统计代写|运筹学作业代写operational research代考|Transportation Models

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

• 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

统计代写|运筹学作业代写operational research代考|Concept of Assignment Problem

The assignment problem is a special class of LP problem. It deals with the situations in which resources are assigned to tasks or other work requirements. Typical examples include assignment of workers to tasks and assignment of machines to jobs. The objective is to yield an optimal matching of resources and tasks. Commonly used criteria are costs, profits, and time. The assignment problem can be described as follows. A company has a group of workers $(i=1,2, \ldots, n)$ and a set of tasks $(j=1,2, \ldots, n)$ to complete. The problem is how to assign $n$ workers to $n$ tasks at the minimum cost, $c_{i j}$. By introducing decision variables $x_{i j}$ to represent the assignment of worker $i$ to task $j$, the assignment model can be written as shown in Model 2.2.1.
Model 2.2.1 Standard assignment model
Minimize $z=\sum_{j=1}^{n} \sum_{j=1}^{n} c_{i j} x_{i j}$
subject to
$$\begin{gathered} \sum_{i=1}^{n} x_{i j}=1 \quad j=1,2, \ldots, n \ \sum_{j=1}^{n} x_{i j}=1 \quad i=1,2, \ldots, n \ \text { All } x_{i j} \geq 0 . \end{gathered}$$
Model 2.2.1 is referred to as the assignment model. Objective function $2.2 .1$ minimizes the total cost associated with worker $i$ performing task $j$.

统计代写|运筹学作业代写operational research代考|Example of Assignment Problem

Figure $2.6$ shows an assignment network in which there are five workers and five tasks. The cost associated with worker $i$ performing task $j$ are shown above the arcs, or arrows. For example, it costs 8 units of dollars for worker 1 to complete task 1 . The capacity of each worker, $s_{j y}$ and the demand of each task, $d_{j}$, are also shown. Because there is only one worker $i$ available for performing a particular task $j$, all $s_{i}$ and $d_{j}$ equal 1 .

This assignment network or problem can be represented by a tableau as shown in Table 2.3. The upper-right corner of each cell in the tableau represents the cost, $c_{i j}$.

By introducing decision variables $x_{i j}$ to represent the assignment of worker $i$ to task $j$, this assignment problem can be formulated as shown in Model $2.2 .2$.
Model 2.2.2 An example of formulation of assignment problem
\text { Minimize } \begin{aligned} & 8 x_{11}+6 x_{12}+2 x_{13}+4 x_{14}+3 x_{15} \ &+6 x_{21}+7 x_{22}+11 x_{23}+10 x_{24}+7 x_{25} \ &+3 x_{31}+5 x_{32}+7 x_{33}+6 x_{34}+4 x_{35} \ &+5 x_{41}+10 x_{42}+12 x_{43}+9 x_{44}+7 x_{45} \ &+7 x_{51}+12 x_{52}+5 x_{53}+7 x_{54}+8 x_{55} \end{aligned}

Constraint sets $2.2 .5$ to $2.2 .9$ ensure that each task is to be performed by exactly one worker. Constraint sets $2.2 .10$ to $2.2 .14$ ensure that each worker is to be assigned to exactly one task. Although Model 2.2.2 is an LP model, the optimal solution must be integral because the assignment model holds the integrality property.

统计代写|运筹学作业代写operational research代考|SAS Code for Assignment Problem

ORASSIGN solves assignment problems, in which one set of items must be assigned to another (e.g., tasks to specific workers) at the lowest total cost (see program “sasor_2_2.sas”).
Figure $2.7$ illustrates the data flow in the ORASSIGN. It shows:

• The cost matrix that is required for ORASSIGN-in this case, the cost associated with worker $i$ performing task $j$
• The macros (\%data, \%model, and \%report)
• The macro variables needed to be set before running the code
• The results datasets that are available for print or can be used for further analysis
In the rest of this section, the procedure for implementing ORASSIGN, together with an example, is explained. The ORASSIGN runs three macros: data-handling (\%data), model-building (\%model), and report-writing (\%report).

统计代写|运筹学作业代写operational research代考|Concept of Assignment Problem

∑一世=1nX一世j=1j=1,2,…,n ∑j=1nX一世j=1一世=1,2,…,n  全部 X一世j≥0.

统计代写|运筹学作业代写operational research代考|Example of Assignment Problem

最小化 8X11+6X12+2X13+4X14+3X15 +6X21+7X22+11X23+10X24+7X25 +3X31+5X32+7X33+6X34+4X35 +5X41+10X42+12X43+9X44+7X45 +7X51+12X52+5X53+7X54+8X55

统计代写|运筹学作业代写operational research代考|SAS Code for Assignment Problem

ORASSIGN 解决了​​分配问题，其中必须以最低的总成本将一组项目分配给另一组（例如，将任务分配给特定的工人）（参见程序“sasor_2_2.sas”）。

• ORASSIGN 所需的成本矩阵——在这种情况下，与工人相关的成本一世执行任务j
• 宏（\%data、\%model 和 \%report）
• 运行代码前需要设置的宏变量
• 可用于打印或可用于进一步分析的结果数据集
在本节的其余部分中，将解释实施 ORASSIGN 的过程以及示例。ORASSIGN 运行三个宏：数据处理 (\%data)、模型构建 (\%model) 和报告编写 (\%report)。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

统计代写|运筹学作业代写operational research代考|Transportation Models

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

• 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

统计代写|运筹学作业代写operational research代考|Concept of Transportation Problem

The transportation problem, first described by Hitchcock in 1941 , is a special class of linear programming (LP) problem. The objective is to yield the leastcost means of shipment through a transportation network in which there is a set of origins providing a commodity to a definite number of destinations. Suppose that a number of suppliers $(i=1,2, \ldots, m)$ provides a commodity to a number of customers $(j=1,2, \ldots, n)$. The transportation problem determines to meet each customer’s requirement, $d_{j}$, while not exceeding the capacity of any supplier, $s_{i}$ at minimum cost, $c_{i j}$. By introducing variables $x_{i j}$ to represent the quantity of the commodity sent from supplier $i$ to customer $j$, the transportation model can be written as shown in Model 2.1.1.
Model 2.1.1 Standard transportation model
Minimize $z=\sum_{i=1}^{m} \sum_{j=1}^{n} c_{i j} x_{i j}$
subject to
$$\sum_{j=1}^{n} x_{i j} \leq s_{i} \quad i=1,2, \ldots, m$$

\begin{aligned} \sum_{i=1}^{m} x_{i j} \geq d_{j} \quad j=1,2, \ldots, n \ & \text { All } x_{i j} \geq 0 \end{aligned}
Model 2.1.1 is referred to as the transportation model. Objective function 2.1.1 minimizes the total transportation cost. Unit transportation costs $c_{i j}$ for shipping 1 unit of commodity from supplier $i$ to customer $j$ are known. These costs are often dependent on the travel distances between supplier $i$ to customer $j$. It is assumed that the cost on a particular route of the transportation network is directly proportional to the number of commodities shipped on that route. If supplier $i$ cannot supply customer $j$, the unit transportation cost $c_{i j}$ is considered infinite $(\infty)$. Constraint set $2.1 .2$ is known as a supply constraint or availability constraint, and constraint set 2.1.3 is known as a demand constraint or requirement constraint. It is assumed that the capacity of each supplier, $s_{j}$, and the demand of each customer, $d_{j}$, are known in advance. If the total supply equals the total demand, then the problem is said to be a balanced transportation problem. In this case, constraint sets $2.1 .2$, and $2.1 .3$ are treated as both equal instead of less than or equal to and greater than or equal to, respectively. If the total supply does not equal the total demand, then the problem is referred to as an unbalanced transportation problem. A dummy customer (when the total supply exceeds the total demand) or a dummy supplier (when the total demand exceeds the total supply) is added to balance the transportation model. Because shipments via the dummy supplier or dummy customer are not real shipments, the unit transportation costs assigned to them are 0 .

统计代写|运筹学作业代写operational research代考|Example of Transportation Problem

Figure $2.1$ shows a transportation network in which there are four suppliers and five customers. The unit transportation costs are shown above the arcs, or arrows. For example, it costs 3 units of dollars to ship 1 unit of commodity from supplier 1 to customer 1 . The capacity of each supplier, $s_{i}$, and the demand of each customer, $d_{j}$, are also shown.

This transportation network or problem can be represented by a tableau as shown in Table 2.1. The upper-right corner of each cell in the tableau represents the unit transportation cost $c_{i j}$.

By introducing variables $x_{i j}$ to represent the quantity of the commodity shipped from supplier $i$ to customer $j$, this transportation problem can be formulated as shown in Model 2.1.2.

统计代写|运筹学作业代写operational research代考|Advanced Options in PROC OPTMODEL

As discussed earlier, PROC OPTMODEL provides a full environment for programming using do-loop, if-then-else, and many other programming statements. We can divide the syntax of PROC OPTMODEL into three types of statements:

1. Options in PROC OPTMODEL
2. Declaration of parameters and variables, as well as objective function and constraints
3. Programming statements
With the option statements, you can control how the optimization model is processed and how results are displayed. The declaration statements define the parameters, variables, constraints, and objectives that describe the model to be solved. All declarations in the PROC OPTMODEL are also saved for later use. The most popular declaration statements are:
• constraint (or con): Defines one or more constraints
• max/min: Declares an objective for the solver
• number (or num): Declares a numeric parameter
• string (or str): Declares a string parameter
• set: Declares a set type parameter
• var: Declares a variable
Parameters and variables can also be initialized using option “init.”

