statistics-lab™ 为您的留学生涯保驾护航 在代写凸优化Convex Optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写凸优化Convex Optimization代写方面经验极为丰富，各种代写凸优化Convex Optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

Since vector limits are computed by taking the limit of each coordinate function, we can write the function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ for a point $x \in \mathbb{R}^{n}$ as follows:
$$\boldsymbol{f}(\mathbf{x})=\left[\begin{array}{c} f_{1}(\mathbf{x}) \ f_{2}(\mathbf{x}) \ \vdots \ f_{m}(\mathbf{x}) \end{array}\right]=\left(f_{1}(\mathbf{x}), f_{2}(\mathbf{x}), \ldots, f_{m}(\mathbf{x})\right)$$
where each $f_{i}(\mathbf{x})$ is a function from $\mathbb{R}^{n}$ to $\mathbb{R}$. Now, $\frac{\partial \boldsymbol{f}(\mathbf{x})}{\partial x_{j}}$ can be defined as
$$\frac{\partial \boldsymbol{f}(\mathbf{x})}{\partial x_{j}}=\left[\begin{array}{c} \frac{\partial f_{1}(\mathbf{x})}{\partial x_{j}} \ \frac{\partial f_{2}(\mathbf{x})}{\partial x_{j}} \ \vdots \ \frac{\partial f_{m}(\mathbf{x})}{\partial x_{j}} \end{array}\right]=\left(\frac{\partial f_{1}(\mathbf{x})}{\partial x_{j}}, \frac{\partial f_{2}(\mathbf{x})}{\partial x_{j}}, \ldots, \frac{\partial f_{m}(\mathbf{x})}{\partial x_{j}}\right)$$
The above vector is a tangent vector at the point $\mathrm{x}$ of the curve $f$ obtained by varying only $x_{j}$ (the $j$ th coordinate of $\mathbf{x}$ ) with $x_{i}$ fixed for all $i \neq j$.

The derivative of a differentiable function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ can be represented by $D \boldsymbol{f}(\mathbf{x})$ – an $m \times n$ matrix defined as
$$\begin{gathered} =\left[\begin{array}{ccc} \frac{\partial \boldsymbol{f}(\mathbf{x})}{\partial x_{1}} \frac{\partial \boldsymbol{f}(\mathbf{x})}{\partial x_{2}} & \cdots & \frac{\partial \boldsymbol{f}(\mathbf{x})}{\partial x_{n}} \end{array}\right]=\left[\begin{array}{c} \nabla f_{1}(\mathbf{x})^{T} \ \nabla f_{2}(\mathbf{x})^{T} \ \vdots \ \nabla f_{m}(\mathbf{x})^{T} \end{array}\right] \ =\left[\begin{array}{ccc} \frac{\partial f_{1}(\mathbf{x})}{\partial x_{1}} \ldots & \frac{\partial f_{1}(\mathbf{x})}{\partial x_{n}} \ \vdots & \vdots \ \frac{\partial f_{m}(\mathbf{x})}{\partial x_{1}} \cdots & \frac{\partial f_{m}(\mathbf{x})}{\partial x_{n}} \end{array}\right] \in \mathbb{R}^{m \times n} \end{gathered}$$
where $\nabla f_{i}(\mathbf{x})$ will be defined below. The above matrix $D \boldsymbol{f}(\mathbf{x})$ is called the Jacobian matrix or derivative matrix of $f$ at the point $\mathbf{x}$.

If the function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ is differentiable, then its gradient $\nabla f(\mathbf{x})$ at a point $\mathbf{x}$ can be defined as
$$f(\mathbf{x})=D f(\mathbf{x})^{T}=\left[\begin{array}{c} \frac{\partial f(\mathbf{x})}{\partial x_{1}} \ \frac{\partial f(\mathbf{x})}{\partial x_{2}} \ \vdots \ \frac{\partial f(\mathbf{x})}{\partial x_{n}} \end{array}\right] \in \mathbb{R}^{n}$$
Note that $\nabla f(\mathbf{x})$ and $\mathbf{x}$ have the same dimension (i.e., both are column vectors of dimension $n$ ). Moreover, if the function $f: \mathbb{R}^{n \times m} \rightarrow \mathbb{R}$ is differentiable, then its gradient $\nabla f(\mathbf{X})$ at a point $\mathbf{X}$ can be defined as
$$\nabla f(\mathbf{X})=D f(\mathbf{X})^{T}=\left[\begin{array}{ccc} \frac{\partial f(\mathbf{X})}{\partial x_{1,1}} & \cdots & \frac{\partial f(\mathbf{X})}{\partial x_{1, m}} \ \vdots & \vdots & \vdots \ \frac{\partial f(\mathbf{X})}{\partial x_{n, 1}} \cdots & \frac{\partial f(\mathbf{X})}{\partial x_{n, m}} \end{array}\right] \in \mathbb{R}^{n \times m}$$
which also has the same dimension with $\mathbf{X} \in \operatorname{dom} f$.

