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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

## 经济代写|博弈论代写Game Theory代考|The KKT-conditions

Lemma $3.1$ indicates the importance of being able to identify equilibria in LAGRANGE games. In order to establish necessary conditions, i.e., conditions which candidates for equilibria must satisfy, we impose further assumptions on problem (22):
(1) $X \subseteq \mathbb{R}^{n}$ is a convex set, i.e., $X$ contains with every $x, x^{\prime}$ also the whole line segment
$$\left[x, x^{\prime}\right]=\left{x+\lambda\left(x^{\prime}-x\right) \mid 0 \leq \lambda \leq 1\right} .$$
(2) The functions $f$ and $g_{i}$ in (22) have continuous partial derivatives $\partial f(x) / \partial x_{j}$ and $\partial g_{i}(x) / \partial x_{j}$ for all $j=1, \ldots, n$.

It follows that also the partial derivatives of the LAGRANGE function $L$ exist. So the marginal change of $L$ into the direction $d$ of the $x$-variables is
\begin{aligned} \nabla_{x} L(x, y) d &=\nabla f(x) d+\sum_{i=1}^{m} y_{i} \nabla g_{i}(x) d \ &=\sum_{j=1}^{n} \frac{\partial f(x)}{\partial x_{j}} d_{j}+\sum_{i=1}^{m} \sum_{j=1}^{n} \frac{\partial g_{i}(x)}{\partial x_{j}} y_{i} d_{j} \end{aligned}
ReMARK $3.3$ (JACOBI matrix). The $(m \times n)$ matrix $D g(x)$ having as coefficients the partial derivatives
$$D g(x){i j}=\partial g{i}(x) / \partial x_{j}$$
of a function $g: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is known as a functional or $\mathrm{JACOBI}^{7}$ matrix. It provides a compact matrix notation for the marginal change of the Lagrange function:
$$\nabla_{x} L(x, y) d=\nabla f(x)+y^{T} D g(x) d .$$

Given $m$ functions $a_{1}, \ldots, a_{m}: \mathbb{R}{+}^{n} \rightarrow \mathbb{R}$ and $m$ scalars $b{1}, \ldots$, $b_{m} \in \mathbb{R}$, the optimization problem
$$\max {x \in \mathbb{R}{+}^{n}} f(x) \quad \text { s.t. } \quad a_{1}(x) \leq b_{1}, \ldots, a_{m}(x) \leq b_{m}$$
is of type (22) with the $m$ restriction functions $g_{i}(x)=b_{i}-a_{i}(x)$ and has the LAGRANGE function
\begin{aligned} L(x, y) &=f(x)+\sum_{i=1}^{m} y_{i}\left(b_{i}-a_{i}(x)\right) \ &=f(x)-\sum_{i=1}^{m} y_{i} a_{i}(x)+\sum_{i=1}^{m} y_{i} b_{i} . \end{aligned}
For an intuitive interpretation of the problem (26), think of the data vector
$$x=\left(x_{1}, \ldots, x_{n}\right)$$
as a plan for $n$ products to be manufactured in quantities $x_{j}$ and of $f(x)$ as the market value of $x$.

Assume that $x$ requires the use of $m$ materials in quantities $a_{1}(x), \ldots, a_{m}(x)$ and that the parameters $b_{1}, \ldots, b_{m}$ describe the quantities of the materials already in the possession of the manufacturer.

If the numbers $y_{1}, \ldots, y_{m}$ represent the market prices (per unit) of the $m$ materials, we find that $L(x, y)$ is the total value of the manufacturer’s assets:
\begin{aligned} L(x, y)=& \text { market value of the production } x \ &+\text { value of the materials left in stock. } \end{aligned}
The manufacturer would like to have that value as high as possible by deciding on an appropriate production plan $x$.

## 经济代写|博弈论代写Game Theory代考|The KKT-conditions

(1) $X \subseteq \mathbb{R}^{n}$ 是一个凸集，即 $X$ 包含每个 $x, x^{\prime}$ 也是整个线段
$\backslash$ left 的分隔符缺失或无法识别
(2) 功能 $f$ 和 $g_{i}$ 在 (22) 中具有连续偏导数 $\partial f(x) / \partial x_{j}$ 和 $\partial g_{i}(x) / \partial x_{j}$ 对所有人 $j=1, \ldots, n$.

$$\nabla_{x} L(x, y) d=\nabla f(x) d+\sum_{i=1}^{m} y_{i} \nabla g_{i}(x) d \quad=\sum_{j=1}^{n} \frac{\partial f(x)}{\partial x_{j}} d_{j}+\sum_{i=1}^{m} \sum_{j=1}^{n} \frac{\partial g_{i}(x)}{\partial x_{j}} y_{i} d_{j}$$

$$D g(x) i j=\partial g i(x) / \partial x_{j}$$

$$\nabla_{x} L(x, y) d=\nabla f(x)+y^{T} D g(x) d .$$

$$\max x \in \mathbb{R}+{ }^{n} f(x) \quad \text { s.t. } \quad a_{1}(x) \leq b_{1}, \ldots, a_{m}(x) \leq b_{m}$$

$$L(x, y)=f(x)+\sum_{i=1}^{m} y_{i}\left(b_{i}-a_{i}(x)\right) \quad=f(x)-\sum_{i=1}^{m} y_{i} a_{i}(x)+\sum_{i=1}^{m} y_{i} b_{i} .$$

$$x=\left(x_{1}, \ldots, x_{n}\right)$$

$$L(x, y)=\text { market value of the production } x \quad+\text { value of the materials left in stock. }$$

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## MATLAB代写

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