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

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• 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

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

Convex (CVX) optimization is an important class of optimization techniques that includes least squares and linear programs as special cases, and has been extensively used in various science and engineering areas. If one can formulate a practical problem as a convex optimization problem, then actually he (she) has solved the original problem (for an optimal solution either analytically or numerically), like least squares (LS) or linear program, (almost) technology. This chapter provides some essential mathematical basics of vector spaces, norms, sets, functions, matrices, and linear algebra, etc., in order to smoothly introduce the CVX optimization theory from fundamentals to applications in each of the following chapters. It is expected that the CVX optimization theory will be more straightforward and readily understood and learned.

In this section, let us introduce all the notations and abbreviations and some mathematical preliminaries that will be used in the remainder of the book. Our notations and abbreviations are standard, following those widely used in convex optimization for signal processing and communications, that are defined, respectively, as follows:
Notations:
$\mathbb{R}, \mathbb{R}^{n}, \mathbb{R}^{m \times n} \quad$ Set of real numbers, $n$-vectors, $m \times n$ matrices
$\mathbb{C}, \mathbb{C}^{n}, \mathbb{C}^{m \times n} \quad$ Set of complex numbers, $n$-vectors, $m \times n$ matrices
$\mathbb{R}{+}, \mathbb{R}{+}^{n}, \mathbb{R}{+}^{m \times n} \quad$ Set of nonnegative real numbers, $n$-vectors, $m \times n$ matrices $\mathbb{R}{++}, \mathbb{R}{++}^{n}, \mathbb{R}{++}^{m \times n} \quad$ Set of positive real numbers, $n$-vectors, $m \times n$ matrices
$\mathbb{Z}, \mathbb{Z}{+}, \mathbb{Z}{++} \quad$ Set of integers, nonnegative integers, positive integers

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

In linear algebra, functional analysis and related areas of mathematics, norm is a function that assigns a strictly positive length or size to all vectors (other than the zero vector) in a vector space. A vector space with a norm is called

a normed vector space. A simple example is the 2-dimensional Euclidean space ” $\mathbb{R}^{2 n}$ equipped with the Euclidean norm or 2 -norm. Elements in this vector space are usually drawn as arrows in a 2-dimensional Cartesian coordinate system starting at the origin $0_{2}$. The Euclidean norm assigns to each vector the length from the origin to the vector end. Because of this, the Euclidean norm is often known as the magnitude of the vector.

Given a vector space $V$ over a subfield $F$ of real (or complex) numbers, norm of a vector in $V$ is a function $|\cdot|: V \rightarrow \mathbb{R}_{+}$with the following axioms: For all $a$ in $F$ and all $\mathbf{u}$ and $v \in V$,

• $|a \mathbf{v}|=|a| \cdot|\mathbf{v}|$ (positive homogeneity or positive scalability).
• $|\mathbf{u}+\mathbf{v}| \leq|\mathbf{u}|+|\mathbf{v}|$ (triangle inequality or subadditivity).
• $|\mathbf{v}|=0$ if and only if $\mathbf{v}$ is the zero vector (positive definiteness).
A simple consequence of the first two axioms, positive homogeneity and the triangle inequality, is $|\mathbf{0}|=0$ and thus $|\mathbf{v}| \geq 0$ (positivity).

The $\ell_{p}$-norm (or $p$-norm) of a vector $v$ is usually denoted as $\left|_{v}\right|_{p}$ and is defined as:
$$|\mathbf{v}|_{p}=\left(\sum_{i=1}^{n}\left|v_{i}\right|^{p}\right)^{1 / p}$$
where $p \geq 1$. The above formula for $0<p<1$ is a well-defined function of $\mathbf{v}$, but it is not a norm of $\mathbf{v}$, because it violates the triangle inequality. For $p=1$ and $p=2$,
\begin{aligned} &|\mathbf{v}|_{1}=\sum_{i=1}^{n}\left|v_{i}\right|, \ &|\mathbf{v}|_{2}=\left(\sum_{i=1}^{n}\left|v_{i}\right|^{2}\right)^{1 / 2} \end{aligned}
When $p=\infty$, the norm is called maximum norm or infinity norm or uniform norm or supremum norm and can be expressed as
$$|\mathbf{v}|_{\infty}=\max \left{\left|v_{1}\right|,\left|v_{2}\right|, \ldots,\left|v_{n}\right|\right} .$$
Note that 1-norm, 2-norm (which is also called the Euclidean norm), and $\infty$ norm have been widely used in various science and engineering problems, while for other values of $p$ (e.g., $p=3,4,5, \ldots$ ) $p$-norm remain theoretical but not yet practical.

