## 数学代写|线性规划作业代写Linear Programming代考|MAT3350

statistics-lab™ 为您的留学生涯保驾护航 在代写线性规划Linear Programming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。我们的专家在代写线性规划Linear Programming代写方面经验极为丰富，各种代写线性规划Linear Programming相关的作业也就用不着说。

## 数学代写|线性规划作业代写Linear Programming代考|Symmetric Dual Problems

A symmetric relation between a primal and its dual problem exists.
Consider a L.P. problem
Find $x_1, x_2, \ldots, x_n$, which
\begin{aligned} & \text { Max. } Z_p=c_1 x_1+c_2 x_2+\ldots+c_n x_n \ & \text { s.t. } \quad a_{11} x_1+a_{12} x_2+\ldots+a_{1 n} x_n \leq b_1 \ & a_{21} x_2+a_{22} x_2+\ldots+a_{2 n} x_n \leq b_2 \ & \left.\begin{array}{cccc} \ldots & \ldots & \ldots & \ldots \ \ldots & \ldots & \ldots & \ldots \ a_{m 1} x_1+a_{m 2} x_2+\ldots+a_{m n} x_n \leq b_m \end{array}\right] \ & x_1, x_2, \ldots \ldots, x_n \geq 0 \ & \end{aligned}
and where the signs of all parameters $a, b$ and $c$ ‘s are arbitrary.
The dual problem of the above L.P. problem is obtained by
(i) Transposing the coefficient matrix.
(ii) Interchanging the role of constant terms and the coefficients of the objective function.
(iii) Reverting the inequalities and
(iv) Minimizing the objective function instead of maximizing it.
The dual problem is as follows :
Find $w_1, w_2, \ldots, w_m$, for which

\begin{aligned}
& \text { and } \
& w_1, w_2, . ., w_m \geq 0 . \
&
\end{aligned}

## 数学代写|线性规划作业代写Linear Programming代考|Standard form of the primal

A L.P. problem is said to be in standard primal form if (i)All the constraints involve the sign $\leq$ if it is a problem of maximization.
or (ii)All the constraints involve the sign $\geq$ if it is a problem of minimization.
86.6. Theorem. Dual of the dual of a given primal, is the primal itself.
[Meerut 95 (BP), 98 (Old); Raj. 85]
Proof. Consider the L.P. problem
Primal. Max. $Z_p=c_1 x_1+c_2 x_2+\ldots+c_n x_n$
\begin{aligned} & \text { s.t. } \quad a_{11} x_1+a_{12} x_2+\ldots+a_{1 n} x_n \leq b_1 \ & a_{21} x_1+a_{22} x_2+\ldots+a_{2 n} x_n \leq b_2 \ & \begin{array}{llllll} \cdots & \cdots & \cdots & \cdots & \cdots \ \cdots & \cdots & \cdots & \cdots & \cdots \end{array} \ & a_{m 1} x_1+a_{m 2} x_2+\ldots+a_{m n} x_n \leq b_m \ & \end{aligned}
and
$$x_1, x_2, \ldots, x_n \geq 0 \text {. }$$
Dual. The dual of the above primal (1) is given by

\begin{aligned}
& \text { and } \quad w_1, w_2, \ldots, w_m \geq 0 \text {. } \
&
\end{aligned}

# 线性规划代写

## 数学代写|线性规划作业代写Linear Programming代考|Symmetric Dual Problems

\begin{aligned} & \text { Max. } Z_p=c_1 x_1+c_2 x_2+\ldots+c_n x_n \ & \text { s.t. } \quad a_{11} x_1+a_{12} x_2+\ldots+a_{1 n} x_n \leq b_1 \ & a_{21} x_2+a_{22} x_2+\ldots+a_{2 n} x_n \leq b_2 \ & \left.\begin{array}{cccc} \ldots & \ldots & \ldots & \ldots \ \ldots & \ldots & \ldots & \ldots \ a_{m 1} x_1+a_{m 2} x_2+\ldots+a_{m n} x_n \leq b_m \end{array}\right] \ & x_1, x_2, \ldots \ldots, x_n \geq 0 \ & \end{aligned}

(i)转置系数矩阵。
(ii)交换常数项和目标函数系数的作用。

(iv)使目标函数最小化，而不是使其最大化。

\begin{aligned}
＆ \text和{}\ ＆ w＿1, w＿2， . .， w＿m \geq 0。\ ＆
\end{aligned}

## 数学代写|线性规划作业代写Linear Programming代考|Standard form of the primal

86.6. 定理。给定原物的对偶的对偶，就是原物本身。
[Meerut 95 (BP)， 98 (Old)];[85]

\begin{aligned} & \text { s.t. } \quad a_{11} x_1+a_{12} x_2+\ldots+a_{1 n} x_n \leq b_1 \ & a_{21} x_1+a_{22} x_2+\ldots+a_{2 n} x_n \leq b_2 \ & \begin{array}{llllll} \cdots & \cdots & \cdots & \cdots & \cdots \ \cdots & \cdots & \cdots & \cdots & \cdots \end{array} \ & a_{m 1} x_1+a_{m 2} x_2+\ldots+a_{m n} x_n \leq b_m \ & \end{aligned}

$$x_1, x_2, \ldots, x_n \geq 0 \text {. }$$

\begin{aligned}
＆ \text和{w＿1, w＿2}, \quad\ldots, w＿m \geq 0 \text。{}\ ＆
\end{aligned}

## 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。

## 数学代写|线性规划作业代写Linear Programming代考|MAT3350

statistics-lab™ 为您的留学生涯保驾护航 在代写线性规划Linear Programming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。我们的专家在代写线性规划Linear Programming代写方面经验极为丰富，各种代写线性规划Linear Programming相关的作业也就用不着说。

## 数学代写|线性规划作业代写Linear Programming代考|Networks

A network consists of two types of objects: nodes and arcs. We shall let $\mathcal{N}$ denote the set of nodes. We let $m$ denote the number of nodes (i.e., the cardinality of the set $\mathcal{N})$.