## 数学代写|凸优化作业代写Convex Optimization代考|Hessian

Suppose that $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ is twice differentiable, and all its second partial derivatives exist and are continuous over the domain of $f$. The Hessian $\nabla^{2} f(\mathbf{x})$ of $f$ is defined as follows:
$\nabla^{2} f(\mathbf{x})=D(\nabla f(\mathbf{x}))=\left{\frac{\partial^{2} f(\mathbf{x})}{\partial x_{i} \partial x_{j}}\right}_{n \times n}$
$=\left[\begin{array}{cccc}\frac{\partial^{2} f(\mathbf{x})}{\partial x_{1}^{2}} & \frac{\partial^{2} f(\mathbf{x})}{\partial x_{1} \partial x_{2}} & \cdots & \frac{\partial^{2} f(\mathbf{x})}{\partial x_{1} \partial x_{n}} \ \frac{\partial^{2} f(\mathbf{x})}{\partial x_{2} \partial x_{1}} & \frac{\partial^{2} f(\mathbf{x})}{\partial x_{2}^{2}} & \ldots & \frac{\partial^{2} f(\mathbf{x})}{\partial x_{2} \partial x_{n}} \ \vdots & \vdots & & \vdots \ \frac{\partial^{2} f(\mathbf{x})}{\partial x_{n} \partial x_{1}} & \frac{\partial^{2} f(\mathbf{x})}{\partial x_{n} \partial x_{2}} & \ldots & \frac{\partial^{2} f(\mathbf{x})}{\partial x_{n}^{2}}\end{array}\right] \in \mathbb{S}^{n}$
The Hessian of a function can be used for verifying the convexity of a twice differentiable function, so its calculation is needed quite often. For instance, assuming that
$$f(\mathbf{x})=\mathbf{x}^{T} \mathbf{P} \mathbf{x}+\mathbf{x}^{T} \mathbf{q}+c,$$
where $\mathbf{P} \in \mathbb{R}^{n \times n}, \mathbf{q} \in \mathbb{R}^{n}$, and $c \in \mathbb{R}$, one can easily obtain
$$\nabla f(\mathbf{x})=\left(\mathbf{P}+\mathbf{P}^{T}\right) \mathbf{x}+\mathbf{q}, \quad \nabla^{2} f(\mathbf{x})=D(\nabla f(\mathbf{x}))=\mathbf{P}+\mathbf{P}^{T} .$$
For the case of $\mathbf{P}=\mathbf{P}^{T} \in \mathbb{S}^{n}$,
$$\nabla f(\mathbf{x})=2 \mathbf{P x}+\mathbf{q}, \quad \nabla^{2} f(\mathbf{x})=D(\nabla f(\mathbf{x}))=2 \mathbf{P}$$

