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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|Selected Applications of the Duality

There are many applications of the duality theorem ranging from economic and algorithmic game theory to optimization model design. We list several of them in this section while putting some others as exercises.

Example 1 (Core of Production Game) In cooperative game theory, the core is the set of feasible allocations from a grand coalition/alliance that cannot be improved upon by a subset of the economy’s agents in the coalition. A subset of agents is said to improve upon if the members of that subset are better off when they form their own smaller allocation. An allocation from the grand coalition is said to have the core property if there is no sub-coalition that can improve upon it so that the grand coalition is stable. The core is the set of all feasible allocations with the core property. From an algorithmic point of view, computing a core element can be difficult in a cooperative game, even just checking whether or not the core is empty is challenging.

Consider a finite set $F$ of firms each of whom has operations that have representations as production linear programs. Suppose the linear program representing the operations of firm $i \in F$ entails choosing an $n$-column vector $\mathbf{x}$ of production levels that
\begin{aligned} &\text { maximize } \mathbf{c}^{T} \mathbf{x} \ &\text { subject to } \mathbf{A x} \leqslant \mathbf{b}^{i}, \ &\mathbf{x} \geqslant \mathbf{0}, \end{aligned}

## 数学代写|线性规划作业代写Linear Programming代考|Robust and Distributionally Robust Optimization

In real applications of linear programming, the data coefficients may vary and be unpredictable. To prepare for the worst, we may have to consider decision making in a robust way. To be more specific, consider the following linear program:
minimize $(\mathbf{c}+C \mathbf{u})^{T} \mathbf{x}$
subject to $\mathbf{A x}=\mathbf{b}$,
$\mathbf{x} \geq \mathbf{0}$
where $\mathbf{C}$ is a known $n \times d$ matrix and uncertain factors are captured in vector $\mathbf{u} \in E^{d}$ that is unknown and uncontrollable to the decision maker. However, $\mathbf{u} \in E^{d}$ is known to be between $\mathbf{0}$ and $\mathbf{1}$.
A Robust Model approach to this problem is to reformulate the problem as
$$\begin{array}{ll} \operatorname{minimize} & \mathbf{c}^{T} \mathbf{x}+\max {\mathbf{0} \leq \mathbf{u} \leq \mathbf{1}}\left[\mathbf{x}^{T} \mathbf{C u}\right] \ \text { subject to } & \mathbf{A x}=\mathbf{b} \ & \mathbf{x} \geq \mathbf{0} \end{array}$$ The decision variables are in $\mathbf{u}$ of the inner problem, and they are selected by an adversary to maximize the objective of given $\mathbf{x}$ : $$\max {\mathbf{u}} \mathbf{x}^{T} \mathbf{C u} \text { s.t. } \mathbf{0} \leq \mathbf{u} \leq \mathbf{1}$$
which is a linear program.

# 线性规划代写

## 数学代写|线性规划作业代写Linear Programming代考|Selected Applications of the Duality

$$\text { maximize } \mathbf{c}^{T} \mathbf{x} \quad \text { subject to } \mathbf{A} \mathbf{x} \leqslant \mathbf{b}^{i}, \mathbf{x} \geqslant \mathbf{0} \text {, }$$

## 数学代写|线性规划作业代写Linear Programming代考|Robust and Distributionally Robust Optimization

$\mathrm{x} \geq 0$

minimize $\quad \mathbf{c}^{T} \mathbf{x}+\max \mathbf{0} \leq \mathbf{u} \leq \mathbf{1}\left[\mathbf{x}^{T} \mathbf{C u}\right]$ subject to $\quad \mathbf{A x}=\mathbf{b} \quad \mathbf{x} \geq \mathbf{0}$

$\max \mathbf{u x}^{T} \mathbf{C u}$ s.t. $\mathbf{0} \leq \mathbf{u} \leq \mathbf{1}$

## 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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