The nodes are connected by arcs. Arcs are assumed to be directed. This means that an arc connecting node $i$ to node $j$ is not the same as an arc connecting node $j$ to node $i$. For this reason, we denote arcs using the standard mathematical notation for ordered pairs. That is, the arc connecting node $i$ to node $j$ is denoted simply as $(i, j)$. We let $\mathcal{A}$ denote the set of all arcs in the network. This set is a subset of the set of all possible arcs:
$$\mathcal{A} \subset{(i, j): i, j \in \mathcal{N}, i \neq j} .$$
In typical networks, the set $\mathcal{A}$ is much smaller than the set of all arcs. In fact, usually each node is only connected to a handful of “nearby” nodes.

The pair $(\mathcal{N}, \mathcal{A})$ is called a network. It is also sometimes called a graph or a digraph (to emphasize the fact that the arcs are directed). Figure 13.1 shows a network having 7 nodes and 14 arcs.

To specify a network flow problem, we need to indicate the supply of (or demand for) material at each node. So, for each $i \in \mathcal{N}$, let $b_i$ denote the amount of material being supplied to the network at node $i$. We shall use the convention that negative supplies are in fact demands. Hence, our problem will be to move the material that sits at the supply nodes over to the demand nodes. The movements must be along the arcs of the network (and adhering to the directions of the arcs). Since, except for the supply and demand, there is no other way for material to enter or leave the system, it follows that the total supply must equal the total demand for the problem to have a feasible solution. Hence, we shall always assume that
$$\sum_{i \in \mathcal{N}} b_i=0$$

## 数学代写|线性规划作业代写Linear Programming代考|Spanning Trees and Bases

Network flow problems can be solved efficiently because the basis matrices have a special structure that can be described nicely in terms of the network. In order to explain this structure, we need to introduce a number of definitions.

First of all, an ordered list of nodes $\left(n_1, n_2, \ldots, n_k\right)$ is called a path in the network if each adjacent pair of nodes in the list is connected by an arc in the network. It is important to note that we do not assume that the arcs point in any particular direction. For example, for nodes $n_i$ and $n_{i+1}$, there must be an arc in the network. It could run either from $n_i$ to $n_{i+1}$ or from $n_{i+1}$ to $n_i$. (One should think about one-way roads – even though cars can only go one way, pedestrians are allowed to walk along the path of the road in either direction.) A network is called connected if there is a path connecting every pair of nodes (see Figure 13.3). For the remainder of this chapter, we make the following assumption:
Assumption. The network is connected.
For any $\operatorname{arc}(i, j)$, we refer to $i$ as its tail and $j$ as its head.
A cycle is a path in which the last node coincides with the first node. A network is called acyclic if it does not contain any cycles (see Figure 13.4).

A network is a tree if it is connected and acyclic (see Figure 13.5). A network $(\tilde{\mathcal{N}}, \tilde{\mathcal{A}})$ is called a subnetwork of $(\mathcal{N}, \mathcal{A})$ if $\tilde{\mathcal{N}} \subset \mathcal{N}$ and $\tilde{\mathcal{A}} \subset \mathcal{A}$. A subnetwork $(\tilde{\mathcal{N}}, \tilde{\mathcal{A}})$ is a spanning tree if it is a tree and $\tilde{\mathcal{N}}=\mathcal{N}$. Since a spanning tree’s node set coincides with the node set of the underlying network, it suffices to refer to a spanning tree by simply giving its arc set.

# 线性规划代写

## 数学代写|线性规划作业代写Linear Programming代考|Networks

$$\mathcal{A} \subset{(i, j): i, j \in \mathcal{N}, i \neq j} .$$

$$\sum_{i \in \mathcal{N}} b_i=0$$

## 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。

## 数学代写|线性规划作业代写Linear Programming代考|IMSE881

statistics-lab™ 为您的留学生涯保驾护航 在代写线性规划Linear Programming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。我们的专家在代写线性规划Linear Programming代写方面经验极为丰富，各种代写线性规划Linear Programming相关的作业也就用不着说。

## 数学代写|线性规划作业代写Linear Programming代考|Revised Simplex Method in Standard Form I.

In revised simplex method, the objective function is treated as if it were another constraint. Whereas in the simplex method we deal with an $m$-dimensional basis. Here in revised simplex method we would deal with $(m+1)$ dimensional basis in standard form I and with a $(m+2)$ dimensional basis in standard form II.
The L.P. P. in its standard form is
Max. $Z=c_1 x_1+c_2 x_2+\ldots+c_n x_n$
\begin{aligned} & \text { s.t. } \quad a_{11} x_1+a_{12} x_2+\ldots+a_{1 n} x_n \leq b_1 \ & a_{21} x_1+a_{22} x_2+\ldots+a_{2 n} x_n \leq b_2 \ & \begin{array}{lllll} \ldots & \ldots & \ldots & \ldots & \ldots \end{array} \ & \begin{array}{lllll} \cdots & \cdots & \cdots & \cdots & \cdots \end{array} \ & a_{m 1} x_1+a_{m 2} x_2+\ldots+a_{m n} x_n \leq b_m \text {. } \ & x_i \geq 0, \quad i=1,2, \ldots, n \text {. } \ & \end{aligned}
and

Considering the objective function as an additional constraint in which $Z$ is as large as possible and unrestricted in sign and introducing the slack and surplus variables, we get the following $(m+1)$ constraints.

## 数学代写|线性规划作业代写Linear Programming代考|To find the inverse of the Basis (i.e. $B_1{ }^{-1}$ ) and the Basic solution in the standard form $I$.