统计代写|运筹学作业代写operational research代考|Basic PROC OPTMODEL

PROC OPTMODEL非常强大，所以我们可以很方便的声明变量和参数，定义目标和约束，解决问题。它还为使用 do-loop、if-then-else 和许多其他编程语句进行编程提供了完整的环境。PROC OPTMODEL 的语法在附录 2 中给出。这里我们给出了在 PROC OPTMODEL 中定义线性规划的一些细节。

1. number：用于定义置信度
2. var：用于定义变量
3. 读取：用于将数据从数据集中加载到相应的参数
1. min/max：用于定义目标函数
2. con：用于定义约束
3. solve：使用选定的求解器解决问题
因为在大多数线性规划中，我们有一个变量向量和一个系数矩阵，PROC OPTMODEL 提供了一个索引工具来更有效地处理这些问题。可以使用整数或一组值来定义索引。例如，
数字 c{1..4}；
曾是⁡X1…4;
定义了四个可以称为的数字C[1],C[2],C[3]， 和C[4]并定义了四个变量，可以称为X[1],X[2],X[3]， 和X[4].

s=和⁡一世 在 1..4X[一世];

统计代写|运筹学作业代写operational research代考|

PROC OPTMODEL 中将数据初始化为参数的另一种方法是使用“读取”语句并使用数据集中保存的数据填充参数。假设 Program 中的数据1.4保存在银行数据集中；以下程序读取数据集并将其加载到相应的变量中： 在此代码中，我们使用了“读取”语句。第一个“读取”加载银行名称以设置行，而第二个“读取”将资本、劳动力和利润的价值加载到每个银行。

统计代写|运筹学作业代写operational research代考|Advanced Options in PROC OPTMODEL

1. PROC OPTMODEL 中的选项
2. 参数和变量的声明，以及目标函数和约束
3. 编程语句
使用选项语句，您可以控制优化模型的处理方式以及结果的显示方式。声明语句定义了描述要求解的模型的参数、变量、约束和目标。PROC OPTMODEL 中的所有声明也被保存以供以后使用。最流行的声明语句是：
• 约束（或con）：定义一个或多个约束
• max/min：为求解器声明一个目标
• number（或 num）：声明一个数字参数
• string（或str）：声明一个字符串参数
• set：声明一个集合类型参数
• var：声明一个变量
参数和变量也可以使用选项“init”进行初始化。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

统计代写|最优控制作业代写optimal control代考|Superdifferential of a semiconcave function

statistics-lab™ 为您的留学生涯保驾护航 在代写最优控制optimal control方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写最优控制optimal control代写方面经验极为丰富，各种代写最优控制Soptimal control相关的作业也就用不着说。

• 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

统计代写|最优控制作业代写optimal control代考|Superdifferential of a semiconcave function

The superdifferential of a semiconcave function enjoys many properties that are not valid for a general Lipschitz continuous function, and that can be regarded as extensions of analogous properties of concave functions. We start with the following basic estimate. Throughout the section $A \subset \mathbb{R}^{n}$ is an open set.

Proposition 3.3.1 Let $u: A \rightarrow \mathbb{R}$ be a semiconcave function with modulus $\omega$ and let $x \in A$. Then, a vector $p \in \mathbb{R}^{n}$ belongs to $D^{+} u(x)$ if and only if
$$u(y)-u(x)-\langle p, y-x\rangle \leq|y-x| \omega(|y-x|)$$
Jor any pont y EA such that $[y, r\rfloor$ s. $_{-}$

Proof – If $p \in \mathbb{R}^{n}$ satisfies (3.18), then, by the very definition of superdifferential, $p \in D^{+} u(x)$. In order to prove the converse, let $p \in D^{+} u(x)$. Then, dividing the semiconcavity inequality $(2.1)$ by $(1-\lambda)|x-y|$, we have
$$\left.\left.\frac{u(y)-u(x)}{|y-x|} \leq \frac{u(x+(1-\lambda)(y-x))-u(x)}{(1-\lambda)|y-x|}+\lambda \omega(|x-y|), \quad \forall \lambda \in\right] 0,1\right] .$$
Hence, taking the limit as $\lambda \rightarrow 1^{-}$, we obtain
$$\frac{u(y)-u(x)}{|y-x|} \leq \frac{\langle p, y-x\rangle}{|y-x|}+\omega(|x-y|),$$
since $p \in D^{+} u(x)$. Estimate (3.18) follows.
Remark 3.3.2 In particular, if $u$ is concave on a convex set $A$. we find that $p \in$ $D^{+} u(x)$ if and only if
$$u(y) \geq u(x)+\langle p, y-x\rangle, \quad \forall y \in A .$$
In convex analysis (see Appendix A. 1) this property is usually taken as the definition of the superdifferential. Therefore, the Fréchet super- and subdifferential coincide with the classical semidifferentials of convex analysis in the case of a concave (resp. convex) function.

Before investigating further properties of the superdifferential, let us show how Proposition 3.3.1 easily yields a compactness property for semiconcave functions.

统计代写|最优控制作业代写optimal control代考|Marginal functions

A function $u: A \rightarrow \mathbb{R}$ is called a marginal function if it can be written in the form
$$u(x)=\inf _{s \in S} F(s, x),$$
where $S$ is some topological space and the function $F: S \times A \rightarrow \mathbb{R}$ depends smoothly on $x$. Functions of this kind appear often in the literature, sometimes with different names (see e.g., the lower $C^{k}$-functions in [123]).

Under suitable regularity assumptions for $F$, a marginal function is semiconcave.
For instance, Corollary $2.1 .6$ immediately implies the following.
Proposition 3.4.1 Let $A \subset \mathbb{R}^{n}$ be open and let $S \subset \mathbb{R}^{m}$ be compact. If $F=F(s, x)$ is continuous in $C(S \times A)$ together with its partial derivatives $D_{x} F$, then the function u defined in (3.34) belongs to $\mathrm{SC}{l o c}(A)$. If $D{x x}^{2} F$ also exists and is continuous in $S \times A$, then $u \in \mathrm{SCL}{l o c}(A)$. We now show that the converse also holds. Theorem 3.4.2 Let $u: A \rightarrow \mathbb{R}$ be a semiconcave function. Then $u$ can be locally written as the minimum of functions of class $C^{1}$. More precisely, for any $K \subset A$ compact, there exists a compact set $S \subset \mathbb{R}^{2 n}$ and a continuous function $F: S \times K \rightarrow$ $\mathbb{R}$ such that $F(s, \cdot)$ is $C^{1}$ for any $s \in S$, the gradients $D{x} F(s, \cdot)$ are equicontinuous, and
$$u(x)=\min _{s \in S} F(s, x), \quad \forall x \in K .$$
If the modulus of semiconcavity of $u$ is linear, then $F$ can be chosen such that $F(s,-)$ is $C^{2}$ for any $s$, with uniformly bounded $C^{2}$ norm.

Proof – Let $\omega$ be the modulus of semiconcavity of $u$ and let $\omega_{1}$ be a function such that $\omega_{1}(0)=0$, that $\omega_{1}(r) \geq \omega(r)$ and that the function $x \rightarrow|x| \omega_{1}(|x|)$ belongs to $C^{1}\left(\mathbb{R}^{n}\right)$. The existence of such an $\omega_{1}$ has been proved in Lemma 3.1.8. If $\omega$ is linear we simply take $\omega_{1} \equiv \omega$.

Let us set $S=\left{(y, p): y \in K, p \in D^{+} u(y)\right}$. By Proposition 3.3.4(a) and the local Lipschitz continuity of $u, S$ is a compact set. Then we define
$$F(y, p, x)=u(y)+\langle p, x-y\rangle+|y-x| \omega_{1}(|y-x|)$$
Then $F$ has the required regularity properties. In addition $F(y, p, x) \geq u(x)$ for all $(y, p, x) \in S \times K$ by Proposition 3.3.1. On the other hand, if $x \in K$, then $D+u(x)$ is nonempty and so thêré exists at lesast a vectō $p$ such that $(x, p) \in S$. Since $F(x, p, x)=u(x)$, we obtain $(3.35)$.

If $u$ is semiconcave with a linear modulus, then it admits another representation as the infimum of regular functions by a procedure that is very similar to the Legendre transformation.

统计代写|最优控制作业代写optimal control代考|Inf-convolutions

Given $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and $\varepsilon>0$, the functions
$$x \rightarrow \inf {y \in \mathbb{R}^{n}}\left(g(y)+\frac{|x-y|^{2}}{2 \varepsilon}\right) \quad x \rightarrow \sup {y \in \mathbb{R}^{n}}\left(g(y)-\frac{|x-y|^{2}}{2 \varepsilon}\right)$$
are called inf- and sup-convolutions of $g$ respectively, due to the formal analogy with the usual convolution. They have been used in various contexts as a way to approximate $g$; one example is the uniqueness theory for viscosity solutions of HamiltonJacobi equations. In some cases it is useful to consider more general expressions, where the quadratic term above is replaced by some other coercive function. In this section we analyze such general convolutions, showing that their regularity properties are strictly related with the properties of semiconcave functions studied in the previous sections.
Definition 3.5.1 Let $g \in C\left(\mathbb{R}^{n}\right)$ satisfy
$$|g(x)| \leq K(1+|x|)$$
for some $K>0$ and let $\phi \in C\left(\mathbb{R}^{n}\right)$ be such that

$$\lim {|q| \rightarrow+\infty} \frac{\phi(q)}{|q|}=+\infty .$$ The inf-convolution of $g$ with kernel $\phi$ is the function $$g \phi(x)=\inf {y \in \mathbb{R}^{a}}[g(y)+\phi(x-y)],$$
while the sup-convolution of $g$ with kernel $\phi$ is defined by
$$g^{\phi}(x)=\sup {y \in \mathbb{R}^{n}}[g(y)-\phi(x-y)] .$$ We observe that the function $u$ given by Hopf’s formula (1.10) is an infconvolution with respect to the $x$ variable for any fixed $t$. In addition, inf-convolutions are a particular case of the marginal functions introduced in the previous section. We give below some regularity properties of the inf-convolutions. The corresponding statements about the sup-convolutions are easily obtained observing that $g^{\phi}=-\left((-g){\phi}\right)$.

