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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Geometric and Economic Interpretations

Suppose that for the linear program in the standard primal 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}
we have the optimal basic feasible solution $\mathbf{x}=\left(\mathbf{x}{\mathbf{B}}, \mathbf{0}\right)$ with corresponding basis B. We shall determine a solution of the dual program $$\begin{array}{ll} \operatorname{maximize} & \mathbf{y}^{T} \mathbf{b} \ \text { subject to } & \mathbf{y}^{T} \mathbf{A} \leqslant \mathbf{c}^{T} \end{array}$$ in terms of $\mathbf{B}$. We partition $\mathbf{A}$ as $\mathbf{A}=[\mathbf{B}, \mathbf{D}]$, where the primal basic feasible solution $\mathbf{x}{\mathbf{B}}=$ $\mathbf{B}^{-1} \mathbf{b}$ is optimal. Now define $\mathbf{y}^{T}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1}$, which is a dual basic solution (the intersection point of $m$ constraints) for the dual of inequality constraints. (Again the components subvector $\mathbf{c}{\mathbf{B}}$ are those of $\mathbf{c}$ associated with the columns of submatrix $\mathbf{B}$ according to the same index order.)

If, in addition, $\mathbf{y}^{T} \mathbf{A} \leqslant \mathbf{c}^{T}$, then $\mathbf{y}$ is feasible and a basic feasible solution for the dual-an extreme point of the dual feasible region. On the other hand,
$$\mathbf{y}^{T} \mathbf{b}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1} \mathbf{b}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{x}_{\mathbf{B}}$$
and thus the value of the dual objective function for this $\mathbf{y}$ is equal to the value of the primal problem. This, in view of Lemma 1, Sect. 3.2, establishes the optimality of $\mathbf{y}$ for the dual.

## 数学代写|线性规划作业代写Linear Programming代考|Dual Multipliers—Shadow Prices

We conclude this section by giving an economic interpretation of the relation between the optimal basis and the vector $\mathbf{y}^{T}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1}$. This vector is not a feasible solution to the dual unless $\mathbf{B}$ is an optimal basis for the primal, but nevertheless, it has an economic interpretation. Furthermore, as we have seen in the development of the simplex method in the next chapter, this $\mathbf{y}$ vector can be used at every step to calculate the relative cost coefficients or reduced gradients. For this reason $\mathbf{y}^{T}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1}$, corresponding to any basis, is often called the vector of simplex multipliers or shadow prices.

Let us pursue the economic interpretation of these simplex multipliers. As usual, denote the columns of $\mathbf{A}$ by $\mathbf{a}{1}, \mathbf{a}{2}, \ldots, \mathbf{a}{n}$ and denote by $\mathbf{e}{1}, \mathbf{e}{2}, \ldots, \mathbf{e}{m}$ the $m$ unit vectors in $E^{m}$. The components of the $\mathbf{a}{j}$ ‘s and $\mathbf{b}$ tell how to construct these vectors from the $\mathbf{e}{i}$ ‘s.

Given any basis $\mathbf{B}$, however, consisting of $m$ columns of $\mathbf{A}$, any other vector can bee constructed (synthëtically) as a linear combination of thesse básis vectors. If there is a unit $\operatorname{cost} c_{j}$ associated with each basis vector $\mathbf{a}{j}$, then the cost of a (synthetic) vector constructed from the basis can be calculated as the corresponding linear combination of the $c{j}$ ‘s associated with the basis. In particular, the cost of the $i$ th unit vector, $\mathbf{e}{i}$, when constructed from the basis $\mathbf{B}$, is $y{i}$, the $i$ th component of $\mathbf{y}^{T}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1}$. Thus the $y{i}$ ‘s can be interpreted as synthetic prices of the unit vectors.
Now, any vector can be expressed in terms of the basis $\mathbf{B}$ in two steps: (1) express the unit vectors in terms of the basis, and then (2) express the desired vector as a linear combination of unit vectors. The corresponding synthetic cost of a vector constructed from the basis $\mathbf{B}$ can correspondingly be computed directly by: (1) finding the synthetic price of the unit vectors, and then (2) using these prices to evaluate the cost of the linear combination of unit vectors. Thus, the simplex multipliers can be used to quickly evaluate the synthetic cost of any vector that is expressed in terms of the unit vectors. The difference between the true cost of this vector and the synthetic cost is the relative cost. The process of calculating the synthetic cost of a vector, with respect to a given basis, by using the simplex multipliers is sometimes referred to as pricing out the vector.

Optimality of the primal corresponds to the situation where every vector $\mathbf{a}{1}, \mathbf{a}{2}$, $\ldots, \mathbf{a}{n}$ is cheaper when constructed from the basis than when purchased directly at its own price. Thus we have $\mathbf{y}^{T} \mathbf{a}{j} \leqslant c_{j}$ for $j=1,2, \ldots, n$ or equivalently $\mathbf{y}^{T} \mathbf{A} \leqslant \mathbf{c}^{T}$

# 线性规划代写

## 数学代写|线性规划作业代写Linear Programming代考|Geometric and Economic Interpretations

$$\operatorname{minimize} \mathbf{c}^{T} \mathbf{x} \quad \text { subject to } \mathbf{A x}=\mathbf{b}, \mathbf{x} \geqslant \mathbf{0}$$
㧴们有最优的基本可行解 $x=(x B, 0)$ 有相应的基 $B$ 。我们将确定对偶程序的解
$$\text { maximize } \mathbf{y}^{T} \mathbf{b} \text { subject to } \mathbf{y}^{T} \mathbf{A} \leqslant \mathbf{c}^{T}$$

$$\mathbf{y}^{T} \mathbf{b}=\mathbf{c B}^{T} \mathbf{B}^{-1} \mathbf{b}=\mathbf{c B}^{T} \mathbf{x}_{\mathbf{B}}$$

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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。

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