(i) To Find $B_1{ }^{-1}$. From $\S 5 \cdot 3$, we have
$$B_1=\left[\begin{array}{cc} 1 & -C_B \ 0 & B \end{array}\right]$$
Since $B_1^{-1}$ exists and is known, therefore using $\S 0 \cdot 13$, the inverse of the matrix $B$, is given by
$$B_1^{-1}=\left[\begin{array}{cc} 1 & C_B B^{-1} \ 0 & B^{-1} \end{array}\right]\left[\begin{array}{l} \text { comparingwith } \S 0.13, \text { we } \ \text { see that here } \ I=1, R=B \text { and } Q=-C_B \end{array}\right]$$
Note. We have seen that we always start with $B=I_m(m \times m$ identity matrix)
$$\therefore B^{-1}=I_m^{-1}=I_m . \quad B_1^{-1}=\left[\begin{array}{cc} 1 & C_B \cdot I_m \ 0 & I_m \end{array}\right]=\left[\begin{array}{ll} 1 & C_B \ 0 & I_m \end{array}\right] .$$
Also if after ensuring that all $b_i \geq 0$, only the slack variables

\begin{aligned} & \text { are } \quad \text { added } \quad \text { and } \quad B=I_m, \quad \text { then } \ & C_B=\left(c_{B 1}, c_{B 2}, \ldots, c_{B m}\right)=(0,0, \ldots, 0)=0 \ & \text { then } \quad B_1^{-1}=\left[\begin{array}{ll} 1 & 0 \ 0 & I \end{array}\right]=I_{m+1} . \end{aligned}
(ii) To find $\alpha_1^{(1)}$ not in the basis matrix $B_1$.

# 线性规划代写

## 数学代写|线性规划作业代写Linear Programming代考|Revised Simplex Method in Standard Form I.

\begin{aligned} & \text { s.t. } \quad a_{11} x_1+a_{12} x_2+\ldots+a_{1 n} x_n \leq b_1 \ & a_{21} x_1+a_{22} x_2+\ldots+a_{2 n} x_n \leq b_2 \ & \begin{array}{lllll} \ldots & \ldots & \ldots & \ldots & \ldots \end{array} \ & \begin{array}{lllll} \cdots & \cdots & \cdots & \cdots & \cdots \end{array} \ & a_{m 1} x_1+a_{m 2} x_2+\ldots+a_{m n} x_n \leq b_m \text {. } \ & x_i \geq 0, \quad i=1,2, \ldots, n \text {. } \ & \end{aligned}

## 数学代写|线性规划作业代写Linear Programming代考|To find the inverse of the Basis (i.e. $B_1{ }^{-1}$ ) and the Basic solution in the standard form $I$.

(i)找到$B_1{ }^{-1}$。从$\S 5 \cdot 3$，我们有
$$B_1=\left[\begin{array}{cc} 1 & -C_B \ 0 & B \end{array}\right]$$

$$B_1^{-1}=\left[\begin{array}{cc} 1 & C_B B^{-1} \ 0 & B^{-1} \end{array}\right]\left[\begin{array}{l} \text { comparingwith } \S 0.13, \text { we } \ \text { see that here } \ I=1, R=B \text { and } Q=-C_B \end{array}\right]$$

$$\therefore B^{-1}=I_m^{-1}=I_m . \quad B_1^{-1}=\left[\begin{array}{cc} 1 & C_B \cdot I_m \ 0 & I_m \end{array}\right]=\left[\begin{array}{ll} 1 & C_B \ 0 & I_m \end{array}\right] .$$

\begin{aligned} & \text { are } \quad \text { added } \quad \text { and } \quad B=I_m, \quad \text { then } \ & C_B=\left(c_{B 1}, c_{B 2}, \ldots, c_{B m}\right)=(0,0, \ldots, 0)=0 \ & \text { then } \quad B_1^{-1}=\left[\begin{array}{ll} 1 & 0 \ 0 & I \end{array}\right]=I_{m+1} . \end{aligned}
(ii)求出不在基矩阵$B_1$中的$\alpha_1^{(1)}$。

## 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。

## 数学代写|线性规划作业代写Linear Programming代考|ISE505

statistics-lab™ 为您的留学生涯保驾护航 在代写线性规划Linear Programming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。我们的专家在代写线性规划Linear Programming代写方面经验极为丰富，各种代写线性规划Linear Programming相关的作业也就用不着说。

## 数学代写|线性规划作业代写Linear Programming代考|Alternative Optimal Solutions

An alternative optimal solution of a L.P.P. is said to exist if the set of variables giving the optimal value of the objective function is not unique.
Conditions for alternative solutions.
Let there be an optimal B.F.S. of a L.P.P. then

1. If for some $\alpha_j$ in $A$ but not in $B, c_j-Z_j=0 ; y_{i j} \leq 0$ for all $i=1,2, \ldots, m$, then a non-basic alternative optima will exist.
2. If for some $\alpha_j$ in $A$ but not in $B, c_j-Z_j=0$ and $y_{i j}>0$ for at least one $i$, then an alternative basic optima will exist.

Proof. 1. We have shown in $\S 3.6$ page 67 , that if we insert (introduce) the column vector $\alpha_j$ in $B$, where $\alpha_j$ is in $A$ but not in $B$ and $y_{i j} \leq 0$ for all $i=1,2, \ldots, m$ then we obtain a non-basic feasible solution $x_B^{\prime}$ with $(m+1)$ number of positive variables given by
.” where
\begin{aligned} & \mathbf{x}B^{\prime}=\left[x{B 1}, x_{B 2}^{\prime}, \ldots, x_{B m}^{\prime}, \lambda\right] \ & x_{B i}^{\prime}=x_{B i}-\lambda y_{i j}, i=1,2, \ldots, m ; \lambda>0 \end{aligned}

and the value of the objective function for this new F.S. is given by
\begin{aligned} & Z^{\prime}=Z+\lambda\left(c_j-Z_j\right) . \ & c_j-Z_j=0, \text { then } Z^{\prime}=Z \end{aligned}
i.e., the value of the objective function for this new non-basic
-F.S. is also equal to $Z^{\prime}$ (optimal value). Hence this new non-basic F.S. is an alternative optimal solution of the given L.P.P.

