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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|组合优化代写Combinatorial optimization代考|Linear Programming

In this chapter we review the most important facts about Linear Programming. Although this chapter is self-contained, it cannot be considered to be a comprehensive treatment of the field. The reader unfamiliar with Linear Programming is referred to the textbooks mentioned at the end of this chapter.
The general problem reads as follows:
\begin{tabular}{|ll|}
\hline LINEAR & PROGRAMMING \
Instance: & A matrix $A \in \mathbb{R}^{m \times n}$ and column vectors $b \in \mathbb{R}^m, c \in \mathbb{R}^n$. \
Task: & Find a column vector $x \in \mathbb{R}^n$ such that $A x \leq b$ and $c^{\top} x$ is maxi- \
& mum, decide that $\left{x \in \mathbb{R}^n: A x \leq b\right}$ is empty, or decide that for all \
& $\alpha \in \mathbb{R}$ there is an $x \in \mathbb{R}^n$ with $A x \leq b$ and $c^{\top} x>\alpha$. \
\hline
\end{tabular}
Here $c^{\top} x$ denotes the scalar product of the vectors. The notion $x \leq y$ for vectors $x$ and $y$ (of equal size) means that the inequality holds in each component. If no sizes are specified, the matrices and vectors are always assumed to be compatible in size. We often omit indicating the transposition of column vectors and write e.g. $c x$ for the scalar product. By 0 we denote the number zero as well as all-zero vectors and all-zero matrices (the order will always be clear from the context).

A lincar program (LP) is an instance of the above problem. We often write a linear program as $\max {c x: A x \leq b}$. A feasible solution of an LP $\max {c x: A x \leq$ $b$ } is a vector $x$ with $A x \leq b$. A feasible solution attaining the maximum is called an optimum solution.

As the problem formulation indicates, there are two possibilities when an LP has no solution: The problem can be infeasible (i.e. $P:=\left{x \in \mathbb{R}^n: A x \leq b\right}=\emptyset$ ) or unbounded (i.e. for all $\alpha \in \mathbb{R}$ there is an $x \in P$ with $c x>\alpha$ ). If an LP is neither infeasible nor unbounded it has an optimum solution:

Proposition 3.1. Let $P=\left{x \in \mathbb{R}^n: A x \leq b\right} \neq \emptyset$ and $c \in \mathbb{R}^n$ with $\delta:=$ $\sup \left{c^{\top} x: x \in P\right}<\infty$. Then there exists a vector $z \in P$ with $c^{\top} z=\delta$.

## 数学代写|组合优化代写Combinatorial optimization代考|The Simplex Algorithm

The oldest and best-known algorithm for LineAR PROGRAMMING is Dantzig’s [1951] simplex method. We first assume that the polyhedron has a vertex, and that some vertex is given as input. Later we shall show how general LPs can be solved with this method.

For a set $J$ of row indices we write $A_J$ for the submatrix of $A$ consisting of the rows in $J$ only, and $b_J$ for the subvector of $b$ consisting of the components with indices in $J$. We abbreviate $a_i:=A_{{i}}$ and $\beta_i:=b_{{i}}$.

Step (1) relies on Proposition $3.9$ and can be implemented with GaussIAN ELIMINATION (Section 4.3). The selection rules for $i$ and $j$ in (3) and (4) (often called pivot rule) are due to Bland [1977]. If one just chose an arbitrary $i$ with $y_i<0$ and an arbitrary $j$ attaining the minimum in (4) the algorithm would run into cyclic repetitions for some instances. Bland’s pivot rule is not the only one that avoids cycling; another one (the so-called lexicographic rule) was proved to avoid cycling already by Dantzig, Orden and Wolfe [1955]. Before proving the correctness of the SiMPI.F.X AI.GORITHM, let us make the following observation (sometimes known as “weak duality”): Proposition 3.13. Let $x$ and $y$ be feasible solutions of the LPS $$\begin{array}{r} \max {c x: A x \leq b} \quad \text { and } \ \min \left{y b: y^{\top} A=c^{\top}, y \geq 0\right}, \end{array}$$ respectively. Then $c x \leq y b$. Proof: $c x=(y A) x=y(A x) \leq y b$. Theorem 3.14. (Dantzig [1951], Dantzig, Orden and Wolfe [1955], Bland [1977]) The SimpleX ALGORITHM terminates after at most $\left(\begin{array}{l}m \ n\end{array}\right)$ iterations. If it returns $x$ and $y$ in (2), these vectors are optimum solutions of the LPs (3.1) and (3.2), respectively, with $c x=y b$. If the algorithm returns $w$ in (3) then $c w>0$ and the $L P(3.1)$ is unbounded.

# 组合优化代考

## 数学代写|组合优化代写combinatoroptimization代考|Linear Programming

. txt

\begin{tabular}{|ll|}
\hline LINEAR & PROGRAMMING \
Instance: & A matrix $A \in \mathbb{R}^{m \times n}$ and column vectors $b \in \mathbb{R}^m, c \in \mathbb{R}^n$. \
Task: & Find a column vector $x \in \mathbb{R}^n$ such that $A x \leq b$ and $c^{\top} x$ is maxi- \
& mum, decide that $\left{x \in \mathbb{R}^n: A x \leq b\right}$ is empty, or decide that for all \
& $\alpha \in \mathbb{R}$ there is an $x \in \mathbb{R}^n$ with $A x \leq b$ and $c^{\top} x>\alpha$. \
\hline
\end{tabular}

lincar程序(LP)是上述问题的一个实例。我们经常编写一个线性程序$\max {c x: A x \leq b}$。LP $\max {c x: A x \leq$$b｝的一个可行解是带有A x \leq b的向量x。达到最大值的可行解称为最优解 正如问题的表述所示，当LP没有解决方案时，有两种可能性:问题可能是不可行(即P:=\left{x \in \mathbb{R}^n: A x \leq b\right}=\emptyset)或无界(即对于所有\alpha \in \mathbb{R}都有一个带c x>\alpha的x \in P)。如果LP既非不可行也非无界，它就有一个最优解: 命题3.1. 让P=\left{x \in \mathbb{R}^n: A x \leq b\right} \neq \emptyset和c \in \mathbb{R}^n加上\delta:=$$\sup \left{c^{\top} x: x \in P\right}<\infty$。然后存在一个带有$c^{\top} z=\delta$的向量$z \in P$ .

## 有限元方法代写

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## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师