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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|The Duality Theorem

To this point the relation between the primal and dual programs has been simply a formal one based on what might appear as an arbitrary definition. In this section, however, the deeper connection between a program and its dual, as expressed by the Duality Theorem, is derived.

The proof of the Duality Theorem given in this section relies on Farkas’ Lemma (Chap. 2, Sect. 2.6) and is therefore somewhat more advanced than previous arguments. It is given here so that the most general form of the Duality Theorem is established directly. An alternative approach is to use the theory of the simplex method to derive the duality result. A simplified version of this alternative approach is given in the next section.
Throughout this section we consider the primal program in standard form
\begin{aligned} &\text { minimize } \mathbf{c}^{T} \mathbf{x} \ &\text { subject to } \mathbf{A x}=\mathbf{b}, \mathbf{x} \geqslant \mathbf{0} \end{aligned}
and its corresponding dual
maximize $\mathbf{y}^{T} \mathbf{b}$
subject to $\mathbf{y}^{T} \mathbf{A} \leqslant \mathbf{c}^{T}$

In this section it is not assumed that $\mathbf{A}$ is necessarily of full rank. The following lemma is easily established and gives us an important relation between the two problems.
Lemma 1 (Weak Duality Lemma) If $\mathbf{x}$ and $\mathbf{y}$ are feasible for (3.3) and (3.4),respectively, then $\mathbf{c}^{T} \mathbf{x} \geqslant \mathbf{y}^{T} \mathbf{b}$.
Proof We have
$$\mathbf{y}^{T} \mathbf{b}=\mathbf{y}^{T} \mathbf{A} \mathbf{x} \leqslant \mathbf{c}^{T} \mathbf{x}$$
the last inequality being valid since $\mathbf{x} \geqslant \mathbf{0}$ and $\mathbf{y}^{T} \mathbf{A} \leqslant \mathbf{c}^{T}$.
This lemma shows that a feasible vector to either problem yields a bound on the value of the other problem. The values associated with the primal are all larger than the values associated with the dual as illustrated in Fig. 3.1. Since the primal seeks a minimum and the dual seeks a maximum, each seeks to reach the other. From this we have an important corollary.

## 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.

# 线性规划代写

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

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

$$\mathbf{y}^{T} \mathbf{b}=\mathbf{y}^{T} \mathbf{A} \mathbf{x} \leqslant \mathbf{c}^{T} \mathbf{x}$$

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

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

$$\text { maximize } \mathbf{y}^{T} \mathbf{b} \text { subject to } \mathbf{y}^{T} \mathbf{A} \leqslant \mathbf{c}^{T}$$

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

## 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。

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