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## 数学代写|线性规划代写Linear Programming代考|Matrix Games

A matrix game is a two-person game defined as follows. Each person first selects, independently of the other, an action from a finite set of choices (the two players in general will be confronted with different sets of actions from which to choose). Then both reveal to each other their choice. If we let $i$ denote the first player’s choice and $j$ denote the second player’s choice, then the rules of the game stipulate that the first player will pay the second player $a_{i j}$ dollars. The array of possible payments
$$A=\left[a_{i j}\right]$$
is presumed known to both players before the game begins. Of course, if $a_{i j}$ is negative for some pair $(i, j)$, then the payment goes in the reverse direction – from the second player to the first. For obvious reasons, we shall refer to the first player as the row player and the second player as the column player. Since we have assumed that the row player has only a finite number of actions from which to choose, we can enumerate these actions and assume without loss of generality that $i$ is simply an integer selected from 1 to $m$. Similarly, we can assume that $j$ is simply an index ranging from 1 to $n$ (in its real-world interpretation, row action 3 will generally have nothing to do with column action 3-the number 3 simply indicates that it is the third action in the enumerated list of choices).

Let us look at a specific familiar example. Namely, consider the game every child knows, called Paper-Scissors-Rock. To refresh the memory of older readers, this is a two-person game in which at the count of three each player declares either Paper, Scissors, or Rock. If both players declare the same object, then the round is a draw. But Paper loses to Scissors (since scissors can cut a piece of paper), Scissors loses to Rock (since a rock can dull scissors), and finally Rock loses to Paper (since a piece of paper can cover up a rock – it’s a weak argument but that’s the way the game is defined). Clearly, for this game, if we enumerate the actions of declaring Paper, Scissors, or Rock as $1,2,3$, respectively, then the payoff matrix is
$$\left[\begin{array}{rrr} 0 & 1 & -1 \ -1 & 0 & 1 \ 1 & -1 & 0 \end{array}\right]$$

## 数学代写|线性规划代写Linear Programming代考|Optimal Strategies

Suppose that the column player adopts strategy $x$ (i.e., decides to play in accordance with the stochastic vector $x$ ). Then the row player’s best defense is to use the strategy $y^*$ that achieves the following minimum:
\begin{aligned} \operatorname{minimize} y^T A x & \ \text { subject to } e^T y & =1 \ y & \geq 0 . \end{aligned}
From the fundamental theorem of linear programming, we know that this problem has a basic optimal solution. For this problem, the basic solutions are simply $y$ vectors that are zero in every component except for one, which is one. That is, the basic optimal solutions correspond to deterministic strategies. This is fairly obvious if we look again at our example. Suppose that
$$x=\left[\begin{array}{l} 1 / 3 \ 1 / 3 \ 1 / 3 \end{array}\right] .$$

## 数学代写|线性规划代写Linear Programming代考|Matrix Games

$$A=\left[a_{i j}\right]$$

$$\left[\begin{array}{rrr} 0 & 1 & -1 \ -1 & 0 & 1 \ 1 & -1 & 0 \end{array}\right]$$

## 数学代写|线性规划代写Linear Programming代考|Optimal Strategies

\begin{aligned} \operatorname{minimize} y^T A x & \ \text { subject to } e^T y & =1 \ y & \geq 0 . \end{aligned}

$$x=\left[\begin{array}{l} 1 / 3 \ 1 / 3 \ 1 / 3 \end{array}\right] .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|线性规划代写Linear Programming代考|The Primal Simplex Method