Consider another example as follows:
\begin{aligned} g(\mathbf{y})=|\mathbf{A} \mathbf{x}-\mathbf{z}|_{2}^{2}, \mathbf{y}=(\mathbf{x}, \mathbf{z}) \in \mathbb{R}^{n+m}, \mathbf{A} \in \mathbb{R}^{m \times n} \ \Longrightarrow \nabla g(\mathbf{y}) &=\left[\begin{array}{c} \nabla_{\mathbf{x}} g(\mathbf{y}) \ \nabla_{\mathbf{z}} g(\mathbf{y}) \end{array}\right]=\left[\begin{array}{cc} 2 \mathbf{A}^{T} \mathbf{A} \mathbf{x}-2 \mathbf{A}^{T} \mathbf{z} \ 2 \mathbf{z}-2 \mathbf{A x} \end{array}\right] \in \mathbb{R}^{n+m} \ \Longrightarrow \nabla^{2} g(\mathbf{y}) &=\left[\begin{array}{cc} D\left(\nabla_{\mathbf{x}} g(\mathbf{y})\right) \ D\left(\nabla_{\mathbf{z}} g(\mathbf{y})\right) \end{array}\right]=\left[\begin{array}{cc} \nabla_{\mathbf{x}}^{2} g(\mathbf{y}) & D_{\mathbf{z}}\left(\nabla_{\mathbf{x}} g(\mathbf{y})\right) \ D_{\mathbf{x}}\left(\nabla_{\mathbf{z}} g(\mathbf{y})\right) & \nabla_{\mathbf{z}}^{2} g(\mathbf{y}) \end{array}\right] \ &=\left[\begin{array}{cc} 2 \mathbf{A}^{T} \mathbf{A} & -2 \mathbf{A}^{T} \ -2 \mathbf{A} & 2 \mathbf{I}{m} \end{array}\right] \in \mathbb{S}^{n+m} \end{aligned} What is the gradient of $f(\mathbf{X})=\log \operatorname{det}(\mathbf{X})$ for $\mathbf{X} \in \mathbb{S}{++}^{n}$ (the set of positive definite matrices)? The answer will be given in Chapter 3 (cf. Remark $3.20$ ).

## 数学代写|凸优化作业代写Convex Optimization代考|Taylor series

Assume that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is $m$ times continuously differentiable. Then
\begin{aligned} f(x+h)=& f(x)+\frac{h}{1 !} f^{(1)}(x)+\frac{h^{2}}{2 !} f^{(2)}(x)+\cdots \ &+\frac{h^{m-1}}{(m-1) !} f^{(m-1)}(x)+R_{m} \end{aligned}
is called the Taylor series expansion, where $f^{(i)}$ is the $i$ th derivative of $f$, and
$$R_{m}=\frac{h^{m}}{m !} f^{(m)}(x+\theta h)$$
is the residual where $\theta \in[0,1]$. If $x=0$, then the series is called Maclaurin series.
On the other hand, if a function is defined as $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and if $f$ is $m$ times continuously differentiable, then the Taylor series expansion is given by
\begin{aligned} f(\mathbf{x}+\mathbf{h})=& f(\mathbf{x})+\frac{d f(\mathbf{x})}{1 !}+\frac{1}{2 !} d^{2} f(\mathbf{x})+\cdots \ &+\frac{1}{(m-1) !} d^{(m-1)} f(\mathbf{x})+R_{m}, \end{aligned}
where
( $h_{i}$ and $x_{i}$, respectively, denoting the $i$ th element of $\mathbf{h}$ and $\mathbf{x}$ ) and
$$R_{m}=\frac{1}{m !} d^{m} f(\mathbf{x}+\theta \mathbf{h})$$
for some $\theta \in[0,1]$.