We have shown in theorem of $\S 3.5$ page 62 that if $y_{i j}>0$ for at least one $i=1,2, \ldots, m$ then by replacing one column $\beta_r$ in $B$ by the column $\alpha_j$ which is in $A$ but not in $B$, we obtain a new B.F.S. $\mathbf{x}B^{\prime}$ given by where $$\mathbf{x}_B^{\prime}=\left[x{B 1}^{\prime}, x_{B 2}^{\prime}, \ldots, x_{B m}^{\prime}\right]$$
and
$$x_{B i}^{\prime}=x_{B i}-\frac{y_{i j}}{y_{r j}} x_{B i}, i=1,2, \ldots, m, i \neq r$$
\begin{aligned} & x_{B r}^{\prime}=\frac{x_{B r}}{y_{r j}} \ & \frac{x_{B r}}{y_{i j}}=\underset{i}{\operatorname{Mini}}\left(\frac{x_{B i}}{y_{i j}}, y_{i j}>0\right) . \end{aligned}
The value of objective function for this new B.F.S. is given by
$$Z^{\prime}=Z+\frac{x_{B r}}{y_{r j}}\left(c_j-Z_j\right)$$
i.e. the value of the objective function for this new B.F.S. is also equal to $Z^{\prime}$ (optimal value). Hence this new B.F.S. is an alternative optimal B.F.S.

## 数学代写|线性规划作业代写Linear Programming代考|Inconsistency and Redundancy in Constraint Equations

By redundancy in constraint equations we mean that the system has more than enough number of constraint equations in other words it has more constraint equations than the number of variables.
This is the situation when $r(A)=r(A b)=k \leq n<m$.
(see $\S 0 \cdot 21$, case $I$ )
In this case there will be $(m-k)$ redundant equations.
Inconsistency. As aiready defind, the set of constraints (linear equations) is said to be inconsistent if $r(A) \neq r(A b)$.

Before solving a L.P. problem by simplex method, we should have $r(A)=r(A b)$ i.e. the constraint equations (after introducing the slack and artificial variables) should be consistent. Since in simplex method we always have $r(A)=r(A b)=m$.

If the system $A x=b$ involves artificial variables. Then we can not say whether this system is consistent of there is any redundancy. Below we give the cases (without proof) to decide about the consistency and redundancy is such systems.

Case I. If the basis $B$ contains no artificial vector and the optimality condition is satisfied (at any iteration) then the current solution is a B.F.S. of the problem.

Case II. If one or more artificial vector appears in the basis $B$ at a zero level i.e. the value of the artificial variables corresponding to artificial vectors in $B$ are zero and the optimality condition is satisfied (at any iteration) then the system is consistent. Further more if $y_{i j}=0 . \forall j$ and $x_{B r}=0$ and $r$ corresponds to the row containing an artificial vector, then the $r$ th constraint equation is redundant.
Case III. If at least one artificial vector appears in the basis $B$ at a positive level i.e. the value of at least one artificial variable corresponding to artificial vector in $B$ is non-zero and the optimality condition is satisfied (at any iteration), then there exists no feasible solution of the problem.

# 线性规划代写

## 数学代写|线性规划作业代写Linear Programming代考|Alternative Optimal Solutions

1.证明;我们在 $\S 3.6$ 第67页，如果我们插入(引入)列向量 $\alpha_j$ 在 $B$，其中 $\alpha_j$ 是在 $A$ 但不是 $B$ 和 $y_{i j} \leq 0$ 对所有人 $i=1,2, \ldots, m$ 得到了一个非基本可行解 $x_B^{\prime}$ 有 $(m+1)$ 给出的正变量数

\begin{aligned} & \mathbf{x}B^{\prime}=\left[x{B 1}, x_{B 2}^{\prime}, \ldots, x_{B m}^{\prime}, \lambda\right] \ & x_{B i}^{\prime}=x_{B i}-\lambda y_{i j}, i=1,2, \ldots, m ; \lambda>0 \end{aligned}

\begin{aligned} & Z^{\prime}=Z+\lambda\left(c_j-Z_j\right) . \ & c_j-Z_j=0, \text { then } Z^{\prime}=Z \end{aligned}

-F.S.也等于$Z^{\prime}$(最优值)。因此，这种新的非基本fss是给定lpp的备选最优解

$$x_{B i}^{\prime}=x_{B i}-\frac{y_{i j}}{y_{r j}} x_{B i}, i=1,2, \ldots, m, i \neq r$$
\begin{aligned} & x_{B r}^{\prime}=\frac{x_{B r}}{y_{r j}} \ & \frac{x_{B r}}{y_{i j}}=\underset{i}{\operatorname{Mini}}\left(\frac{x_{B i}}{y_{i j}}, y_{i j}>0\right) . \end{aligned}

$$Z^{\prime}=Z+\frac{x_{B r}}{y_{r j}}\left(c_j-Z_j\right)$$

## 数学代写|线性规划作业代写Linear Programming代考|Inconsistency and Redundancy in Constraint Equations

(见$\S 0 \cdot 21$，案例$I$)

## 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。

## 数学代写|线性规划作业代写Linear Programming代考|MAT3100

statistics-lab™ 为您的留学生涯保驾护航 在代写线性规划Linear Programming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。我们的专家在代写线性规划Linear Programming代写方面经验极为丰富，各种代写线性规划Linear Programming相关的作业也就用不着说。

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

Having reduced the computation of the optimal strategies $x^$ and $y^$ to the solution of linear programs, it is now a simple matter to show that they are consistent with each other. The next theorem, which establishes this consistency, is called the Minimax Theorem:
THEOREM 11.1. There exist stochastic vectors $x^$ and $y^$ for which
$$\max _x y^{* T} A x=\min _y y^T A x^* .$$
PROOF. The proof follows trivially from the observation that (11.4) is the dual of (11.3). Therefore, $v^=u^$. Furthermore,
$$v^=\min _i e_i^T A x^=\min _y y^T A x^*$$
and similarly,
$$u^=\max _j e_j^T A^T y^=\max _x x^T A^T y^=\max _x y^{ T} A x .$$

## 数学代写|线性规划作业代写Linear Programming代考|Poker

Some card games such as poker involve a round of bidding in which the players at times bluff by increasing their bid in an attempt to coerce their opponents into backing down, even though if the challenge is accepted they will surely lose. Similarly, they will sometimes underbid to give their opponents false hope. In this section, we shall study a simplified version of poker (the real game is too hard to analyze) to see if bluffing and underbidding are justified bidding strategies.