统计代写|最优控制作业代写optimal control代考|Superdifferential of a semiconcave function

Jor 任何 pont y EA 使得[是的,r⌋s。−

统计代写|最优控制作业代写optimal control代考|Marginal functions

F(是的,p,X)=在(是的)+⟨p,X−是的⟩+|是的−X|ω1(|是的−X|)

统计代写|最优控制作业代写optimal control代考|Inf-convolutions

X→信息是的∈Rn(G(是的)+|X−是的|22e)X→支持是的∈Rn(G(是的)−|X−是的|22e)

|G(X)|≤ķ(1+|X|)

Gφ(X)=支持是的∈Rn[G(是的)−φ(X−是的)].我们观察到函数在Hopf 的公式 (1.10) 给出的是关于X任何固定的变量吨. 此外，inf-convolutions 是上一节介绍的边缘函数的一个特例。我们在下面给出了 inf 卷积的一些规律性属性。观察到关于上卷积的相应陈述很容易获得Gφ=−((−G)φ).

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

statistics-lab™ 为您的留学生涯保驾护航 在代写最优控制optimal control方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写最优控制optimal control代写方面经验极为丰富，各种代写最优控制Soptimal control相关的作业也就用不着说。

• 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

In the last decades a branch of mathematics has developed called nonsmooth analysis, whose object is to generalize the basic tools of calculus to functions that are not differentiable in the classical sense. For this purpose, one introduces suitable notions of generalized differentials, which are extensions of the usual gradient; the best known example is the subdifferential of convex analysis. The motivation for this study is that in more and more fields of analysis, like the optimization problems considered in this book, the functions that come into play are often nondifferentiable.
For semiconcave functions, the analysis of generalized gradients is important in view of applications to control theory. As we have already seen in a special case (Corollary 1.5.10), if the value function of a control problem is smooth, then one can design the optimal trajectories knowing the differential of the value function. In the general case, where the value function is not smooth but only semiconcave, one can try to follow a similar procedure starting from its generalized gradient.

In Section $3.1$ we define the generalized differentials which are relevant for our purposes and recall basic properties and equivalent characterizations of these objects. Then, we restrict ourselves to semiconcave functions. In Section $3.2$ we show that semiconcave functions possess one-sided directional derivatives everywhere, while in Section $3.3$ we describe the special properties of the superdifferential of a semiconcave function; in particular, we show that it is nonempty at every point and that it is a singleton exactly at the points of differentiability. These properties are classical in the case of concave functions; here we prove that they hold for semiconcave functions with arbitrary modulus.

Section $3.4$ is devoted to the so-called marginal functions, which are obtained as the infimum of smooth functions. We show that semiconcave functions can be characterized as suitable classes of marginal functions. In addition, we describe the semi-differentials of a marginal function using the general results of the previous sections. In Section $3.5$ we study the so-called inf-convolutions. They are marginal functions defined by a process which is a generalization of Hopf’s formula, and provide approximations to a given function which enjoy useful properties. Finally, in Section $3.6$ we introduce proximal gradients and proximally smooth sets, and we analyze how these notions are related to semiconcavity.

统计代写|最优控制作业代写optimal control代考|Generalized differentials

We begin with the definitions of some generalized differentials and derivatives from nonsmooth analysis. In this section $u$ is a real-valued function defined on an open set $A \subset \mathbb{R}^{n}$.
Definition 3.1.1 For any $x \in A$, the sets
\begin{aligned} D^{-} u(x) &=\left{p \in \mathbb{R}^{n}: \liminf {y \rightarrow x} \frac{u(y)-u(x)-\langle p, y-x\rangle}{|y-x|} \geq 0\right} \ D^{+} u(x) &=\left{p \in \mathbb{R}^{n}: \limsup {y \rightarrow x} \frac{u(y)-u(x)-\langle p, y-x\rangle}{|y-x|} \leq 0\right} \end{aligned}
are called, respectively, the (Fréchet) superdifferential and subdifferential of $u$ at $x$.
From the definition it follows that, for any $x \in A$,
$$D^{-}(-u)(x)=-D^{+} u(x) .$$
Example 3.1.2
Let $A=\mathbb{R}$ and let $u(x)=|x|$. Then it is easily seen that $D^{+} u(0)=\emptyset$ whereas $D^{-} u(0)=[-1,1] .$
Let $A=\mathbb{R}$ and let $u(x)=\sqrt{|x|}$. Then, $D^{+} u(0)=\emptyset$ whereas $D^{-} u(0)=\mathbb{R}$.
Let $A=\mathbb{R}^{2}$ and $u(x, y)=|x|-|y|$. Then, $D^{+} u(0,0)=D^{-} u(0,0)=\emptyset$.
Definition 3.1.3 Let $x \in A$ and $\theta \in \mathbb{R}^{n}$. The upper and lower Dini derivatives of $u$ at $x$ in the direction $\theta$ are defined as
$$\partial^{+} u(x, \theta)=\lim {h \rightarrow 0^{+}, \theta^{\prime} \rightarrow \theta} \frac{u\left(x+h \theta^{\prime}\right)-u(x)}{h}$$ and $$\partial^{-} u(x, \theta)=\liminf {h \rightarrow 0^{+}, \theta^{\prime} \rightarrow \theta} \frac{u\left(x+h \theta^{\prime}\right)-u(x)}{h},$$
respectively.
It is readily seen that, for any $x \in A$ and $\theta \in \mathbb{R}^{n}$
$$\partial^{-}(-u)(x, \theta)=-\partial^{+} u(x, \theta) .$$
Remark 3.1.4 Whenever $u$ is Lipschitz continuous in a neighborhood of $x$, the lower Dini derivative reduces to
$$\partial^{-} u(x, \theta)=\liminf _{h \rightarrow 0+} \frac{u(x+h \theta)-u(x)}{h}$$
for any $\theta \in \mathbb{R}^{n}$. Indeed, if $L>0$ is the Lipschitz constant of $u$ we have
$$\left|\frac{u\left(x+h \theta^{\prime}\right)-u(x)}{h}-\frac{u(x+h \theta)-u(x)}{h}\right| \leq L\left|\theta^{\prime}-\theta\right|,$$
and (3.5) easily follows. A similar property holds for the upper Dini derivative.

统计代写|最优控制作业代写optimal control代考|Directional derivatives

We begin our exposition of the differential properties of semiconcave functions showing that they possess (one-sided) directional derivatives
$$\partial u(x, \theta):=\lim {h \rightarrow 0^{+}} \frac{u(x+h \theta)-u(x)}{h}$$ at any point $x$ and in any direction $\theta$. Theorem 3.2.1 Let $u: A \rightarrow \mathbb{R}$ be semiconcave. Then, for any $x \in A$ and $\theta \in \mathbb{R}^{n}$, $$\partial u(x, \theta)=\partial^{-} u(x, \theta)=\partial^{+} u(x, \theta)=u{-}^{0}(x, \theta) .$$
Proof – Let $\delta>0$ be fixed so that $B_{\delta|\theta|}(x) \subset A$. Then, for any pair of numbers $h_{1}, h_{2}$ satisfying $0<h_{1} \leq h_{2}<\delta$, estimate (2.1) yields
$$\left(1-\frac{h_{1}}{h_{2}}\right) u(x)+\frac{h_{1}}{h_{2}} u\left(x+h_{2} \theta\right)-u\left(x+h_{1} \theta\right) \leq h_{1}\left(1-\frac{h_{1}}{h_{2}}\right)|\theta| \omega\left(h_{2}|\theta|\right) .$$
Hence,
\begin{aligned} &\frac{u\left(x+h_{1} \theta\right)-u(x)}{h_{1}} \ &\geq \frac{u\left(x+h_{2} \theta\right)-u(x)}{h_{2}}-\left(1-\frac{h_{1}}{h_{2}}\right)|\theta| \omega\left(h_{2}|\theta|\right) . \end{aligned}
Taking the liminf as $h_{1} \rightarrow 0^{+}$in both sides of the above inequality, we obtain

$$\partial^{-} u(x, \theta) \geq \frac{u\left(x+h_{2} \theta\right)-u(x)}{h_{2}}-|\theta| \omega\left(h_{2}|\theta|\right)$$
Now, taking the limsup as $h_{2} \rightarrow 0^{+}$, we conclude that
$$\partial^{-} u(x, \theta) \geq \partial^{+} u(x, \theta) .$$
So, $\partial u(x, \theta)$ exists and coincides with the lower and upper Dini derivatives.
To complete the proof of $(3.15)$ it suffices to show that
$$\partial^{+} u(x, \theta) \leq u_{-}^{0}(x, \theta),$$
since the reverse inequality holds by definition and by Remark 3.1.4. For this purpose, let $\varepsilon>0, \lambda \in] 0, \delta[$ be fixed. Since $u$ is continuous, we can find $\alpha \in$ ] $0,(\delta-\lambda) \theta$ [ such that
$$\frac{u(x+\lambda \theta)-u(x)}{\lambda} \leq \frac{u(y+\lambda \theta)-u(y)}{\lambda}+\varepsilon, \quad \forall y \in B_{\alpha}(x) .$$
Using inequality (3.16) with $x$ replaced by $y$, we obtain
$$\left.\frac{u(y+\lambda \theta)-u(y)}{\lambda} \leq \frac{u(y+h \theta)-u(y)}{h}+|\theta| \omega(\lambda|\theta|), \quad \forall h \in\right] 0, \lambda[.$$
Therefore,
$$\frac{u(x+\lambda \theta)-u(x)}{\lambda} \leq \inf {y \in B{u}(x), h \in|0, \lambda|} \frac{u(y+h \theta)-u(y)}{h}+|\theta| \omega(\lambda|\theta|)+\varepsilon .$$
This implies, by definition of $u_{-}^{0}(x, \theta)$, that
$$\frac{u(x+\lambda \theta)-u(x)}{\lambda} \leq u_{-}^{0}(x, \theta)+|\theta| \omega(\lambda|\theta|)+\varepsilon .$$
Hence, taking the limit as $\varepsilon, \lambda \rightarrow 0$, we obtain inequality (3.17).