It is easiest to illustrate the ideas with an example:
$$\begin{array}{rrr} \operatorname{maximize} & 3 x_1-x_2 \ \text { subject to } & 1 \leq-x_1+x_2 \leq 5 \ 2 \leq-3 x_1+2 x_2 \leq 10 \ & 2 x_1-x_2 \leq 0 \ -2 \leq \quad & x_1 \ & 0 \leq \quad x_2 \leq 6 . \end{array}$$
With this formulation, zero no longer plays the special role it once did. Instead, that role is replaced by the notion of a variable or a constraint being at its upper or lower bound. Therefore, instead of defining slack variables for each constraint, we use $w_i$ simply to denote the value of the $i$ th constraint:
\begin{aligned} & w_1=-x_1+x_2 \ & w_2=-3 x_1+2 x_2 \ & w_3=2 x_1-x_2 . \end{aligned}
The constraints can then be interpreted as upper and lower bounds on these variables. Now when we record our problem in a dictionary, we will have to keep explicit track of the upper and lower bound on the original $x_j$ variables and the new $w_i$ variables. Also, the value of a nonbasic variable is no longer implicit; it could be at either its upper or its lower bound. Hence, we shall indicate which is the case by putting a box around the relevant bound. Finally, we need to keep track of the values of the basic variables. Hence, we shall write our dictionary as follows:

## 数学代写|线性规划代写Linear Programming代考|The Dual Simplex Method

The problem considered in the previous section had an initial dictionary that was feasible. But as always, we must address the case where the initial dictionary is not feasible. That is, we must define a Phase I algorithm. Following the ideas presented in Chapter 5, we base our Phase I algorithm on a dual simplex method. To this end, we need to introduce the dual of (9.1). So first we rewrite (9.1) as
\begin{aligned} \operatorname{maximize} & c^T x \ \text { subject to } \quad A x & \leq b \ -A x & \leq-a \ x & \leq u \ -x & \leq-l, \end{aligned}
and adding slack variables, we have
\begin{aligned} & \operatorname{maximize} \quad c^T x \ & \text { subject to } \quad A x+f=b \ &-A x+p=-a \ & x+t=u \ &-x+g=-l \ & f, p, t, g \geq 0 . \end{aligned}
We see immediately from the inequality form of the primal that the dual can be written as
$$\begin{array}{rc} \operatorname{minimize} & b^T v-a^T q+u^T s-l^T h \ \text { subject to } & A^T(v-q)-(h-s)=c \ v, q, s, h \geq 0 . \end{array}$$

## 数学代写|线性规划代写Linear Programming代考|The Primal Simplex Method

$$\begin{array}{rrr} \operatorname{maximize} & 3 x_1-x_2 \ \text { subject to } & 1 \leq-x_1+x_2 \leq 5 \ 2 \leq-3 x_1+2 x_2 \leq 10 \ & 2 x_1-x_2 \leq 0 \ -2 \leq \quad & x_1 \ & 0 \leq \quad x_2 \leq 6 . \end{array}$$

\begin{aligned} & w_1=-x_1+x_2 \ & w_2=-3 x_1+2 x_2 \ & w_3=2 x_1-x_2 . \end{aligned}

## 数学代写|线性规划代写Linear Programming代考|The Dual Simplex Method

\begin{aligned} \operatorname{maximize} & c^T x \ \text { subject to } \quad A x & \leq b \ -A x & \leq-a \ x & \leq u \ -x & \leq-l, \end{aligned}

\begin{aligned} & \operatorname{maximize} \quad c^T x \ & \text { subject to } \quad A x+f=b \ &-A x+p=-a \ & x+t=u \ &-x+g=-l \ & f, p, t, g \geq 0 . \end{aligned}

$$\begin{array}{rc} \operatorname{minimize} & b^T v-a^T q+u^T s-l^T h \ \text { subject to } & A^T(v-q)-(h-s)=c \ v, q, s, h \geq 0 . \end{array}$$

## 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。

Posted on Categories:Linear Programming, 数学代写, 线性规划

## avatest™帮您通过考试

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## 数学代写|线性规划代写Linear Programming代考|Two-Phase Methods

