### 数学代写|线性规划作业代写Linear Programming代考|MATH7232

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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|线性规划作业代写Linear Programming代考|Basic Feasible Solutions

Consider the system of equalities
$$\mathbf{A x}=\mathbf{b},$$
where $\mathbf{x}$ is an $n$-vector, $\mathbf{b}$ is an $m$-vector, and $\mathbf{A}$ is an $m \times n$ matrix. Suppose that from the $n$ columns of $\mathbf{A}$ we select a set of $m$ linearly independent columns (such a set exists if the rank of $\mathbf{A}$ is $m$ ). For notational simplicity assume that we select the first $m$ columns of $\mathbf{A}$ and denote the $m \times m$ matrix determined by these columns by B. The matrix $\mathbf{B}$ is then nonsingular and we may uniquely solve the equation.
$$\mathbf{B x}{\mathbf{B}}=\mathbf{b} \quad \text { or } \quad \mathbf{x}{\mathbf{B}}=\mathbf{B}^{-1} \mathbf{b}$$
for the $m$-vector $\mathbf{x}{\mathbf{B}}$ whose components are associated with the columns of submatrix $\mathbf{B}$ according to the same index order. By putting $\mathbf{x}=\left(\mathbf{x}{\mathbf{B}}, \mathbf{0}\right)$ (that is, setting the first $m$ components of $\mathbf{x}$ equal to those of $\mathbf{x}{\mathbf{B}}$ and the remaining components equal to zero), we obtain a solution to $\mathbf{A x}=\mathbf{b}$. This leads to the following definition. Definition Given the set of $m$ simultaneous linear equations in $n$ unknowns (2.10), let $\mathbf{B}$ be any nonsingular $m \times m$ submatrix made up of columns of $\mathbf{A}$. Then, if all $n-m$ components of $\mathbf{x}$ not associated with columns of $\mathbf{B}$ are set equal to zero, the solution to the resulting set of equations is said to be a basic solution to (2.10) with respect to basis $\mathbf{B}$. The components of $\mathbf{x}$ associated with the columns of $\mathbf{B}$. denoted by subvector $\mathbf{X}{\mathbf{R}}$ according to the same column index order in $\mathbf{B}$ throughout this book, are called basic variables.
In the above definition we refer to $\mathbf{B}$ as a basis, since $\mathbf{B}$ consists of $m$ linearly independent columns that can be regarded as a basis for the space $E^{m}$. The basic solution corresponds to an expression for the vector $\mathbf{b}$ as a linear combination of these basis vectors. This interpretation is discussed further in the next section.

In general, of course, Eq. (2.10) may have no basic solutions. However, we may avoid trivialities and difficulties of a nonessential nature by making certain elementary assumptions regarding the structure of the matrix $\mathbf{A}$. First, we usually assume that $n>m$, that is, the number of variables $x_{j}$ exceeds the number of equality constraints. Second, we usually assume that the rows of $\mathbf{A}$ are linearly independent, corresponding to linear independence of the $m$ equations. A linear dependency among the rows of $\mathbf{A}$ would lead either to contradictory constraints and hence no solutions to $(2.10)$, or to a redundancy that could be eliminated. Formally, we explicitly make the following assumption in our development, unless noted otherwise.

## 数学代写|线性规划作业代写Linear Programming代考|The Fundamental Theorem of Linear Programming

In this section, through the fundamental theorem of linear programming, we establish the primary importance of basic feasible solutions in solving linear programs. The method of proof of the theorem is in many respects as important as the result itself, since it represents the beginning of the development of the simplex method. The theorem (due to Carathéodory) itself shows that it is necessary only to consider basic feasible solutions when seeking an optimal solution to a linear program because the optimal value is always achieved at such a solution.
Corresponding to a linear program in standard form
\begin{aligned} &\operatorname{minimize} \mathbf{c}^{T} \mathbf{x} \ &\text { subject to } \mathbf{A x}=\mathbf{b}, \mathbf{x} \geqslant \mathbf{0} \end{aligned}
a feasible solution to the constraints that achieves the minimum value of the objective function subject to those constraints is said to be an optimal feasible solution. If this solution is basic, it is an optimal basic feasible solution.
Fundamental Theorem of Linear Programming Given a linear program in standard form (2.13) where $\mathbf{A}$ is an $m \times n$ matrix of rank $m$,
i) if there is a feasible solution, there is a basic feasible solution;
ii) if there is an optimal feasible solution, there is an optimal basic feasible solution.
Proof of (i) Denote the columns of $\mathbf{A}$ by $\mathbf{a}{1}, \mathbf{a}{2}, \ldots, \mathbf{a}{n}$. Suppose $\mathbf{x}=$ $\left(x{1}, x_{2}, \ldots, x_{n}\right)$ is a feasible solution. Then, in terms of the columns of $\mathbf{A}$, this solution satisfies:
$$x_{1} \mathbf{a}{1}+x{2} \mathbf{a}{2}+\cdots+x{n} \mathbf{a}{n}=\mathbf{b} .$$ Assume that exactly $p$ of the variables $x{i}$ are greater than zero, and for convenience, that they are the first $p$ variables. Thus
$$x_{1} \mathbf{a}{1}+x{2} \mathbf{a}{2}+\cdots+x{p} \mathbf{a}_{p}=\mathbf{b}$$

## 数学代写|线性规划作业代写Linear Programming代考|Basic Feasible Solutions

$$\mathbf{A} \mathbf{x}=\mathbf{b},$$

$$\mathbf{B} \mathbf{x} \mathbf{B}=\mathbf{b} \quad \text { or } \quad \mathbf{x B}=\mathbf{B}^{-1} \mathbf{b}$$

## 数学代写|线性规划作业代写Linear Programming代考|The Fundamental Theorem of Linear Programming

minimize $\mathbf{c}^{T} \mathbf{x} \quad$ subject to $\mathbf{A} \mathbf{x}=\mathbf{b}, \mathbf{x} \geqslant \mathbf{0}$

i) 如果有可行的解决方案，则有一个基本的可行解决方案；
ii) 如果有最佳的可行解决方案，则有一个最佳的基本可行解决方案。
(i) 表示的证明 $\mathbf{A}$ 经过 $\mathbf{a} 1, \mathbf{a} 2, \ldots, \mathbf{a}$. 认为 $\mathbf{x}=\left(x 1, x_{2}, \ldots, x_{n}\right)$ 是一个可行的解决方案。然后，就 $\mathbf{A}$ ，该解 决方案满足:
$$x_{1} \mathbf{a} 1+x 2 \mathbf{a} 2+\cdots+x n \mathbf{a} n=\mathbf{b} .$$

$$x_{1} \mathbf{a} 1+x 2 \mathbf{a} 2+\cdots+x p \mathbf{a}_{p}=\mathbf{b}$$

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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。