统计代写|最优控制作业代写optimal control代考|Generalized differentials

\begin{对齐} D^{-} u(x) &=\left{p \in \mathbb{R}^{n}: \liminf {y \rightarrow x} \frac{u(y)-u( x)-\langle p, yx\rangle}{|yx|} \geq 0\right} \ D^{+} u(x) &=\left{p \in \mathbb{R}^{n}： \limsup {y \rightarrow x} \frac{u(y)-u(x)-\langle p, yx\rangle}{|yx|} \leq 0\right} \end{aligned}\begin{对齐} D^{-} u(x) &=\left{p \in \mathbb{R}^{n}: \liminf {y \rightarrow x} \frac{u(y)-u( x)-\langle p, yx\rangle}{|yx|} \geq 0\right} \ D^{+} u(x) &=\left{p \in \mathbb{R}^{n}： \limsup {y \rightarrow x} \frac{u(y)-u(x)-\langle p, yx\rangle}{|yx|} \leq 0\right} \end{aligned}

D−(−在)(X)=−D+在(X).

∂+在(X,θ)=林H→0+,θ′→θ在(X+Hθ′)−在(X)H和∂−在(X,θ)=林恩夫H→0+,θ′→θ在(X+Hθ′)−在(X)H,

∂−(−在)(X,θ)=−∂+在(X,θ).

∂−在(X,θ)=林恩夫H→0+在(X+Hθ)−在(X)H

|在(X+Hθ′)−在(X)H−在(X+Hθ)−在(X)H|≤大号|θ′−θ|,

统计代写|最优控制作业代写optimal control代考|Directional derivatives

∂在(X,θ):=林H→0+在(X+Hθ)−在(X)H在任何时候X并且在任何方向θ. 定理 3.2.1 令在:一种→R是半凹的。那么，对于任何X∈一种和θ∈Rn,∂在(X,θ)=∂−在(X,θ)=∂+在(X,θ)=在−0(X,θ).

(1−H1H2)在(X)+H1H2在(X+H2θ)−在(X+H1θ)≤H1(1−H1H2)|θ|ω(H2|θ|).

∂−在(X,θ)≥∂+在(X,θ).

∂+在(X,θ)≤在−0(X,θ),

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

统计代写|实验设计作业代写experimental design代考|CORRELATION FORM

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• 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

统计代写|实验设计作业代写experimental design代考|CORRELATION FORM

When the main concern is to decide which variables to include in the model, a very useful transformation of the data is to scale each variable, predictors and dependent variables alike, so that the normal equations can be written in correlation form. This enables us to identify important variables which should be included in the model and it also reveals some of the dependenoles between the predictor variables.

As usual, we consider the variables to be in deviation form. The correlation coefficient between $x_{1}$ and $x_{2}$ is
$$\left.\left.r_{12}=s_{12} / \sqrt{\left(s_{11}\right.} s_{22}\right)=\sum x_{1} x_{2} / \sqrt{\left(s_{11}\right.} s_{22}\right)$$
If we divide each variable $x_{1}$ by $\sqrt{S}{11}$ and denote the result as $$x{1}^{}=x_{1} / \sqrt{s}{1 i}$$ then $x{i}^{}$ is said to be in correlation form. Notice that
$$\Sigma x_{i}^{}=0$$ $$\Sigma\left(x_{i}^{}\right)^{2}=1$$
$$\Sigma x_{i}^{} x_{j}^{}=r_{1 j}$$
We have transformed the model from

$$y=B_{1} x_{1}+B_{2} x_{2}+\varepsilon \text { to } y^{}=\alpha_{1} x_{1}^{}+\alpha_{2} x_{2}^{*}+\varepsilon$$
and the normal equations simplify from
\begin{aligned} &s_{11} b_{1}+s_{12} b_{2}=s_{y 1} \ &s_{12} b_{1}+s_{22} b_{2}=s_{y 2} \end{aligned} \text { to } \quad r_{12} a_{1}+r_{12}+a_{2}=r_{y 1}=r_{y 2} \quad \text { (3.5.3) }

统计代写|实验设计作业代写experimental design代考|VARIABLE SELECTION – ALL POSSIBLE REGRESSIONS

In many situations, researchers know which variables may be included in the predictor model. There is some advantage in reducing the number of predictor variables to form a more parsimonious model. One way to achieve this is to run all possible regressions and to consider such statistics as the coefficient of determination, $R^{2}=$ SSR/SST.
We will use the heart data of Section 3.5, again relabelling the variables as A through $F$. With the variables in correlation form, $R^{2}=S S R$, the sum of squares for regression, and this is given for each possible combination of predictor variables in Table $3.6 .1$.

To assist the choice of the best subset, C.L. Mallows suggested fitting all possible models and evaluating the statistic
$$C_{p}=S S E_{p} / s^{2}-(n-2 p)$$
Here, $n$ is the number of observations and $p$ is the number of predictor variables in the subset, including a constant term. For each subset, the value of Mallows’ statistio can be evaluated from the correponding value of SSR. The complete set of these statistics are listed in Table 3.6.2. For each subset we use the mean squared error, MSE, of the full model as an estimate of the variance.

Suppose that the true model has q predictor variables.

统计代写|实验设计作业代写experimental design代考|VARIABLE SELECTION – SEQUENTIAL METHODS

When the number of possible variables in a model is large, it may be inappropriate to run every possible regression and evaluate Mallows’ statistic for each one, even though short cuts can be taken to evaluate such statistios by adding or subtracting terms rather than by evaluating each one from scratch.

Another approach is to add, or remove, variables, sequentially. We have seen that adding a variable will increase SSR, the sum of squares for regression. From Section $3.4$ we could perform an F-test to decide if the increase in SSR is si gnificant. The first method we consider is that of forward selection.

统计代写|实验设计作业代写experimental design代考|CORRELATION FORM

r12=s12/(s11s22)=∑X1X2/(s11s22)

ΣX一世=0Σ(X一世)2=1
ΣX一世Xj=r1j

$$y=B_{1} x_{1}+B_{2} x_{2}+\varepsilon \text { to } y^{ }=\alpha_{1} x_{1}^{ }+\alpha_{ 2} x_{2}^{*}+\varepsilon 一种nd吨H和n这r米一种l和q在一种吨一世这nss一世米pl一世F是的Fr这米 s11b1+s12b2=s是的1 s12b1+s22b2=s是的2\text { to } \quad r_{12} a_{1}+r_{12}+a_{2}=r_{y 1}=r_{y 2} \quad \text { (3.5.3) }$$

统计代写|实验设计作业代写experimental design代考|VARIABLE SELECTION – ALL POSSIBLE REGRESSIONS

Cp=小号小号和p/s2−(n−2p)

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

统计代写|实验设计作业代写experimental design代考|WHICH VARIABLES SHOULD BE INCLUDED IN THE MODEL

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• 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

统计代写|实验设计作业代写experimental design代考|INTRODUCTION

When a model can be formed by including some, or all, of the predictor variables, there is a problem in deciding how many variables to include. The decision we arrive at will depend to some extent on the purpose we have in mind. If we merely wish to explain the variation of the dependent variable in the sample, then $1 \mathrm{t}$ would seem obvious that as many predictor variables as possible should be included. This can be seen with the lactation curve of Example $2.11$. If enough powers of $w$ were added to the model the curve would pass through every observed value, but it would be so jagged and complicated it would be difficult to understand what was happening. On the other hand, a small model has the advantage that it is easy to understand the relationships between the variables. Further more, a small model will usually yield estimators which are less influenced by peculiarites of the sample and so are more stable. Another important decision which must be made is whether to use the original predictor variables or to transform them in some way, often by taking a linear combination. For example, the cost of a particular kind of fencing for a rectangular field may largely depend on the length and breadth of the field. If all the fields in the

sample are in the same proportions then only one variable (length or breadth) would be needed. Even if they are not in the same proportions, one variable may be sufficient, namely the sum of the length and the breadth or, indeed, the perimeter. This is our ideal solution, reducing the number of predictor variables from two to one and at the same time obtaining a predictor variable which has physi= cal meaning, With a particular data set, the predicted value of the cost may be $y=1.11+0.9$ b so that the best single variable would be the $r$ ight hand side with $1=l$ ength and $b=b r e a d t h$, but this particular linear combination would have no physical meaning. We need to keep both aspects in mind, balancing statistioal optimum against physical meaning.
In the first section we shall 11 mit our discussion to orthogonal predictor variables, Although this may seem an unnecessarily strong restriction to place on the model, orthogonal variables of ten exist in experimental design situations. Indeed the values of the variables in the sample are often deliberately chosen to be orthogonal. We explain the advantages of this in section $3.2$, while in section $3.4$ we show that 1 t is possible to transform variables, for any data set, so that they are orthogonal.

统计代写|实验设计作业代写experimental design代考|ORTHOGONAL PREDICTOR VARIABLES

If the variables in a model are expressed as deviations from their means and if there are $k$ predictor variables, the sum of squares for regression is given by
\begin{aligned} \mathrm{SSR} &=b_{1} s_{y 1}+b_{2} s_{y 2}+\cdots+b_{k} s_{y k} \ &=s_{y 1}^{2} / s_{11}+s_{y 2}^{2} / s_{22}+\cdots+s_{y k}^{2} / s_{k k} \end{aligned}
The total sum of squares is
$$\text { SST }-s_{y y}=\sum y_{i}^{2}$$
By subtraction, we find the sum of squares for error (residual) is

$$\mathrm{SSE}=S S T-S S R$$
In this seotion, we assume that the predicton variables are orthog= onal and explore the implications of the number of variables included in the model.