Simplified poker involves two players, A and B, and a deck having three cards, 1, 2 , and 3 . At the beginning of a round, each player “antes up” $\$ 1$and is dealt one card from the deck. A bidding session follows in which each player in turn, starting with A, either (a) bets and adds$\$1$ to the “kitty” or (b) passes. Bidding terminates when
a bet is followed by a bet,
a pass is followed by a pass, or
a bet is followed by a pass.
In the first two cases, the winner of the round is decided by comparing cards, and the kitty goes to the player with the higher card. In the third case, bet followed by pass, the player who bet wins the round independently of who had the higher card (in real poker, the player who passes is said to fold).

With these simplified betting rules, there are only five possible betting scenarios:
\begin{tabular}{|c|c|c|c|}
\hline A passes, & B passes: & & $\$ 1$to holder of higher card \ \hline A passes, & B bets, & A passes: &$\$1$ to $B$ \
\hline A passes, & B bets, & A bets: & $\$ 2$to holder of higher card \ \hline A bets, & B passes: & &$\$1$ to $\mathrm{A}$ \
\end{tabular}

## 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。

## 数学代写|线性规划作业代写Linear Programming代考|INE701

statistics-lab™ 为您的留学生涯保驾护航 在代写线性规划Linear Programming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。我们的专家在代写线性规划Linear Programming代写方面经验极为丰富，各种代写线性规划Linear Programming相关的作业也就用不着说。

## 数学代写|线性规划作业代写Linear Programming代考|Convex Sets

Given a finite set of points, $z_1, z_2, \ldots, z_n$, in $\mathbb{R}^m$, a point $z$ in $\mathbb{R}^m$ is called a convex combination of these points if ${ }^{\prime}$
$$z=\sum_{j=1}^n t_j z_j,$$
where $t_j \geq 0$ for each $j$ and $\sum_{j=1}^n t_j=1$. It is called a strict convex combination if none of the $t_j$ ‘s vanish. For $n=2$, the set of all convex combinations of two points is simply the line segment connecting them.

A subset $S$ of $\mathbb{R}^m$ is called convex if, for every $x$ and $y$ in $S, S$ also contains all points on the line segment connecting $x$ and $y$. That is, $t x+(1-t) y \in S$, for every $0<t<1$. See Figure 10.1 .

Certain elementary properties of convex sets are trivial to prove. For example, the intersection of an arbitrary collection of convex sets is convex. Indeed, let $S_\alpha$, $\alpha \in I$, denote a collection of convex sets indexed by some set $I$. Then the claim is that $\cap_{\alpha \in I} S_\alpha$ is convex. To see this, consider an arbitrary pair of points $x$ and $y$ in the intersection. It follows that $x$ and $y$ are in each $S_\alpha$. By the convexity of $S_\alpha$ it follows that $S_\alpha$ contains the line segment connecting $x$ and $y$. Since each of these sets contains the line segment, so does their intersection. Hence, the intersection is convex.
Here is another easy one:
THEOREM 10.1. A set $C$ is convex if and only if it contains all convex combinations of points in $C$.

Proof. Let $C$ be a convex set. By definition, $C$ contains all convex combinations of pairs of points in $C$. The first nontrivial step is to show that $C$ contains all convex combinations of triples of points in $C$. To see this, fix $z_1, z_2$, and $z_3$ in $C$ and consider
$$z=t_1 z_1+t_2 z_2+t_3 z_3$$
where $t_j \geq 0$ for each $j$ and $\sum_{j=1}^3 t_j=1$. If any of the $t_j$ ‘s vanish, then $z$ is really just a convex combination of two points and so belongs to $C$. Hence, suppose that each of the $t_j$ ‘s is strictly positive. Rewrite $z$ as follows:
\begin{aligned} z & =\left(1-t_3\right)\left(\frac{t_1}{1-t_3} z_1+\frac{t_2}{1-t_3} z_2\right)+t_3 z_3 \ & =\left(1-t_3\right)\left(\frac{t_1}{t_1+t_2} z_1+\frac{t_2}{t_1+t_2} z_2\right)+t_3 z_3 . \end{aligned}

## 数学代写|线性规划作业代写Linear Programming代考|Carathéodory’s Theorem

In the previous section, we showed that the convex hull of a set $S$ can be constructed by forming all convex combinations of finite sets of points from $S$. In 1907,Carathéodory showed that it is not necessary to use all finite sets. Instead, $m+1$ points suffice:

THEOREM 10.3. The convex hull $\operatorname{conv}(S)$ of a set $S$ in $\mathbb{R}^m$ consists of all convex combinations of $m+1$ points from $S$ :
$$\operatorname{conv}(S)=\left{z=\sum_{j=1}^{m+1} t_j z_j: z_j \in S \text { and } t_j \geq 0 \text { for all } j \text {, and } \sum_j t_j=1\right} \text {. }$$
Proof. Let $H$ denote the set on the right. From Theorem 10.2, we see that $H$ is contained in $\operatorname{conv}(S)$. Therefore, it suffices to show that every point in $\operatorname{conv}(S)$ belongs to $H$. To this end, fix a point $z$ in $\operatorname{conv}(S)$. By Theorem 10.2, there exists a collection of, say, $n$ points $z_1, z_2, \ldots, z_n$ in $S$ and associated nonnegative multipliers $t_1, t_2, \ldots, t_n$ summing to one such that
$$z=\sum_{j=1}^n t_j z_j$$
Let $A$ denote the matrix consisting of the points $z_1, z_2, \ldots, z_n$ as the columns of $A$ :
$$A=\left[\begin{array}{llll} z_1 & z_2 & \cdots & z_n \end{array}\right] .$$
Also, let $x^$ denote the vector consisting of the multipliers $t_1, t_2, \ldots, t_n$ : $$x^=\left[\begin{array}{c} t_1 \ t_2 \ \vdots \ t_n \end{array}\right]$$

# 线性规划代写

## 数学代写|线性规划作业代写Linear Programming代考|Convex Sets

$$z=\sum_{j=1}^n t_j z_j,$$

$\mathbb{R}^m$的子集$S$称为凸if，因为$S, S$中的每个$x$和$y$也包含连接$x$和$y$的线段上的所有点。也就是$t x+(1-t) y \in S$对应于$0<t<1$。如图10.1所示。

$$z=t_1 z_1+t_2 z_2+t_3 z_3$$

\begin{aligned} z & =\left(1-t_3\right)\left(\frac{t_1}{1-t_3} z_1+\frac{t_2}{1-t_3} z_2\right)+t_3 z_3 \ & =\left(1-t_3\right)\left(\frac{t_1}{t_1+t_2} z_1+\frac{t_2}{t_1+t_2} z_2\right)+t_3 z_3 . \end{aligned}

## 数学代写|线性规划作业代写Linear Programming代考|Carathéodory’s Theorem

$$\operatorname{conv}(S)=\left{z=\sum_{j=1}^{m+1} t_j z_j: z_j \in S \text { and } t_j \geq 0 \text { for all } j \text {, and } \sum_j t_j=1\right} \text {. }$$

$$z=\sum_{j=1}^n t_j z_j$$

$$A=\left[\begin{array}{llll} z_1 & z_2 & \cdots & z_n \end{array}\right] .$$

## 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。

## 数学代写|线性规划作业代写Linear Programming代考|Math269

statistics-lab™ 为您的留学生涯保驾护航 在代写线性规划Linear Programming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。我们的专家在代写线性规划Linear Programming代写方面经验极为丰富，各种代写线性规划Linear Programming相关的作业也就用不着说。

## 数学代写|线性规划作业代写Linear Programming代考|Sensitivity Analysis

One often needs to solve not just one linear programming problem but several closely related problems. There are many reasons that this need might arise. For example, the data that define the problem may have been rather uncertain and one may wish to consider various possible data scenarios. Or perhaps the data are known accurately but change from day to day, and the problem must be resolved for each new day. Whatever the reason, this situation is quite common. So one is led to ask whether it is possible to exploit the knowledge of a previously obtained optimal solution to obtain more quickly the optimal solution to the problem at hand. Of course, the answer is often yes, and this is the subject of this section.

We shall treat a number of possible situations. All of them assume that a problem has been solved to optimality. This means that we have at our disposal the final, optimal dictionary:
\begin{aligned} \zeta & =\zeta^-z_{\mathcal{N}}^{ T} x_{\mathcal{N}} \ x_{\mathcal{B}} & =x_{\mathcal{B}}^*-B^{-1} N x_{\mathcal{N}} . \end{aligned}

Suppose we wish to change the objective coefficients from $c$ to, say, $\tilde{c}$. It is natural to ask how the dictionary at hand could be adjusted to become a valid dictionary for the new problem. That is, we want to maintain the current classification of the variables into basic and nonbasic variables and simply adjust $\zeta^, z_{\mathcal{N}}^$, and $x_{\mathcal{B}}^$ appropriately. Recall from (6.7), (6.8), and (6.9) that \begin{aligned} x_{\mathcal{B}}^ & =B^{-1} b, \ z_{\mathcal{N}}^* & =\left(B^{-1} N\right)^T c_{\mathcal{B}}-c_{\mathcal{N}}, \ \zeta^* & =c_{\mathcal{B}}^T B^{-1} b . \end{aligned}
Hence, the change from $c$ to $\tilde{c}$ requires us to recompute $z_{\mathcal{N}}^$ and $\zeta^$, but $x_{\mathcal{B}}^$ remains unchanged. Therefore, after recomputing $z_{\mathcal{N}}^$ and $\zeta^*$, the new dictionary is still primal feasible, and so there is no need for a Phase I procedure: we can jump straight into the primal simplex method, and if $\tilde{c}$ is not too different from $c$, we can expect to get to the new optimal solution in a relatively small number of steps.

## 数学代写|线性规划作业代写Linear Programming代考|Parametric Analysis and the Homotopy Method

In this section, we illustrate the notion of parametric analysis by applying a technique called the homotopy method to get a new algorithm for solving linear programming problems. The homotopy method is a general technique in which one creates a continuous deformation that changes a given difficult problem into a related but trivially solved problem and then attempts to work backwards from the trivial problem to the difficult problem by solving (hopefully without too much effort) all the problems in between. Of course, there is a continuum of problems between the hard one and the trivial one, and so we shouldn’t expect that this technique will be effective in every situation; but for linear programming and for many other problem domains, it turns out to yield efficient algorithms.