Let us summarize the algorithm obtained by applying the dual simplex method as a Phase I procedure followed by the primal simplex method as a Phase II. Initially, we set
$$\mathcal{B}={n+1, n+2, \ldots, n+m} \quad \text { and } \quad \mathcal{N}={1,2, \ldots, n} .$$
Then from (6.1) we see that $A=[N B]$, where
$$N=\left[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1 n} \ a_{21} & a_{22} & \ldots & a_{2 n} \ \vdots & \vdots & & \vdots \ a_{m 1} & a_{m 2} & \ldots & a_{m n} \end{array}\right], \quad B=\left[\begin{array}{cccc} 1 & & & \ & 1 & & \ & & \ddots & \ & & 1 \end{array}\right],$$
and from (6.2) we have
$$c_{\mathcal{N}}=\left[\begin{array}{c} c_1 \ c_2 \ \vdots \ c_n \end{array}\right] \quad \text { and } \quad c_{\mathcal{B}}=\left[\begin{array}{c} 0 \ 0 \ \vdots \ 0 \end{array}\right]$$
Substituting these expressions into the definitions of $x_{\mathcal{B}}^, z_{\mathcal{N}}^$, and $\zeta^$, we find that \begin{aligned} x_{\mathcal{B}}^ & =B^{-1} b=b \ z_{\mathcal{N}}^* & =\left(B^{-1} N\right)^T c_{\mathcal{B}}-c_{\mathcal{N}}=-c_{\mathcal{N}} \ \zeta^* & =0 . \end{aligned}
\begin{aligned} \zeta & =c_{\mathcal{N}}^T x_{\mathcal{N}} \ x_{\mathcal{B}} & =b-N x_{\mathcal{N}} . \end{aligned}

## 数学代写|线性规划代写Linear Programming代考|Negative Transpose Property

We emphasized the symmetry between the primal problem and its dual. This symmetry can be easily summarized by saying that the dual of a standard-form linear programming problem is the negative transpose of the primal problem. Now, in this chapter, the symmetry appears to have been lost. For example, the basis matrix is an $m \times m$ matrix. Why $m \times m$ and not $n \times n$ ? It seems strange. In fact, if we had started with the dual problem, added slack variables to it, and introduced a basis matrix on that side it would be an $n \times n$ matrix. How are these two basis matrices related? It turns out that they are not themselves related in any simple way, but the important matrix $B^{-1} N$ is still the negative transpose of the analogous dual construct. The purpose of this section is to make this connection clear.
Consider a standard-form linear programming problem
\begin{aligned} & \operatorname{maximize} c^T x \ & \text { subject to } A x \leq b \ & x \geq 0, \end{aligned}
and its dual
\begin{aligned} \operatorname{minimize} b^T y & \ \text { subject to } A^T y & \geq c \ y & \geq 0 . \end{aligned}

Let $w$ be a vector containing the slack variables for the primal problem, let $z$ be a slack vector for the dual problem, and write both problems in equality form:
\begin{aligned} & \text { maximize } c^T x \ & \text { subject to } A x+w=b \ & x, w \geq 0 \text {, } \ & \end{aligned}
and
\begin{aligned} & \text { minimize } b^T y \ & \text { subject to } A^T y-z=c \ & y, z \geq 0 \text {. } \ & \end{aligned}
Introducing three new notations,
$$\bar{A}=\left[\begin{array}{ll} A & I \end{array}\right], \quad \bar{c}=\left[\begin{array}{l} c \ 0 \end{array}\right], \quad \text { and } \quad \bar{x}=\left[\begin{array}{l} x \ w \end{array}\right],$$
the primal problem can be rewritten succinctly as follows:
$$\begin{array}{ll} \operatorname{maximize} & \bar{c}^T \bar{x} \ \text { subject to } & \bar{A} \bar{x}=b \ \bar{x} \geq 0 . \end{array}$$

## 数学代写|线性规划代写Linear Programming代考|Two-Phase Methods

$$\mathcal{B}={n+1, n+2, \ldots, n+m} \quad \text { and } \quad \mathcal{N}={1,2, \ldots, n} .$$

$$N=\left[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1 n} \ a_{21} & a_{22} & \ldots & a_{2 n} \ \vdots & \vdots & & \vdots \ a_{m 1} & a_{m 2} & \ldots & a_{m n} \end{array}\right], \quad B=\left[\begin{array}{cccc} 1 & & & \ & 1 & & \ & & \ddots & \ & & 1 \end{array}\right],$$

$$c_{\mathcal{N}}=\left[\begin{array}{c} c_1 \ c_2 \ \vdots \ c_n \end{array}\right] \quad \text { and } \quad c_{\mathcal{B}}=\left[\begin{array}{c} 0 \ 0 \ \vdots \ 0 \end{array}\right]$$

\begin{aligned} \zeta & =c_{\mathcal{N}}^T x_{\mathcal{N}} \ x_{\mathcal{B}} & =b-N x_{\mathcal{N}} . \end{aligned}