Remark $1.12$ The first-order and second-order Taylor series expansions of a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ are given by
\begin{aligned} f(\mathbf{x}+\mathbf{h}) &=f(\mathbf{x})+\nabla f\left(\mathbf{x}+\theta_{1} \mathbf{h}\right)^{T} \mathbf{h}=f(\mathbf{x})+D f\left(\mathbf{x}+\theta_{1} \mathbf{h}\right) \mathbf{h} \ &=f(\mathbf{x})+\nabla f(\mathbf{x})^{T} \mathbf{h}+\frac{1}{2} \mathbf{h}^{T} \nabla^{2} f\left(\mathbf{x}+\theta_{2} \mathbf{h}\right) \mathbf{h} \end{aligned}
for some $\theta_{1}, \theta_{2} \in[0,1]$. When $f: \mathbb{R}^{n \times m} \rightarrow \mathbb{R}$, let $\mathbf{X}=\left{x_{i j}\right}_{n \times m}=\left[\mathbf{x}{1}, \ldots, \mathbf{x}{m}\right]$ and $\mathbf{H}=\left[\mathbf{h}{1}, \ldots, \mathbf{h}{m}\right] \in \mathbb{R}^{n \times m}$. Then the first-order and second-order Taylor series expansions of $f$ are given by
\begin{aligned} f(\mathbf{X}+\mathbf{H})=& f(\mathbf{X})+\operatorname{Tr}\left(\nabla f\left(\mathbf{X}+\theta_{1} \mathbf{H}\right)^{T} \mathbf{H}\right) \ =& f(\mathbf{X})+\operatorname{Tr}\left(D f\left(\mathbf{X}+\theta_{1} \mathbf{H}\right) \mathbf{H}\right) \ =& f(\mathbf{X})+\operatorname{Tr}\left(\nabla f(\mathbf{X})^{T} \mathbf{H}\right)+\sum_{j=1}^{m} \sum_{l=1}^{m} \mathbf{h}{j}^{T} D{\mathbf{x}{l}}\left(\nabla{\mathbf{x}{j}} f\left(\mathbf{X}+\theta{2} \mathbf{H}\right)\right) \mathbf{h}{l} \ =& f(\mathbf{X})+\operatorname{Tr}\left(\nabla f(\mathbf{X})^{T} \mathbf{H}\right) \ &+\sum{j=1}^{m} \sum_{l=1}^{m} \mathbf{h}{j}^{T}\left{\frac{\partial^{2} f\left(\mathbf{X}+\theta{2} \mathbf{H}\right)}{\partial x_{i j} \partial x_{k l}}\right}_{n \times n} \mathbf{h}{l} \end{aligned} for some $\theta{1}, \theta_{2} \in[0,1]$. Moreover, (1.53) and (1.55) are also the corresponding first-order Taylor series approximations, and (1.54) and (1.56) are the corresponding second-order Taylor series approximations, if $\theta_{1}$ and $\theta_{2}$ are set to zero.
For a differentiable function $\boldsymbol{f}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$, the first-order Taylor series expansion is given by
$$\boldsymbol{f}(\mathrm{x}+\mathbf{h})=\boldsymbol{f}(\mathrm{x})+(D \boldsymbol{f}(\mathrm{x}+\theta \mathbf{h})) \mathbf{h}$$
for some $\theta \in[0,1]$, which is also the corresponding first-order Taylor series approximation of $\boldsymbol{f}(\mathbf{x})$ if $\theta$ is set to zero.

## 凸优化代写

F(X)=[F1(X) F2(X) ⋮ F米(X)]=(F1(X),F2(X),…,F米(X))

∂F(X)∂Xj=[∂F1(X)∂Xj ∂F2(X)∂Xj ⋮ ∂F米(X)∂Xj]=(∂F1(X)∂Xj,∂F2(X)∂Xj,…,∂F米(X)∂Xj)

=[∂F(X)∂X1∂F(X)∂X2⋯∂F(X)∂Xn]=[∇F1(X)吨 ∇F2(X)吨 ⋮ ∇F米(X)吨] =[∂F1(X)∂X1…∂F1(X)∂Xn ⋮⋮ ∂F米(X)∂X1⋯∂F米(X)∂Xn]∈R米×n

F(X)=DF(X)吨=[∂F(X)∂X1 ∂F(X)∂X2 ⋮ ∂F(X)∂Xn]∈Rn

∇F(X)=DF(X)吨=[∂F(X)∂X1,1⋯∂F(X)∂X1,米 ⋮⋮⋮ ∂F(X)∂Xn,1⋯∂F(X)∂Xn,米]∈Rn×米

## 数学代写|凸优化作业代写Convex Optimization代考|Hessian

\nabla^{2} f(\mathbf{x})=D(\nabla f(\mathbf{x}))=\left{\frac{\partial^{2} f(\mathbf{x})} {\partial x_{i} \partial x_{j}}\right}_{n \times n}\nabla^{2} f(\mathbf{x})=D(\nabla f(\mathbf{x}))=\left{\frac{\partial^{2} f(\mathbf{x})} {\partial x_{i} \partial x_{j}}\right}_{n \times n}
=[∂2F(X)∂X12∂2F(X)∂X1∂X2⋯∂2F(X)∂X1∂Xn ∂2F(X)∂X2∂X1∂2F(X)∂X22…∂2F(X)∂X2∂Xn ⋮⋮⋮ ∂2F(X)∂Xn∂X1∂2F(X)∂Xn∂X2…∂2F(X)∂Xn2]∈小号n

F(X)=X吨磷X+X吨q+C,

∇F(X)=(磷+磷吨)X+q,∇2F(X)=D(∇F(X))=磷+磷吨.