Remark 1.2 Every norm is a convex function (which will be introduced in Chapter 3). As a result, finding a global optimum of a norm-based objective function is often tractable.

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

In mathematics, a matrix norm is a natural extension of the notion of a vector norm to matrices. Some useful matrix norms needed throughout the book are introduced next.
The Frobenius norm of an $m \times n$ matrix $\mathbf{A}$ is defined as
$$|\mathbf{A}|_{\mathrm{F}}=\left(\sum_{i=1}^{m} \sum_{j=1}^{n}\left|[\mathbf{A}]{i j}\right|^{2}\right)^{1 / 2}=\sqrt{\operatorname{Tr}\left(\mathbf{A}^{T} \mathbf{A}\right)}$$ where $$\operatorname{Tr}(\mathbf{X})=\sum{i=1}^{n}[\mathbf{X}]_{i i}$$
denotes the trace of a square matrix $\mathbf{X} \in \mathbb{R}^{n \times n}$. As $n=1$, A reduces to a column vector of dimension $m$ and its Frobenius norm also reduces to the 2-norm of the vector.

The other class of norm is known as the induced norm or operator norm. Suppose that $|\cdot|_{a}$ and $|\cdot|_{b}$ are norms on $\mathbb{R}^{m}$ and $\mathbb{R}^{n}$, respectively. Then the operator/induced norm of $\mathbf{A} \in \mathbb{R}^{m \times n}$, induced by the norms $|\cdot|_{a}$ and $|\cdot|_{b}$, is defined as
$$|\mathbf{A}|_{a, b}=\sup \left{|\mathbf{A} \mathbf{u}|_{a} \mid|\mathbf{u}|_{b} \leq 1\right},$$
where $\sup (C)$ denotes the least upper bound of the set $C$. As $a=b$, we simply denote $|\mathbf{A}|_{a, b}$ by $|\mathbf{A}|_{a}$.
Commonly used induced norms of an $m \times n$ matrix
$$\mathbf{A}=\left{a_{i j}\right}_{m \times n}=\left[\mathbf{a}{1}, \ldots, \mathbf{a}{n}\right]$$
are as follows:
$$|\mathbf{A}|_{1}=\max {|\mathbf{u}|{1} \leq 1}\left|\sum_{j=1}^{n} u_{j} \mathbf{a}{j}\right|{1}, \quad(a=b=1)$$
$\leq \max {|\mathbf{u}|{1 \leq 1}} \sum_{j=1}^{n}\left|u_{j}\right| \cdot\left|\mathbf{a}{j}\right|{1}$ (by triangle inequality)
$$=\max {1 \leq j \leq n}\left|\mathbf{a}{j}\right|_{1}=\max {1 \leq j \leq n} \sum{i=1}^{m}\left|a_{i j}\right|$$
(with the inequality to hold with equality for $\mathbf{u}=\mathbf{e}{l}$ where $l=$ $\arg \max {1 \leq j \leq n}\left|\mathbf{a}{j}\right|{1}$ ) which is simply the maximum absolute column sum

of the matrix.
$|\mathbf{A}|_{\infty}=\max {|\mathbf{u}|{\infty \leq 1}}\left{\max {1 \leq i \leq m}\left|\sum{j=1}^{n} a_{i j} u_{j}\right|\right}, \quad(a=b=\infty)$
$=\max {1 \leq i \leq m} \sum{j=1}^{n}\left|a_{i j}\right|, \quad$ (i.e., $u_{j}=\operatorname{sgn}\left{a_{i j}\right} \forall j$ )
which is simply the maximum absolute row sum of the matrix.
In the special case of $a=b=2$, the induced norm is called the spectral norm or $\ell_{2}$ norm. The spectral norm of a matrix $\mathbf{A}$ is the largest singular value of A or the square root of the largest eigenvalue of the positive semidefinite matrix $\mathbf{A}^{T} \mathbf{A}$, i.e..
$$|\mathbf{A}|_{2}=\sup \left{|\mathbf{A} \mathbf{u}|_{2} \mid|\mathbf{u}|_{2} \leq 1\right}=\sigma_{\max }(\mathbf{A})=\sqrt{\lambda_{\max }\left(\mathbf{A}^{T} \mathbf{A}\right)}$$
The singular values of a matrix A will be defined in (1.109) and their relation to the corresponding eigenvalues of $\mathbf{A}^{T} \mathbf{A}$ (or $\mathbf{A} \mathbf{A}^{T}$ ) is given by (1.116) in Subsection 1.2.6 later.

