## 数学代写|凸优化作业代写Convex Optimization代考|MATH3204

2023年4月3日

couryes-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
couryes™为您提供可以保分的包课服务

## 数学代写|凸优化作业代写Convex Optimization代考|Euclidean distance problems

In a Euclidean distance problem, we are concerned only with the distances between the vectors, $d_{i j}$, and do not care about the lengths of the vectors, or about the angles between them. These distances, of course, are invariant not only under orthogonal transformations, but also translation: The configuration $\tilde{a}1=a_1+b, \ldots, \tilde{a}_n=a_n+b$ has the same distances as the original configuration, for any $b \in \mathbf{R}^n$. In particular, for the choice $$b=-(1 / n) \sum{i=1}^n a_i=-(1 / n) A \mathbf{1},$$
we see that $\tilde{a}i$ have the same distances as the original configuration, and also satisfy $\sum{i=1}^n \tilde{a}_i=0$. It follows that in a Euclidean distance problem, we can assume, without any loss of generality, that the average of the vectors $a_1, \ldots, a_n$ is zero, i.e., $A 1=0$.

We can solve Euclidean distance problems by considering the lengths (which cannot occur in the objective or constraints of a Euclidean distance problem) as free variables in the optimization problem. Here we rely on the fact that there is a configuration with distances $d_{i j} \geq 0$ if and only if there are lengths $l_1, \ldots, l_n$ for which $G \succeq 0$, where $G_{i j}=\left(l_i^2+l_j^2-d_{i j}^2\right) / 2$.

We define $z \in \mathbf{R}^n$ as $z_i=l_i^2$, and $D \in \mathbf{S}^n$ by $D_{i j}=d_{i j}^2$ (with, of course, $\left.D_{i i}=0\right)$. The condition that $G \succeq 0$ for some choice of lengths can be expressed as
$$G=\left(z \mathbf{1}^T+\mathbf{1} z^T-D\right) / 2 \succeq 0 \text { for some } z \succeq 0,$$
which is an LMI in $D$ and $z$. A matrix $D \in \mathbf{S}^n$, with nonnegative elements, zero diagonal, and which satisfies (8.8), is called a Euclidean distance matrix. A matrix is a Euclidean distance matrix if and only if its entries are the squares of the Euclidean distances between the vectors of some configuration. (Given a Euclidean distance matrix $D$ and the associated length squared vector $z$, we can reconstruct one, or all, configurations with the given pairwise distances using the method described above.)

The condition (8.8) turns out to be equivalent to the simpler condition that $D$ is negative semidefinite on $\mathbf{1}^{\perp}$, i.e.,
\begin{aligned} (8.8) & \Longleftrightarrow u^T D u \leq 0 \text { for all } u \text { with } \mathbf{1}^T u=0 \ & \Longleftrightarrow\left(I-(1 / n) \mathbf{1 1} 1^T\right) D\left(I-(1 / n) \mathbf{1 1} 1^T\right) \preceq 0 . \end{aligned}

## 数学代写|凸优化作业代写Convex Optimization代考|Extremal volume ellipsoids

Suppose $C \subseteq \mathbf{R}^n$ is bounded and has nonempty interior. In this section we consider the problems of finding the maximum volume ellipsoid that lies inside $C$, and the minimum volume ellipsoid that covers $C$. Both problems can be formulated as convex programming problems, but are tractable only in special cases.

The minimum volume ellipsoid that contains a set $C$ is called the Löwner-John ellipsoid of the set $C$, and is denoted $\mathcal{E}{\mathrm{lj}}$. To characterize $\mathcal{E}{\mathrm{lj}}$, it will be convenient to parametrize a general ellipsoid as
$$\mathcal{E}=\left{v \mid|A v+b|_2 \leq 1\right}$$
i.e., the inverse image of the Euclidean unit ball under an affine mapping. We can assume without loss of generality that $A \in \mathbf{S}{++}^n$, in which case the volume of $\mathcal{E}$ is proportional to $\operatorname{det} A^{-1}$. The problem of computing the minimum volume ellipsoid containing $C$ can be expressed as $$\begin{array}{ll} \text { minimize } & \log \operatorname{det} A^{-1} \ \text { subject to } & \sup {v \in C}|A v+b|_2 \leq 1, \end{array}$$
where the variables are $A \in \mathbf{S}^n$ and $b \in \mathbf{R}^n$, and there is an implicit constraint $A \succ 0$. The objective and constraint functions are both convex in $A$ and $b$, so the problem (8.10) is convex. Evaluating the constraint function in (8.10), however, involves solving a convex maximization problem, and is tractable only in certain special cases.

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|Euclidean distance problems

$$G=\left(z \mathbf{1}^T+\mathbf{1} z^T-D\right) / 2 \succeq 0 \text { for some } z \succeq 0 \text {, }$$

$(8.8) \Longleftrightarrow u^T D u \leq 0$ for all $u$ with $\mathbf{1}^T u=0$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。