∇F(X)=2磷X+q,∇2F(X)=D(∇F(X))=2磷

G(是)=|一种X−和|22,是=(X,和)∈Rn+米,一种∈R米×n ⟹∇G(是)=[∇XG(是) ∇和G(是)]=[2一种吨一种X−2一种吨和 2和−2一种X]∈Rn+米 ⟹∇2G(是)=[D(∇XG(是)) D(∇和G(是))]=[∇X2G(是)D和(∇XG(是)) DX(∇和G(是))∇和2G(是)] =[2一种吨一种−2一种吨 −2一种2一世米]∈小号n+米什么是梯度F(X)=日志⁡这⁡(X)为了X∈小号++n（一组正定矩阵）？答案将在第 3 章中给出（参见备注3.20 ).

## 数学代写|凸优化作业代写Convex Optimization代考|Taylor series

F(X+H)=F(X)+H1!F(1)(X)+H22!F(2)(X)+⋯ +H米−1(米−1)!F(米−1)(X)+R米

R米=H米米!F(米)(X+θH)

F(X+H)=F(X)+dF(X)1!+12!d2F(X)+⋯ +1(米−1)!d(米−1)F(X)+R米,

（H一世和X一世，分别表示一世第一个元素H和X） 和
R米=1米!d米F(X+θH)

F(X+H)=F(X)+∇F(X+θ1H)吨H=F(X)+DF(X+θ1H)H =F(X)+∇F(X)吨H+12H吨∇2F(X+θ2H)H

\begin{aligned} f(\mathbf{X}+\mathbf{H})=& f(\mathbf{X})+\operatorname{Tr}\left(\nabla f\left(\mathbf{X}+ \theta_{1} \mathbf{H}\right)^{T} \mathbf{H}\right) \ =& f(\mathbf{X})+\operatorname{Tr}\left(D f\left( \mathbf{X}+\theta_{1} \mathbf{H}\right) \mathbf{H}\right) \ =& f(\mathbf{X})+\operatorname{Tr}\left(\nabla f (\mathbf{X})^{T} \mathbf{H}\right)+\sum_{j=1}^{m} \sum_{l=1}^{m} \mathbf{h}{j} ^{T} D{\mathbf{x}{l}}\left(\nabla{\mathbf{x}{j}} f\left(\mathbf{X}+\theta{2} \mathbf{H} \right)\right) \mathbf{h}{l} \ =& f(\mathbf{X})+\operatorname{Tr}\left(\nabla f(\mathbf{X})^{T} \mathbf {H}\right) \ &+\sum{j=1}^{m} \sum_{l=1}^{m} \mathbf{h}{j}^{T}\left{\frac{\部分^{2} f\left(\mathbf{X}+\theta{2} \mathbf{H}\right)}{\partial x_{i j} \partial x_{k l}}\right}_{n \次 n} \mathbf{h}{l} \end{aligned}\begin{aligned} f(\mathbf{X}+\mathbf{H})=& f(\mathbf{X})+\operatorname{Tr}\left(\nabla f\left(\mathbf{X}+ \theta_{1} \mathbf{H}\right)^{T} \mathbf{H}\right) \ =& f(\mathbf{X})+\operatorname{Tr}\left(D f\left( \mathbf{X}+\theta_{1} \mathbf{H}\right) \mathbf{H}\right) \ =& f(\mathbf{X})+\operatorname{Tr}\left(\nabla f (\mathbf{X})^{T} \mathbf{H}\right)+\sum_{j=1}^{m} \sum_{l=1}^{m} \mathbf{h}{j} ^{T} D{\mathbf{x}{l}}\left(\nabla{\mathbf{x}{j}} f\left(\mathbf{X}+\theta{2} \mathbf{H} \right)\right) \mathbf{h}{l} \ =& f(\mathbf{X})+\operatorname{Tr}\left(\nabla f(\mathbf{X})^{T} \mathbf {H}\right) \ &+\sum{j=1}^{m} \sum_{l=1}^{m} \mathbf{h}{j}^{T}\left{\frac{\部分^{2} f\left(\mathbf{X}+\theta{2} \mathbf{H}\right)}{\partial x_{i j} \partial x_{k l}}\right}_{n \次 n} \mathbf{h}{l} \end{aligned}对于一些θ1,θ2∈[0,1]. 此外，(1.53) 和 (1.55) 也是对应的一阶泰勒级数逼近，而 (1.54) 和 (1.56) 是对应的二阶泰勒级数逼近，如果θ1和θ2被设置为零。

F(X+H)=F(X)+(DF(X+θH))H

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## MATLAB代写

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