We start with an example. Suppose we wish to solve the following linear programming problem:
\begin{aligned} \operatorname{maximize}-2 x_1+3 x_2 & \ \text { subject to } \quad-x_1+x_2 & \leq-1 \ -x_1-2 x_2 & \leq-2 \ x_2 & \leq 1 \ x_1, x_2 & \geq 0 . \end{aligned}

The starting dictionary is
\begin{aligned} \zeta & =-2 x_1-(-3) x_2 \ \hline x_3 & =-1+x_1-x_2 \ x_4 & =-2+x_1+\quad 2 x_2 \ x_5 & =1 \quad-\quad x_2 . \end{aligned}
This dictionary is neither primal nor dual feasible. Let’s perturb it by adding a positive real number $\mu$ to each right-hand side and subtracting it from each objective function coefficient. We now arrive at a family of dictionaries, parametrized by $\mu$ :
\begin{aligned} & \frac{\zeta=\quad-(2+\mu) x_1-(-3+\mu) x_2}{x_3=-1+\mu+\quad x_1-\quad x_2} \ & x_4=-2+\mu+\quad x_1+\quad 2 x_2 \ & x_5=1+\mu \quad-\quad x_2 \text {. } \ & \end{aligned}

# 线性规划代写

## 数学代写|线性规划作业代写Linear Programming代考|Sensitivity Analysis

\begin{aligned} \zeta & =\zeta^-z_{\mathcal{N}}^{ T} x_{\mathcal{N}} \ x_{\mathcal{B}} & =x_{\mathcal{B}}^*-B^{-1} N x_{\mathcal{N}} . \end{aligned}

## 数学代写|线性规划作业代写Linear Programming代考|Parametric Analysis and the Homotopy Method

\begin{aligned} \operatorname{maximize}-2 x_1+3 x_2 & \ \text { subject to } \quad-x_1+x_2 & \leq-1 \ -x_1-2 x_2 & \leq-2 \ x_2 & \leq 1 \ x_1, x_2 & \geq 0 . \end{aligned}

\begin{aligned} \zeta & =-2 x_1-(-3) x_2 \ \hline x_3 & =-1+x_1-x_2 \ x_4 & =-2+x_1+\quad 2 x_2 \ x_5 & =1 \quad-\quad x_2 . \end{aligned}

\begin{aligned} & \frac{\zeta=\quad-(2+\mu) x_1-(-3+\mu) x_2}{x_3=-1+\mu+\quad x_1-\quad x_2} \ & x_4=-2+\mu+\quad x_1+\quad 2 x_2 \ & x_5=1+\mu \quad-\quad x_2 \text {. } \ & \end{aligned}

## 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。

## 数学代写|线性规划作业代写Linear Programming代考|Complementary Slackness

statistics-lab™ 为您的留学生涯保驾护航 在代写线性规划Linear Programming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。我们的专家在代写线性规划Linear Programming代写方面经验极为丰富，各种代写线性规划Linear Programming相关的作业也就用不着说。

## 数学代写|线性规划作业代写Linear Programming代考|Complementary Slackness

Sometimes it is necessary to recover an optimal dual solution when only an optimal primal solution is known. The following theorem, known as the Complementary Slackness Theorem, can help in this regard.
THEOREM 5.3. Suppose that $x=\left(x_1, x_2, \ldots, x_n\right)$ is primal feasible and that $y=\left(y_1, y_2, \ldots, y_m\right)$ is dual feasible. Let $\left(w_1, w_2, \ldots, w_m\right)$ denote the corresponding primal slack variables, and let $\left(z_1, z_2, \ldots, z_n\right)$ denote the corresponding dual slack variables. Then $x$ and $y$ are optimal for their respective problems if and only if
$$\begin{array}{ll} x_j z_j=0, & \text { for } j=1,2, \ldots, n, \ w_i y_i=0, & \text { for } i=1,2, \ldots, m . \end{array}$$
Proof. We begin by revisiting the chain of inequalities used to prove the weak duality theorem:
\begin{aligned} \sum_j c_j x_j & \leq \sum_j\left(\sum_i y_i a_{i j}\right) x_j \ & =\sum_i\left(\sum_j a_{i j} x_j\right) y_i \ & \leq \sum_i b_i y_i . \end{aligned}
Recall that the first inequality arises from the fact that each term in the left-hand sum is dominated by the corresponding term in the right-hand sum. Furthermore, this domination is a consequence of the fact that each $x_j$ is nonnegative and
$$c_j \leq \sum_i y_i a_{i j} .$$

## 数学代写|线性规划作业代写Linear Programming代考|The Dual Simplex Method

In this section, we study what happens if we apply the simplex method to the dual problem. As we saw in our discussion of the strong duality theorem, one can actually apply the simplex method to the dual problem without ever writing down the dual problem or its dictionaries. Instead, the so-called dual simplex method is seen simply as a new way of picking the entering and leaving variables in a sequence of primal dictionaries.
We begin with an example:
\begin{aligned} \operatorname{maximize} \quad-x_1-x_2 & \ \text { subject to } \quad-2 x_1-x_2 & \leq 4 \ -2 x_1+4 x_2 & \leq-8 \ -x_1+3 x_2 & \leq-7 \ x_1, x_2 & \geq 0 . \end{aligned}

The dual of this problem is
\begin{aligned} \operatorname{minimize} & 4 y_1-8 y_2-7 y_3 \ \text { subject to }-2 y_1-2 y_2-y_3 & \geq-1 \ -y_1+4 y_2+3 y_3 & \geq-1 \ y_1, y_2, y_3 & \geq 0 . \end{aligned}
Introducing variables $w_i, i=1,2,3$, for the primal slacks and $z_j, j=1,2$, for the dual slacks, we can write down the initial primal and dual dictionaries:
(P)
\begin{aligned} \zeta & =-x_1-x_2 \ \hline w_1 & =4+2 x_1+x_2 \ w_2 & =-8+2 x_1-4 x_2 \ w_3 & =-7+x_1-3 x_2 \end{aligned}
(D)
\begin{aligned} -\xi & =-4 y_1+8 y_2+7 y_3 \ \hline z_1 & =1-2 y_1-2 y_2-y_3 \ z_2 & =1-y_1+4 y_2+3 y_3 . \end{aligned}