We consider now the effect of adding another variable, $x_{k+1}$, to the model and assume that this variable $1 s$ also orthogonal to the other predfotor variables. The SST will not be affected by adding $x_{k+1}$ to the model. We introduce the notation that SSR(k) is the sum of squares for negression when the variables $x_{1}, x_{2}, \cdots x_{k}$ are in the model. It is clear that
(i) $\operatorname{SSR}(k+1) \geq \operatorname{SSR}(k)$
This follows from $(3.2 .1)$ as each term in the sum cannot be negative so that adding a further variable cannot decrease the sum of squares for regression.
(ii) $\operatorname{SSE}(k+1) \leq \operatorname{SSE}(k)$
This is the other side of the coin and follows from $(3.2 .2)$.
(111)
$$R(k+1)^{2}=\operatorname{SSR}(k+1) / \operatorname{SST} \geq \mathrm{R}(k)^{2}=\operatorname{SSR}(k) / \mathrm{SST}$$
SSR $(k+1)$ can be thought of as the amount of variation in $y$ explained by the $(k+1)$ predictor variables, and $R(k+1)^{2}$ is the proportion of the variation in y explained by these variables. These monotone properties are illustrated by the diagrams in figure $3.2 .1$.

Although orthogonal predictor variables are the ideal, they will rarely occur in practice with observational data. If some of the predictor variables are highly correlated, the matrix $X \mathrm{~T} X$ will be nearly singular. This could raise statistical and numerical problems, particularly if there is interest in estimating the coefficients of the model. We nave more to say on this in the next section and in a later section on Ridge Estimators.

Moderate correlations between predictor variables will cause few problems. While it is not essential to convert predictor variables to others which are orthogonal, it is instruotive to do so as it gives insight into the meaning of the coefficients and the tests of significance based on them.

In Problem 1.5, we considered predicting the outcome of a student in the mathematics paper 303 (which we denoted by y) by marks

recelved in the papers 201 and 203 (denoted by $x_{1}$ and $x_{2}$, respectively). The actual numbers of these papers are not relevant, but, for interest sake, the paper 201 was a calculus paper and 203 an al gebra paper, both at second year university level and 303 was a third year paper in algebra. The sum of squares for regression when $y$ is regressed singly and together on the $x$ variables (and the $F^{2}$ values) are:
$\begin{array}{lll}\text { SSR on } 201 \text { alone : } & 1433.6 & (.405) \ \text { SSR on } 203 \text { alone : } & 2129.2 & (.602) \ \text { SSR on } 201 \text { and } 203: & 2265.6 & (.641)\end{array}$
Clearly, the two $x$ variables are not orthogonal (and, in fact, the correlation coefficient between them is 0.622) as the individual sums of squares for regression do not add to that given by the model with both variables included. Once we have regressed the 303 marks on the 201 marks, the additional sum of squares due to $2031 \mathrm{~s}$ (2265.6 $1433.6)=832$. In this section we show how to adjust one variable for another so that they are orthogonal, and, as a consequence, their sums of squares for regression add to that given by the model with both variables included.
$\begin{array}{ll}\text { SSR for } 201 & =1433.6=\text { SSR for } x_{1} \ \text { SSR for } 203 \text { adjusted for } 201=832.0=\text { SSR for } z_{2} \ \text { SSR for } 201 \text { and } 203 & =2265.6\end{array}$

统计代写|实验设计作业代写experimental design代考|ORTHOGONAL PREDICTOR VARIABLES

SST −s是的是的=∑是的一世2

（一）固态硬盘⁡(ķ+1)≥固态硬盘⁡(ķ)

(二)上证所⁡(ķ+1)≤上证所⁡(ķ)

(111)
R(ķ+1)2=固态硬盘⁡(ķ+1)/SST≥R(ķ)2=固态硬盘⁡(ķ)/小号小号吨

SSR 开启 201 独自的 ： 1433.6(.405)  SSR 开启 203 独自的 ： 2129.2(.602)  SSR 开启 201 和 203:2265.6(.641)

SSR 为 201=1433.6= SSR 为 X1  SSR 为 203 调整为 201=832.0= SSR 为 和2  SSR 为 201 和 203=2265.6

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

统计代写|最优控制作业代写optimal control代考|Special properties of SCL

statistics-lab™ 为您的留学生涯保驾护航 在代写最优控制optimal control方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写最优控制optimal control代写方面经验极为丰富，各种代写最优控制Soptimal control相关的作业也就用不着说。

• 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

统计代写|最优控制作业代写optimal control代考|Special properties of SCL

While many properties of semiconcave functions are valid in the case of an arbitrary modulus of semiconcavity, there are some results which are peculiar to the case of a linear modulus; we collect in this section some important ones, in addition to those already given in Proposition 1.1.3.

We have seen in Proposition 1.1.3 that semiconcave functions with a linear modulus can be regarded as $C^{2}$ perturbations of concave functions. This allows to extend immediately some well-known properties of concave functions, such as the following.

Theorem 2.3.1 Let $u \in \mathrm{SCL}(A)$, with $A \subset \mathbb{R}^{n}$ open. Then the following properties hold.
(i) (Alexandroff’s Theorem) $u$ is twice differentiable a.e, that is, for a.e. every $x_{0} \in A$, there exist a vector $p \in \mathbb{R}^{n}$ and a symmetric matrix $B$ such that
$$\lim {x \rightarrow x{0}} \frac{u(x)-u\left(x_{0}\right)-\left\langle p, x-x_{0}\right)+\left\langle B\left(x-x_{0}\right), x-x_{0}\right\rangle}{\left|x-x_{0}\right|^{2}}=0 .$$
(ii) The gradient of u, defined almost everywhere in A, belongs to the class $\mathrm{BV}_{\text {loc }}\left(A, \mathbb{R}^{n}\right)$.
Proof – Properties (i) and (ii) hold for a convex function (see e.g., $[72$, Ch. 6.3]). Since $u$ is the difference of a smooth function and a convex one, $u$ also satisfies these properties.

The following result shows that semiconcave functions with linear modulus exhibit a behavior similar to $C^{2}$ functions near a minimum point.

Theorem 2.3.2 Let $u \in \mathrm{SCL}(A)$, with $A \subset \mathbb{R}^{n}$ open, and let $x_{0} \in A$ be a point of local minimum for $u$. Then there exist a sequence $\left{x_{h}\right} \subset A$ and an infinitesimal sequence $\left{\varepsilon_{h}\right} \subset \mathbb{R}+$ such that $u$ is $t$ wice differentiable in $x_{h}$ and that
$$\lim {h \rightarrow \infty} x{h}=x_{0}, \quad \lim {h \rightarrow \infty} D u\left(x{h}\right)=0, \quad D^{2} u\left(x_{h}\right) \geq-\varepsilon_{h} I \quad \forall h .$$
The proof of this theorem is based on the following result.

统计代写|最优控制作业代写optimal control代考|A differential Harnack inequality

Let us consider the parabolic Hamilton-Jacobi equation
$$\partial_{f} u(t, x)+|\nabla u(t, x)|^{2}=\Delta u(t, x), \quad t \geq 0, x \in \mathbb{R}^{n} .$$
We have seen in Proposition 2.2.6 that the solutions to this equation are semiconcave. We now show how such a semiconcavity result is related to the classical Harnack inequality for the heat equation.

A remarkable feature of equation $(2.15)$ is that it can be reduced to the heat equation by a change of unknown called the Cole-Hopf transformation, or logarithmic transformation. In fact, if we set $w(t, x)=\exp (-u(t, x))$, a direct computation shows that $u$ satisfies $(2.15)$ if and only if $\partial_{t} w=\Delta w$. Let us investigate the properties of $w$ which follow from the semiconcavity of $u$.

Proposition 2.4.1 Let $w$ be a positive solution of the heat equation in $[0, T] \times \mathbb{R}^{n}$ whose first and second derivatives are bounded. Then w satisfies
$$\nabla^{2} w+\frac{w}{2 t} I-\frac{\nabla w \otimes \nabla w}{w} \geq 0$$
Here $\nabla^{2} w$ denotes the hessian matrix of $w$ with respect to the space variables; inequality (2.16) means that the matrix on the left-hand side is positive semidefinite.
Proof – It is not restrictive to assume that $w$ is greater than some positive constant; if this is not the case, we can replace $w$ by $w+\varepsilon$ and then let $\varepsilon \rightarrow 0^{+}$. Let us set $u(t, x)=-\ln (w(t, x))$. Then $u$ is a solution of equation (2.15). In addition, $u$ is bounded together with its first and second derivatives. Therefore, by Proposition $2.2 .6, u\left(t,{ }^{-}\right)$is semiconcave with modulus $\omega(\rho)=\rho /(4 t)$. Using the equivalent formulations of Proposition 1.1.3, we can restate this property as
$$\nabla^{2} u \leq \frac{1}{2 t} I$$
On the other hand, an easy computation shows that
$$\nabla^{2} u=-\frac{\nabla^{2} w}{w}+\frac{\nabla w \otimes \nabla w}{w^{2}}$$
and this proves (2.16). Taking the trace of the left-hand side of (2.16), we obtain
$$\Delta w+\frac{n w}{2 t}-\frac{|\nabla w|^{2}}{w} \geq 0$$
which implies $(2.17)$, since $w$ solves the heat equation.
Inequality (2.17) is called a differential Harnack estimate. The connection with the classical Harnack inequality is explained by the following result.

统计代写|最优控制作业代写optimal control代考|A generalized semiconcavity estimate

In this section we compare the semiconcavity estimate with another one-sided estimate, a priori weaker, which was introduced in [46]. We prove here that the two estimates are in some sense equivalent, and this has applications for the study of certain Hamilton-Jacobi equations, as we will see in the following (see Theorem $5.3 .7)$.

