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# 数学代写|数学分析作业代写Mathematical Analysis代考|Convexity Criterion for Differentiable Functions

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Convexity Criterion for Differentiable Functions

Convexity Criterion for Differentiable Functions Let $A$ be an open and convex set in $\mathbb{R}^n$ and $f: A \rightarrow \mathbb{R}$ a differentiable function on $A$. Then $f$ is convex on $A$ if and only if
$$f(x) \geq f\left(x_0\right)+\left(D f\left(x_0\right), x-x_0\right), \quad \forall x, x_0 \in A$$
or if and only if
$$(D f(y)-D f(x), y-x) \geq 0, \quad \forall x, y \in A$$
Condition (3.86) is interpreted geometrically by saying that the graph of $f$ lies above the tangent plane, in correspondence to any point $x \in A$ and for any tangent point $x_0 \in A$. In view of (3.87) we say that the Jacobian $D f: A \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^n$ of a convex function is a monotone operator on $\mathbb{R}^n$, thus extending the monotonicity, well known for $n=1$, of the derivatives of convex functions in one real variable.
Proof If $f$ is convex, by definition
$$f\left(t x+(1-t) x_0\right) \leq t f(x)+(1-t) f\left(x_0\right)$$

for any $x, x_0 \in A$ and $t \in[0,1]$. We rewrite this as
$$f\left(x_0+t\left(x-x_0\right)\right) \leq f\left(x_0\right)+t\left[f(x)-f\left(x_0\right)\right]$$
whence
$$\frac{f\left(x_0+t\left(x-x_0\right)\right)-f\left(x_0\right)}{t} \leq f(x)-f\left(x_0\right) .$$
The limit as $t \rightarrow 0^{+}$of the left-hand side is the derivative of $\varphi(t)=f\left(x_0+t\left(x-x_0\right)\right)$ at $t=0$. By the chain rule, when $t \rightarrow 0^{+}$we have
$$\left(D f\left(x_0\right), x-x_0\right) \leq f(x)-f\left(x_0\right)$$

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|onvexity Criterion for $C^2$ Functions

Convexity Criterion for $C^2$ Functions Let $A$ be an open convex set of $\mathbb{R}^n$ and $f$ : $A \rightarrow \mathbb{R}$ a function of class $C^2(A)$. Then $f$ is convex on $A$ if and only if the Hessian matrix $D^2 f(x)$ is positive semi-definite for any $x \in A$.

Proof Suppose $D^2 f(x)$ is positive semi-definite for any $x \in A$. Given $x, x_0 \in A$, Taylor’s formula of order two with Lagrange remainder ensures that there is a real number $\vartheta \in(0,1)$ such that
$$f(x)=f\left(x_0\right)+\left(D f\left(x_0\right), x-x_0\right)+\frac{1}{2}\left(D^2 f\left(x_0+\vartheta\left(x-x_0\right)\right) \cdot\left(x-x_0\right), x-x_0\right)$$
As the Hessian matrix is positive semi-definite at every point in the convex set $A \subset \mathbb{R}^n$, and since
$$x_0+\vartheta\left(x-x_0\right)=\vartheta x+(1-\vartheta) x_0$$
is a convex combination of $x, x_0 \in A$, and hence belongs to $A$, it follows that
\begin{aligned} & f(x)-f\left(x_0\right)-\left(D f\left(x_0\right), x-x_0\right)= \ = & \frac{1}{2}\left(D^2 f\left(x_0+\vartheta\left(x-x_0\right)\right) \cdot\left(x-x_0\right), x-x_0\right) \geq 0 . \end{aligned}

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Convexity Criterion for Differentiable Functions

$$f(x) \geq f\left(x_0\right)+\left(D f\left(x_0\right), x-x_0\right), \quad \forall x, x_0 \in A$$

$$(D f(y)-D f(x), y-x) \geq 0, \quad \forall x, y \in A$$

$$f\left(t x+(1-t) x_0\right) \leq t f(x)+(1-t) f\left(x_0\right)$$

$$f\left(x_0+t\left(x-x_0\right)\right) \leq f\left(x_0\right)+t\left[f(x)-f\left(x_0\right)\right]$$

$$\frac{f\left(x_0+t\left(x-x_0\right)\right)-f\left(x_0\right)}{t} \leq f(x)-f\left(x_0\right) .$$

$$\left(D f\left(x_0\right), x-x_0\right) \leq f(x)-f\left(x_0\right)$$

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|onvexity Criterion for $C^2$ 功能

$$f(x)=f\left(x_0\right)+\left(D f\left(x_0\right), x-x_0\right)+\frac{1}{2}\left(D^2 f\left(x_0+\vartheta\left(x-x_0\right)\right) \cdot\left(x-x_0\right), x-x_0\right)$$

$$x_0+\vartheta\left(x-x_0\right)=\vartheta x+(1-\vartheta) x_0$$

$$f(x)-f\left(x_0\right)-\left(D f\left(x_0\right), x-x_0\right)==\quad \frac{1}{2}\left(D^2 f\left(x_0+\vartheta\left(x-x_0\right)\right) \cdot\left(x-x_0\right), x-x_0\right) \geq 0$$

## 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。