# 线性规划代写

## 数学代写|线性规划作业代写Linear Programming代考|Complementary Slackness

$$\begin{array}{ll} x_j z_j=0, & \text { for } j=1,2, \ldots, n, \ w_i y_i=0, & \text { for } i=1,2, \ldots, m . \end{array}$$

\begin{aligned} \sum_j c_j x_j & \leq \sum_j\left(\sum_i y_i a_{i j}\right) x_j \ & =\sum_i\left(\sum_j a_{i j} x_j\right) y_i \ & \leq \sum_i b_i y_i . \end{aligned}

$$c_j \leq \sum_i y_i a_{i j} .$$

## 数学代写|线性规划作业代写Linear Programming代考|The Dual Simplex Method

\begin{aligned} \operatorname{maximize} \quad-x_1-x_2 & \ \text { subject to } \quad-2 x_1-x_2 & \leq 4 \ -2 x_1+4 x_2 & \leq-8 \ -x_1+3 x_2 & \leq-7 \ x_1, x_2 & \geq 0 . \end{aligned}

\begin{aligned} \operatorname{minimize} & 4 y_1-8 y_2-7 y_3 \ \text { subject to }-2 y_1-2 y_2-y_3 & \geq-1 \ -y_1+4 y_2+3 y_3 & \geq-1 \ y_1, y_2, y_3 & \geq 0 . \end{aligned}

(p)
\begin{aligned} \zeta & =-x_1-x_2 \ \hline w_1 & =4+2 x_1+x_2 \ w_2 & =-8+2 x_1-4 x_2 \ w_3 & =-7+x_1-3 x_2 \end{aligned}
(d)
\begin{aligned} -\xi & =-4 y_1+8 y_2+7 y_3 \ \hline z_1 & =1-2 y_1-2 y_2-y_3 \ z_2 & =1-y_1+4 y_2+3 y_3 . \end{aligned}

## 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。

## 数学代写|线性规划作业代写Linear Programming代考|Measuring the Size of a Problem

statistics-lab™ 为您的留学生涯保驾护航 在代写线性规划Linear Programming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。我们的专家在代写线性规划Linear Programming代写方面经验极为丰富，各种代写线性规划Linear Programming相关的作业也就用不着说。

## 数学代写|线性规划作业代写Linear Programming代考|Measuring the Size of a Problem

Before looking at worst cases, we must discuss two issues. First, how do we specify the size of a problem? Two parameters come naturally to mind: $m$ and $n$.

However, we should mention some drawbacks associated with this choice. First of all, it would be preferable to use only one number to indicate size. Since the data for a problem consist of the constraint coefficients together with the right-hand side and objective function coefficients, perhaps we should use the total number of data elements, which is roughly $m n$.

The product $m n$ isn’t bad, but what if many or even most of the data elements are zero? Wouldn’t one expect such a problem to be easier to solve? Efficient implementations do indeed take advantage of the presence of lots of zeros, and so an analysis should also account for this. Hence, a good measure might be simply the number of nonzero data elements. This would definitely be an improvement, but one can go further. On a computer, floating-point numbers are all the same size and can be multiplied in the same amount of time. But if a person is to solve a problem by hand (or use unlimited precision computation on a computer), then certainly multiplying 23 by 7 is a lot easier than multiplying 23453.2352 by 86833.245643 . So perhaps the best measure of a problem’s size is not the number of data elements, but the actual number of bits needed to store all the data on a computer. This measure is popular among most computer scientists and is usually denoted by $L$.

However, with a little further abstraction, the size of the data, $L$, is seen to be ambiguous. As we saw in Chapter 1, real-world problems, while generally large and sparse, usually can be described quite simply and involve only a small amount of true input data that gets greatly expanded when setting the problem up with a constraint matrix, right-hand side, and objective function. So should $L$ represent the number of bits needed to specify the nonzero constraint coefficients, objective coefficients, and right-hand sides, or should it be the number of bits in the original data set plus the number of bits in the description of how this data represents a linear programming problem? No one currently uses this last notion of problem size, but it seems fairly reasonable that they should (or at least that they should seriously consider it). Anyway, our purpose here is merely to mention that these important issues are lurking about, but, as stated above, we shall simply focus on $m$ and $n$ to characterize the size of a problem.

## 数学代写|线性规划作业代写Linear Programming代考|Measuring the Effort to Solve a Problem

The second issue to discuss is how one should measure the amount of work required to solve a problem. The best answer is the number of seconds of computer time required to solve the problem, using the computer sitting on one’s desk. Unfortunately, there are (hopefully) many readers of this text, not all of whom use the exact same computer. Even if they did, computer technology changes rapidly, and a few years down the road everyone would be using something entirely different. It would be nice if the National Institute of Standards and Technology (the government organization in charge of setting standards, such as how many threads/inch a standard light bulb should have) would identify a standard computer for the purpose of benchmarking algorithms, but, needless to say, this is not very likely. So the time needed to solve a problem, while the most desirable measure, is not the most practical one here. Fortunately, there is a fairly reasonable substitute. Algorithms are generally iterative processes, and the time to solve a problem can be factored into the number of iterations required to solve the problem times the amount of time required to do each iteration. The first factor, the number of iterations, does not depend on the computer and so is a reasonable surrogate for the actual time. This surrogate is useful when comparing various algorithms within the same general class of algorithms, in which the time per iteration can be expected to be about the same among the algorithms; however, it becomes meaningless when one wishes to compare two entirely different algorithms. For now, we shall measure the amount of effort to solve a linear programming problem by counting the number of iterations needed to solve it.

# 线性规划代写

## 数学代写|线性规划作业代写Linear Programming代考|Measuring the Size of a Problem

\begin{aligned} \zeta & =\bar{\zeta}+\sum_{j \in \mathcal{N}} \bar{c}j x_j \ x_i & =\bar{b}i-\sum{j \in \mathcal{N}} \bar{a}{i j} x_j \quad i \in \mathcal{B} . \end{aligned}

## 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。