Let us consider a function $u: A \rightarrow \mathbb{R}$, with $A \subset \mathbb{R}^{n}$ open. Given $x 0 \in A$, we set, for $0<\delta<\operatorname{dist}\left(x_{0}, \partial A\right), x \in B_{1}$,
$$u_{x_{0}, \delta}(x)=\frac{u\left(x_{0}+\delta x\right)-u\left(x_{0}\right)}{\delta}$$

Definition 2.5.1 Let $C \subset A$ be a compact set. We say that u satisfies a generalized one-sided estimate in $C$ if there exist $\left.K \geq 0, \delta_{0} \in\right] 0$, $\operatorname{dist}(C, \partial A)[$ and a nondecreasing upper semicontinuous function $\tilde{\omega}:[0,1] \rightarrow \mathbb{R}{+}$, such that $\lim {h \rightarrow 0} \tilde{\omega}(h)=0$ and
\begin{aligned} &\lambda u_{x_{0}, \delta}(x)+(1-\lambda) u_{x_{0}, \delta}(y)-u_{x_{0}, \delta}(\lambda x+(1-\lambda) y) \ &\leq \lambda(1-\lambda)|x-y|{K \delta+\widetilde{\omega}(|x-y|)} \end{aligned}
for all $\left.x_{0} \in C, \delta \in\right] 0, \delta_{0}\left[, x, y \in B_{1}, \lambda \in[0,1]\right.$.
It is easily seen that, if $u$ is semiconcave in $A$, then the above property is satisfied taking $\tilde{\omega}$ equal to a modulus of semiconcavity of $u$ in $A$ and $K=0$. Conversely, semiconcavity can be deduced from the one-sided estimate above, as the next result shows.

Theorem 2.5.2 Let $u: A \rightarrow \mathbb{R}$, with A open and let $C$ be a compact subset of $A$. If u satisfies a generalized one-sided estimate in $C$, then $u$ is semiconcave in $C$.

Proof – By hypothesis inequality $(2.20)$ holds for some $K, \delta_{0}, \tilde{\omega}$. Let us take $x, y \in$ $C$ such that $[x, y] \subset C$ and $\lambda \in[0,1]$. It is not restrictive to assume $|x-y|<\delta_{0} / 2$. For any $\delta$ with $|x-y|<\delta<\delta_{0}$, we set
$$x_{0}=\lambda x+(1-\lambda) y, x^{\prime}=\delta^{-1}(1-\lambda)(x-y), y^{\prime}=\delta^{-1} \lambda(y-x) .$$
From $(2.19)$ and $(2.20)$ we obtain
\begin{aligned} &\lambda u(x)+(1-\lambda) u(y)-u(\lambda x+(1-\lambda) y) \ &=\delta\left{\lambda u_{x_{0}, \delta}\left(x^{\prime}\right)+(1-\lambda) u_{x_{0}, \delta}\left(y^{\prime}\right)-u_{x_{0}, \delta}\left(\lambda x^{\prime}+(1-\lambda) y^{\prime}\right)\right} \ &\leq \delta \lambda(1-\lambda)\left|x^{\prime}-y^{\prime}\right|\left{K \delta+\widetilde{\omega}\left(\left|x^{\prime}-y^{\prime}\right|\right)\right} \ &=\lambda(1-\lambda)|x-y|\left{K \delta+\widetilde{\omega}\left(\delta^{-1}|x-y|\right)\right} . \end{aligned}
Therefore
$$\lambda u(x)+(1-\lambda) u(y)-u(\lambda x+(1-\lambda) y) \leq \lambda(1-\lambda)|x-y| \omega(|x-y|)$$
where $\omega(\rho):=\inf {\delta \in\rfloor \rho, \delta{0}[}\left{K \delta+\tilde{\omega}\left(\delta^{-1} \rho\right)\right}$. The function $\omega$ is upper semicontinuous and nondecreasing. The conclusion will follow if we show that $\lim {h \rightarrow 0} \omega(h)=0$. Given $\varepsilon \in 10.2 K \delta$ o $[$. we choose $\eta \in] 0$. 1[ such that $\tilde{\omega}(s)<\varepsilon / 2$ for $0{0}[$; therefore
$$\omega(\rho) \leq\left{K \frac{\varepsilon}{2 K}+\tilde{\omega}\left(\frac{2 K}{\varepsilon} \rho\right)\right}<\varepsilon .$$
This shows that $\lim _{\rho \rightarrow 0} \omega(\rho)=0$ and concludes the proof.

统计代写|最优控制作业代写optimal control代考|Special properties of SCL

(i) (Alexandroff 定理)在是二次可微的ae，也就是说，对于ae，每X0∈一种, 存在一个向量p∈Rn和一个对称矩阵乙这样

(ii) u 的梯度，在 A 中几乎处处定义，属于类乙在地方 (一种,Rn).

统计代写|最优控制作业代写optimal control代考|A differential Harnack inequality

∂F在(吨,X)+|∇在(吨,X)|2=Δ在(吨,X),吨≥0,X∈Rn.

∇2在+在2吨一世−∇在⊗∇在在≥0

∇2在≤12吨一世

∇2在=−∇2在在+∇在⊗∇在在2

Δ在+n在2吨−|∇在|2在≥0

统计代写|最优控制作业代写optimal control代考|A generalized semiconcavity estimate

X0=λX+(1−λ)是的,X′=d−1(1−λ)(X−是的),是的′=d−1λ(是的−X).

\begin{对齐} &\lambda u(x)+(1-\lambda) u(y)-u(\lambda x+(1-\lambda) y) \ &=\delta\left{\lambda u_{x_ {0}, \delta}\left(x^{\prime}\right)+(1-\lambda) u_{x_{0}, \delta}\left(y^{\prime}\right)-u_ {x_{0}, \delta}\left(\lambda x^{\prime}+(1-\lambda) y^{\prime}\right)\right} \ &\leq \delta \lambda(1- \lambda)\left|x^{\prime}-y^{\prime}\right|\left{K \delta+\widetilde{\omega}\left(\left|x^{\prime}-y^{ \prime}\right|\right)\right} \ &=\lambda(1-\lambda)|xy|\left{K \delta+\widetilde{\omega}\left(\delta^{-1}|xy |\right)\right} 。\结束{对齐}\begin{对齐} &\lambda u(x)+(1-\lambda) u(y)-u(\lambda x+(1-\lambda) y) \ &=\delta\left{\lambda u_{x_ {0}, \delta}\left(x^{\prime}\right)+(1-\lambda) u_{x_{0}, \delta}\left(y^{\prime}\right)-u_ {x_{0}, \delta}\left(\lambda x^{\prime}+(1-\lambda) y^{\prime}\right)\right} \ &\leq \delta \lambda(1- \lambda)\left|x^{\prime}-y^{\prime}\right|\left{K \delta+\widetilde{\omega}\left(\left|x^{\prime}-y^{ \prime}\right|\right)\right} \ &=\lambda(1-\lambda)|xy|\left{K \delta+\widetilde{\omega}\left(\delta^{-1}|xy |\right)\right} 。\结束{对齐}

λ在(X)+(1−λ)在(是的)−在(λX+(1−λ)是的)≤λ(1−λ)|X−是的|ω(|X−是的|)

\omega(\rho) \leq\left{K \frac{\varepsilon}{2 K}+\波浪号{\omega}\left(\frac{2 K}{\varepsilon} \rho\right)\right} <\伐普西隆。\omega(\rho) \leq\left{K \frac{\varepsilon}{2 K}+\波浪号{\omega}\left(\frac{2 K}{\varepsilon} \rho\right)\right} <\伐普西隆。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

统计代写|最优控制作业代写optimal control代考|Semiconcave Functions

statistics-lab™ 为您的留学生涯保驾护航 在代写最优控制optimal control方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写最优控制optimal control代写方面经验极为丰富，各种代写最优控制Soptimal control相关的作业也就用不着说。

• 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

统计代写|最优控制作业代写optimal control代考|Semiconcave Functions

This chapter and the following two are devoted to the general properties of semiconcave functions. We begin here by studying the direct consequences of the definition and some basic examples, while the next chapters deal with generalized differentials and singularities. At this stage we study semiconcave functions without referring to specific applications; later in the book we show how the results obtained here can be applied to Hamilton-Jacobi equations and optimization problems.

The chapter is structured as follows. In Section $2.1$ we define semiconcave functions in full generality, and study some direct consequences of the definition, like the Lipschitz continuity and the relationship with the differentiability. Then we consider some examples in Section 2.2, like the distance function from a set, or the solutions to certain partial differential equations. We give an account of the vanishing viscosity method for Hamilton-Jacobi equations, where semiconcavity estimates play an important role. In Section $2.3$ we recall some properties which are peculiar to semiconcave functions with a linear modulus, like Alexandroff’s theorem or Jensen’s lemma. In Section $2.4$ we investigate the relation between viscous Hamilton-Jacobi equations and the heat equation induced by the Cole-Hopf transformation, showing that semiconcavity corresponds to the Li-Yau differential Harnack inequality for the heat equation. Finally, in Section $2.5$ we analyze the relation between semiconcavity and a generalized one-sided estimate, a property which will be applied later in the book to prove semiconcavity of viscosity solutions.

