|\mathbf{q}|=0当且仅当\mathbf{q}=\mathbf{0}。 |\alpha q|=|\alpha| \cdot|\mathbf{q}| 对于所有实数\alpha。 \left|q^{(1)}+q^{(2)}\right| \leq\left|q^{(1)}\right|+\left|q^{(2)}\right|． 当我们说两点\mathbf{q}^{(1)}和\mathbf{q}^{(2)}在一起时，我们的意思是 \left|\mathbf{q}^{(1)}-\mathbf{q}^{(2)}\right|很小。 例4.1-5。二维欧几里得空间合适的范数是什么?很容易证实$$ |\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| $$满足属性(4.1-14)。现在假设指定了一个点\mathbf{q}^{(1)}，并且要求\left|\mathbf{q}^{(2)}-\mathbf{q}^{(1)}\right|<\delta。q^{(2)}可接受的位置是什么?如果使用|q|_2作为范数，则\mathbf{q}^{(2)}必须位于以\mathbf{q}^{(1)}为圆心、半径为\delta的圆内，如图4-2(a)所示。另一方面，如果以|\mathbf{q}|_1为标准，则\mathbf{q}^{(2)}的可接受位置如图4-2(b)所示。 接下来，让我们定义函数的范数。 定义4-6 函数的范数是一个对应的规则，它分配给每个函数\mathbf{x} \in \Omega，为t \in\left[t_0, t_f\right]定义一个实数。\mathbf{x}的范数用|\mathbf{x}|表示，满足以下性质: |\mathbf{x}| \geq 0$$|\mathbf{x}|=0$当且仅当$\mathbf{x}(t)=\mathbf{0}$适用于所有人 $t \in\left[t_0, t_f\right]$
$(4.1-15 a)$

$|\alpha \mathbf{x}|=|\alpha| \cdot|\mathbf{x}|$ 对于所有实数$\alpha$。

$\left|\mathbf{x}^{(1)}+\mathbf{x}^{(2)}\right| \leq\left|\mathbf{x}^{(1)}\right|+\left|\mathbf{x}^{(2)}\right|$．
$(4.1-15 c)$

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

$$\Delta f \triangleq f(\mathbf{q}+\Delta \mathbf{q})-f(\mathbf{q}) .$$

$$f(\mathbf{q})=q_1^2+2 q_1 q_2 \text { for all real } q_1, q_2 .$$
$f$的增量为
\begin{aligned} \Delta f= & f(\mathbf{q}+\Delta \mathbf{q})-f(\mathbf{q})=\left[q_1+\Delta q_1\right]^2 \ & \left.+2\left[q_1+\Delta q_1\right] 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}

$$\Delta J \triangleq J(\mathbf{x}+\delta \mathbf{x})-J(\mathbf{x}) .$$

## MATLAB代写

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