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

R,Rn,R米×n实数集，n-向量，米×n矩阵
C,Cn,C米×n复数集，n-向量，米×n矩阵
R+,R+n,R+米×n一组非负实数，n-向量，米×n矩阵R++,R++n,R++米×n一组正实数，n-向量，米×n矩阵

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

• |一种在|=|一种|⋅|在|（正同质性或正可扩展性）。
• |在+在|≤|在|+|在|（三角不等式或次可加性）。
• |在|=0当且仅当在是零向量（正定性）。
前两个公理，正同质性和三角不等式的简单推论是|0|=0因此|在|≥0（积极性）。

|在|p=(∑一世=1n|在一世|p)1/p

|在|1=∑一世=1n|在一世|, |在|2=(∑一世=1n|在一世|2)1/2

|\mathbf{v}|_{\infty}=\max \left{\left|v_{1}\right|,\left|v_{2}\right|, \ldots,\left|v_{n} \对|\对} 。|\mathbf{v}|_{\infty}=\max \left{\left|v_{1}\right|,\left|v_{2}\right|, \ldots,\left|v_{n} \对|\对} 。

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

|一种|F=(∑一世=1米∑j=1n|[一种]一世j|2)1/2=Tr⁡(一种吨一种)在哪里Tr⁡(X)=∑一世=1n[X]一世一世

|\mathbf{A}|_{a, b}=\sup \left{|\mathbf{A} \mathbf{u}|_{a} \mid|\mathbf{u}|_{b} \leq 1\右},|\mathbf{A}|_{a, b}=\sup \left{|\mathbf{A} \mathbf{u}|_{a} \mid|\mathbf{u}|_{b} \leq 1\右},

\mathbf{A}=\left{a_{i j}\right}_{m \times n}=\left[\mathbf{a}{1}, \ldots, \mathbf{a}{n}\right]\mathbf{A}=\left{a_{i j}\right}_{m \times n}=\left[\mathbf{a}{1}, \ldots, \mathbf{a}{n}\right]

|一种|1=最大限度|在|1≤1|∑j=1n在j一种j|1,(一种=b=1)
≤最大限度|在|1≤1∑j=1n|在j|⋅|一种j|1（通过三角不等式）
=最大限度1≤j≤n|一种j|1=最大限度1≤j≤n∑一世=1米|一种一世j|
（不等式成立在=和l在哪里l= 参数⁡最大限度1≤j≤n|一种j|1) 这只是最大绝对列总和

|\mathbf{A}|_{\infty}=\max {|\mathbf{u}|{\infty \leq 1}}\left{\max {1 \leq i \leq m}\left|\sum {j=1}^{n} a_{i j} u_{j}\right|\right}, \quad(a=b=\infty)|\mathbf{A}|_{\infty}=\max {|\mathbf{u}|{\infty \leq 1}}\left{\max {1 \leq i \leq m}\left|\sum {j=1}^{n} a_{i j} u_{j}\right|\right}, \quad(a=b=\infty)
=最大限度1≤一世≤米∑j=1n|一种一世j|,（IE，u_{j}=\operatorname{sgn}\left{a_{i j}\right} \forall ju_{j}=\operatorname{sgn}\left{a_{i j}\right} \forall j)

|\mathbf{A}|_{2}=\sup \left{|\mathbf{A} \mathbf{u}|_{2} \mid|\mathbf{u}|_{2} \leq 1\右}=\sigma_{\max }(\mathbf{A})=\sqrt{\lambda_{\max }\left(\mathbf{A}^{T} \mathbf{A}\right)}|\mathbf{A}|_{2}=\sup \left{|\mathbf{A} \mathbf{u}|_{2} \mid|\mathbf{u}|_{2} \leq 1\右}=\sigma_{\max }(\mathbf{A})=\sqrt{\lambda_{\max }\left(\mathbf{A}^{T} \mathbf{A}\right)}

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