统计代写|最优控制作业代写optimal control代考|Definition and basic properties

Throughout the section $S$ will be a subset of $\mathbb{R}^{n}$.
Definition 2.1.1 We say that a function $u: S \rightarrow \mathbb{R}$ is semiconcave if there exists a nondecreasing upper semicontinuous function $\omega: \mathbb{R}{+} \rightarrow \mathbb{R}{+}$such that $\lim _{\rho \rightarrow 0^{+}} \omega(\rho)=0$ and
$$\lambda u(x)+(1-\lambda) u(y)-u(\lambda x+(1-\lambda) y) \leq \lambda(1-\lambda)|x-y| \omega(|x-y|)$$

for any pair $x, y \in S$, such that the segment $[x, y]$ is contained in $S$ and for any $\lambda \in[0,1]$. We call $\omega a$ modulus of semiconcavity for $u$ in $S$. A function $v$ is called semiconvex in $S$ if $-v$ is semiconcave.

In the case of $\omega$ linear, we recover the class of semiconcave functions introduced in the previous chapter (see Definition 1.1.1 and Proposition 1.1.3). We recall that, if $\omega(\rho)=\frac{C}{2} \rho$, for some $C \geq 0$, then $C$ is called a semiconcavity constant for $u$ in $S$.
We denote by $\mathrm{SC}(S)$ the space of all semiconcave functions in $S$ and by $\mathrm{SCL}(S)$ the functions which are semiconcave in $S$ with a linear modulus. A usual, we use the notation $S C_{l o c}(S)$ or $S C L_{l o c}(S)$ for the functions which are semiconcave (with a linear modulus) locally in $S$, i.e., on every compact subset of $S$.

As we have remarked in Chapter 1 , semiconcave functions with a linear modulus are the most common in the literature. Although they are a smaller class, they are sufficient for many applications; in addition, they enjoy stronger properties than general semiconcave functions and are easier to analyze, since they are more closely related to concave functions. Nevertheless, it is interesting to consider semiconcave functions with a general modulus, since they are a larger class, sharing many of the properties of the case of a linear modulus.

An interesting consequence of the general definition of semiconcavity given above is that any $C^{1}$ function is semiconcave, without any assumption on its second derivatives, as the next result shows.

统计代写|最优控制作业代写optimal control代考|Examples

A first interesting example of a semiconcave function is provided by the distance function. We recall that the distance function from a given nonempty closed set $C \subset$ $\mathbb{R}^{n}$ is defined by
$$d_{C}(x)=\min {y \in C}|y-x|, \quad\left(x \in \mathbb{R}^{n}\right)$$ As we show below, $d{C}$ is not semiconcave in the whole space $\mathbb{R}^{n}$, but is semiconcave on the complement of $C$, at least locally. On the other hand, the square of the distance function is semiconcave in $\mathbb{R}^{\pi}$. Before proving this result, let us introduce a property of sets which is useful for the analysis of the semiconcavity of $d_{C}$.

Definition 2.2.1 We say that a set $C \subset \mathbb{R}^{n}$ satisfies an interior sphere condition for some $r>0$ if $C$ is the union of closed spheres of radius $r$, i.e., for any $x \in C$ there exists y such that $x \in \overline{B_{r}(y)} \subset C$.

Proposition 2.2.2 Let $C \subset \mathbb{R}^{n}$ be a closed set, $C \neq \emptyset, \mathbb{R}^{n}$. Then the distance funcrion $d_{C}$ satisfies the following properties:
(i) $d_{C}^{2} \in \mathrm{SCL}\left(\mathbb{R}^{n}\right)$ with semiconcavity constant 2 .
(ii) $d_{C} \in \mathrm{SCL}{\text {loc }}\left(\mathbb{R}^{n} \backslash C\right.$ ). More precisely, given a set $S$ (not necessarily compact) such that dist $(S, C)>0, d{C}$ is semiconcave in $S$ with semiconcavity constant equal to $\operatorname{dist}(S, C)^{-1}$.
(iii) If C satisfies an interior sphere condition for some $r>0$, then $d c \in \mathrm{SCL}\left(\overline{\mathbb{R}^{n} \backslash C}\right)$ with semiconcavity constant equal to $r^{-1}$.
(iv) $d_{C}$ is not locally semiconcave in the whole space $\mathbb{R}^{n}$.
Proof – (i) For any $x \in \mathbb{R}^{n}$ we have
$$d_{C}^{2}(x)-|x|^{2}=\inf {y \in C}|x-y|^{2}-|x|^{2}=\inf {y \in C}|y|^{2}-2\langle x, y\rangle$$
Since the infimum of linear functions is concave we deduce, by Proposition 1.1.3, that property (i) holds.
(ii) Let us first observe that, given $z, h \in \mathbb{R}^{n}, z \neq 0$, we have
\begin{aligned} &(|z+h|+|z-h|)^{2} \ &\leq 2\left(|z+h|^{2}+|z-h|^{2}\right)=4\left(|z|^{2}+|h|^{2}\right) \leq\left(2|z|+\frac{|h|^{2}}{|z|}\right)^{2} \end{aligned}

统计代写|最优控制作业代写optimal control代考|Examples

dC(X)=分钟是的∈C|是的−X|,(X∈Rn)正如我们在下面展示的，dC整个空间都不是半凹的Rn, 但在的补码上是半凹的C，至少在本地。另一方面，距离函数的平方是半凹的R圆周率. 在证明这个结果之前，让我们介绍一个集合的性质，它有助于分析dC.

(i)dC2∈小号C大号(Rn)半凹常数为 2 。
(二)dC∈小号C大号地方 (Rn∖C)。更准确地说，给定一个集合小号（不一定紧凑）使得 dist(小号,C)>0,dC是半凹的小号半凹常数等于距离⁡(小号,C)−1.
(iii) 如果 C 满足某个内部球体条件r>0， 然后dC∈小号C大号(Rn∖C¯)半凹常数等于r−1.
(四)dC在整个空间中不是局部半凹的Rn.

$$d_{C}^{2}(x)-|x|^{2}=\inf {y \in C}|xy|^ {2}-|x|^{2}=\inf {y \in C}|y|^{2}-2\langle x, y\rangle 小号一世nC和吨H和一世nF一世米在米这Fl一世n和一种rF在nC吨一世这ns一世sC这nC一种在和在和d和d在C和,b是的磷r这p这s一世吨一世这n1.1.3,吨H一种吨pr这p和r吨是的(一世)H这lds.(一世一世)大号和吨在sF一世rs吨这bs和r在和吨H一种吨,G一世在和n和,H∈Rn,和≠0,在和H一种在和 (|和+H|+|和−H|)2 ≤2(|和+H|2+|和−H|2)=4(|和|2+|H|2)≤(2|和|+|H|2|和|)2$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

统计代写|最优控制作业代写optimal control代考|Hamilton–Jacobi equations

statistics-lab™ 为您的留学生涯保驾护航 在代写最优控制optimal control方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写最优控制optimal control代写方面经验极为丰富，各种代写最优控制Soptimal control相关的作业也就用不着说。

• 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

统计代写|最优控制作业代写optimal control代考|Hamilton–Jacobi equations

In this section we introduce a partial differential equation which is solved by the value function of our variational problem. We assume throughout that hypotheses (1.9) are satisfied. We use the notation
$$u_{t}=\frac{\partial u}{\partial t}, \quad \nabla u=\left(\frac{\partial u}{\partial x_{1}}, \ldots, \frac{\partial u}{\partial x_{n}}\right)$$
Theorem 1.4.1 Let $u$ be differentiable at a point $(t, x) \in Q_{T}$. Then
$$u_{t}(t, x)+H(\nabla u(t, x))=0$$
where
$$H(p)=\sup _{q \in \mathbb{R}^{n}}[p \cdot q-L(q)] .$$
Equation (1.14) is called the Hamilton-Jacobi equation of our problem in the calculus of variations. In the terminology of control theory, such an equation is also called Bellman’s equation or dynamic programming equation. The function $H$ is called the hamiltonian. In general, a function defined as in (1.15) is called the Legendre transform of $L$ (see Appendix A.1).

统计代写|最优控制作业代写optimal control代考|Method of characteristics

We describe in this section the method of characteristics, which is a classical approach to the study of first order partial differential equations like the HamiltonJacobi equation (1.16). This method explains why such equations do not possess in general smooth solutions for all times, and has some interesting connections with the variational problem associated to the equation. A more general treatment of these topics will be given in Section 5.1.

Suppose that $H, u_{0}$ are in $C^{2}\left(\mathbb{R}^{n}\right)$, and suppose that we already know that problem (1.16) has a solution $u$ of class $C^{2}$ in some strip $Q_{T}$. For fixed $z \in \mathbb{R}^{n}$, let us denote by $X(t ; z)$ the solution of the ordinary differential equation (here the dot denotes differentiation with respect to $t$ )
$$\dot{X}=D H(\nabla u(t, X)), \quad X(0)=z$$
Such a solution is defined in some maximal interval $\left[0, T_{z}[\right.$ (although it will later turn out that $T_{z}=T$ for all $\left.z\right)$. The curve $t \rightarrow(t, X(t ; z))$ is called the characteristic curve associated with $u$ and starting from the point $(0, z)$. Let us now set
$$U(t ; z)=u(t, X(t ; z)), \quad P(t ; z)=\nabla u(t, X(t ; z)) .$$
Then, using the fact that $u$ solves problem (1.16) we find that
$$\begin{gathered} \dot{U}=u_{t}(t, X)+\nabla u(t, X) \cdot \dot{X}=-H(P)+D H(P) \cdot P \ \dot{P}=\nabla u_{t}(t, X)+\nabla^{2} u(t, X) \dot{X}=\nabla\left(u_{t}+H(\nabla u)\right)(t, X)=0 \end{gathered}$$
Therefore $P$ is constant, and so the right-hand side of (1.19) is also constant. Thus, $X$ is defined in $[0, T$ [ and we can compute explicitly $X, U, P$ obtaining
$$\left{\begin{array}{l} P(t ; z)=D u_{0}(z) \ X(t ; z)=z+t D H\left(D u_{0}(z)\right) \ U(t ; z)=u_{0}(z)+t\left[D H\left(D u_{0}(z)\right) \cdot D u_{0}(z)-H\left(D u_{0}(z)\right)\right] \end{array}\right.$$
Observe that the right-hand side of $(1.21)$ is no longer defined in terms of the solution $u$, but only depends on the initial value $u_{0}$. This suggests that, even without assuming in advance the existence of a solution, one can use these formulas to define one. As we are now going to show, such a construction can be in general carried out only locally in time.

We need the following classical result about the global invertibility of maps (see e.g., [11, Th. 3.1.8]).

统计代写|最优控制作业代写optimal control代考|Semiconcavity of Hopf’s solution

In this section we show that the semiconcavity property characterizes the value function among all possible Lipschitz continuous solutions of the Hamilton-Jacobi equation (1.16).
Theorem 1.6.1 Let $L, u_{0}$ satisfy assumptions (1.9). Suppose in addition that
(i) $L \in C^{2}\left(\mathbb{R}^{n}\right), D^{2} L(q) \leq \frac{2}{\alpha} I \quad \forall q \in \mathbb{R}^{n}$
(ii) $u_{0}(x+h)+u_{0}(x-h)-2 u_{0}(x) \leq C_{0}|h|^{2}, \quad \forall x, h \in \mathbb{R}^{n}$

for suitable constants $\alpha>0, C_{0} \geq 0$. Then there exists a constant $C_{1} \geq 0$ such that
\begin{aligned} &u(t+s, x+h)+u(t-s, x-h)-2 u(t, x) \ &\leq \frac{2 t C_{0}}{2 t+\alpha\left(t^{2}-s^{2}\right) C_{0}}\left(|h|+C_{1}|s|\right)^{2} \end{aligned}
for all $t>0, s \in]-t, t\left[, x, h \in \mathbb{R}^{n}\right.$.
Proof – For fixed $t, s, x, h$ as in the statement of the theorem, let us choose $\hat{x} \in \mathbb{R}^{n}$ such that
$$u(t, x)=t L\left(\frac{x-\hat{x}}{t}\right)+u_{0}(\hat{x}) .$$
Such a $\hat{x}$ exists by Hopf’s formula; in addition, by (1.13), there exists $C_{1}$, depending only on $L$, such that
$$\frac{|x-\hat{x}|}{t} \leq C_{1} .$$
We set, for $\lambda \geq 0$,
$$x_{\lambda}^{+}=\hat{x}+\lambda\left(h-s \frac{x-\hat{x}}{t}\right), \quad x_{\lambda}^{-}=\hat{x}-\lambda\left(h-s \frac{x-\hat{x}}{t}\right) .$$
Then we have
$$\frac{x_{\lambda}^{+}+x_{\lambda}^{-}}{2}=\hat{x}, \quad \frac{x_{\lambda}^{+}-x_{\lambda}^{-}}{2}=\lambda\left(h-s \frac{x-\hat{x}}{t}\right) .$$
By (1.29) we have
$$\frac{\left|x_{\lambda}^{+}-x_{\lambda}^{-}\right|}{2} \leq \lambda\left(|h|+C_{1}|s|\right) .$$
By Hopf’s formula (1.10) we have
$$u(t \pm s, x \pm h) \leq(t \pm s) L\left(\frac{x \pm h-x_{\lambda}^{\pm}}{t \pm s}\right)+u_{0}\left(x_{\lambda}^{\pm}\right) .$$

统计代写|最优控制作业代写optimal control代考|Hamilton–Jacobi equations

H(p)=支持q∈Rn[p⋅q−大号(q)].

统计代写|最优控制作业代写optimal control代考|Method of characteristics

X˙=DH(∇在(吨,X)),X(0)=和

$$\left{磷(吨;和)=D在0(和) X(吨;和)=和+吨DH(D在0(和)) 在(吨;和)=在0(和)+吨[DH(D在0(和))⋅D在0(和)−H(D在0(和))]\对。$$

统计代写|最优控制作业代写optimal control代考|Semiconcavity of Hopf’s solution

(i)大号∈C2(Rn),D2大号(q)≤2一种一世∀q∈Rn
(二)在0(X+H)+在0(X−H)−2在0(X)≤C0|H|2,∀X,H∈Rn

|X−X^|吨≤C1.

Xλ+=X^+λ(H−sX−X^吨),Xλ−=X^−λ(H−sX−X^吨).

Xλ++Xλ−2=X^,Xλ+−Xλ−2=λ(H−sX−X^吨).

|Xλ+−Xλ−|2≤λ(|H|+C1|s|).

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

统计代写|运筹学作业代写operational research代考|Basic PROC OPTMODEL

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

• 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

统计代写|运筹学作业代写operational research代考|Basic PROC OPTMODEL

PROC OPTMODEL is very powerful, so we can easily declare variables and parameters, define objective and constraints, and solve the problem. It also provides a full environment for programming using do-loop, if-then-else, and many other programming statements. The syntax of PROC OPTMODEL is given in Appendix 2 . Here we give some details defining a linear programming within PROC OPTMODEL.

In most cases of defining a linear programming, we need to use the following six statements:

1. number: For defining confidences
2. var: For defining variables
1. min/max: For defining an objective function
2. con: For defining a constraint
3. solve: For solving the problem using selected solver
Because in most linear programming we have a vector of variables and a matrix of coefficients, PROC OPTMODEL provides an indexing facility to handle these more efficiently. The index can be defined using integer numbers or a set of values. For example,
number c{1..4};
$\operatorname{var} x{1 \ldots 4}$;
defines four numbers that can be referred to as $c[1], c[2], c[3]$, and $c[4]$ and defines four variables that can be referred to as $x[1], x[2], x[3]$, and $x[4]$.

Using the index provides an easy environment for working with parameters. For example, the following statement finds the sum of the four parameters just mentioned:
$$s=\operatorname{sum}{i \text { in } 1 . .4} x[i] ;$$

统计代写|运筹学作业代写operational research代考|

Generally, parameters and expressions can have numerical or character values. For example, the number statement used in the previous code declares a numerical variable, while with the set statement we can define both numerical and string variables. Consider the following codes that define two sets of rows and columns and initialize the bank data with the following matrix

Using this definition enables us to refer to the elements of the bank matrix using index variables. For example Bank[“Bank2”, “Labor”] equals 50 and Bank[“Bank3”, “Capital”] equals 25,000.

An alternative initialization of the data to parameters in PROC OPTMODEL is using the “read” statement and populating parameter with the data saved in a dataset. Assume that the data in Program $1.4$ are saved in a bank dataset; the following program reads the dataset and loads it to the corresponding variables: In this code, we used a “read” statement. The first “read” loads the bank names to set row while the second “read” loads the value of capital, labor, and profit to each bank.

统计代写|运筹学作业代写operational research代考|Advanced Options in PROC OPTMODEL

As discussed earlier, PROC OPTMODEL provides a full environment for programming using do-loop, if-then-else, and many other programming statements. We can divide the syntax of PROC OPTMODEL into three types of statements:

1. Options in PROC OPTMODEL
2. Declaration of parameters and variables, as well as objective function and constraints
3. Programming statements
With the option statements, you can control how the optimization model is processed and how results are displayed. The declaration statements define the parameters, variables, constraints, and objectives that describe the model to be solved. All declarations in the PROC OPTMODEL are also saved for later use. The most popular declaration statements are:
• constraint (or con): Defines one or more constraints
• max/min: Declares an objective for the solver
• number (or num): Declares a numeric parameter
• string (or str): Declares a string parameter
• set: Declares a set type parameter
• var: Declares a variable
Parameters and variables can also be initialized using option “init.”

统计代写|运筹学作业代写operational research代考|Basic PROC OPTMODEL

PROC OPTMODEL非常强大，所以我们可以很方便的声明变量和参数，定义目标和约束，解决问题。它还为使用 do-loop、if-then-else 和许多其他编程语句进行编程提供了完整的环境。PROC OPTMODEL 的语法在附录 2 中给出。这里我们给出了在 PROC OPTMODEL 中定义线性规划的一些细节。

1. number：用于定义置信度
2. var：用于定义变量
3. 读取：用于将数据从数据集中加载到相应的参数
1. min/max：用于定义目标函数
2. con：用于定义约束
3. solve：使用选定的求解器解决问题
因为在大多数线性规划中，我们有一个变量向量和一个系数矩阵，PROC OPTMODEL 提供了一个索引工具来更有效地处理这些问题。可以使用整数或一组值来定义索引。例如，
数字 c{1..4}；
曾是⁡X1…4;
定义了四个可以称为的数字C[1],C[2],C[3]， 和C[4]并定义了四个变量，可以称为X[1],X[2],X[3]， 和X[4].

s=和⁡一世 在 1..4X[一世];

统计代写|运筹学作业代写operational research代考|

PROC OPTMODEL 中将数据初始化为参数的另一种方法是使用“读取”语句并使用数据集中保存的数据填充参数。假设 Program 中的数据1.4保存在银行数据集中；以下程序读取数据集并将其加载到相应的变量中： 在此代码中，我们使用了“读取”语句。第一个“读取”加载银行名称以设置行，而第二个“读取”将资本、劳动力和利润的价值加载到每个银行。

统计代写|运筹学作业代写operational research代考|Advanced Options in PROC OPTMODEL

1. PROC OPTMODEL 中的选项
2. 参数和变量的声明，以及目标函数和约束
3. 编程语句
使用选项语句，您可以控制优化模型的处理方式以及结果的显示方式。声明语句定义了描述要求解的模型的参数、变量、约束和目标。PROC OPTMODEL 中的所有声明也被保存以供以后使用。最流行的声明语句是：
• 约束（或con）：定义一个或多个约束
• max/min：为求解器声明一个目标
• number（或 num）：声明一个数字参数
• string（或str）：声明一个字符串参数
• set：声明一个集合类型参数
• var：声明一个变量
参数和变量也可以使用选项“init”进行初始化。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。