## 数学代写|线性规划代写Linear Programming代考|Negative Transpose Property

\begin{aligned} & \operatorname{maximize} c^T x \ & \text { subject to } A x \leq b \ & x \geq 0, \end{aligned}

\begin{aligned} \operatorname{minimize} b^T y & \ \text { subject to } A^T y & \geq c \ y & \geq 0 . \end{aligned}

\begin{aligned} & \text { maximize } c^T x \ & \text { subject to } A x+w=b \ & x, w \geq 0 \text {, } \ & \end{aligned}

\begin{aligned} & \text { minimize } b^T y \ & \text { subject to } A^T y-z=c \ & y, z \geq 0 \text {. } \ & \end{aligned}

$$\bar{A}=\left[\begin{array}{ll} A & I \end{array}\right], \quad \bar{c}=\left[\begin{array}{l} c \ 0 \end{array}\right], \quad \text { and } \quad \bar{x}=\left[\begin{array}{l} x \ w \end{array}\right],$$

$$\begin{array}{ll} \operatorname{maximize} & \bar{c}^T \bar{x} \ \text { subject to } & \bar{A} \bar{x}=b \ \bar{x} \geq 0 . \end{array}$$

## 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。

Posted on Categories:Linear Programming, 数学代写, 线性规划

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|线性规划代写Linear Programming代考|The Weak Duality Theorem

As we saw in our example, the dual problem provides upper bounds for the primal objective function value. This result is true in general and is referred to as the Weak Duality Theorem:

THEOREM 5.1. If $\left(x_1, x_2, \ldots, x_n\right)$ is feasible for the primal and $\left(y_1, y_2, \ldots, y_m\right)$ is feasible for the dual, then
$$\sum_j c_j x_j \leq \sum_i b_i y_i .$$
Proof. The proof is a simple chain of obvious inequalities:
\begin{aligned} \sum_j c_j x_j & \leq \sum_j\left(\sum_i y_i a_{i j}\right) x_j \ & =\sum_{i j} y_i a_{i j} x_j \ & =\sum_i\left(\sum_j a_{i j} x_j\right) y_i \ & \leq \sum_i b_i y_i, \end{aligned}
where the first inequality follows from the fact that each $x_j$ is nonnegative and each $c_j$ is no larger than $\sum_i y_i a_{i j}$. The second inequality, of course, holds for similar reasons.

Consider the subset of the real line consisting of all possible values for the primal objective function, and consider the analogous subset associated with the dual problem. The weak duality theorem tells us that the set of primal values lies entirely to the left of the set of dual values. As we shall see shortly, these sets are both closed intervals (perhaps of infinite extent), and the right endpoint of the primal set butts up against the left endpoint of the dual set (see Figure 5.1). That is, there is no gap between the optimal objective function value for the primal and for the dual. The lack of a gap between primal and dual objective values provides a convenient tool for verifying optimality. Indeed, if we can exhibit a feasible primal solution $\left(x_1^, x_2^, \ldots, x_n^\right)$ and a feasible dual solution $\left(y_1^, y_2^, \ldots, y_m^\right)$ for which
$$\sum_j c_j x_j^=\sum_i b_i y_i^,$$
then we may conclude that each of these solutions is optimal for its respective problem. To see that the primal solution is optimal, consider any other feasible solution $\left(x_1, x_2, \ldots, x_n\right)$. By the weak duality theorem, we have that
$$\sum_j c_j x_j \leq \sum_i b_i y_i^=\sum_j c_j x_j^$$

## 数学代写|线性规划代写Linear Programming代考|The Strong Duality Theorem

The fact that for linear programming there is never a gap between the primal and the dual optimal objective values is usually referred to as the Strong Duality Theorem:
THEOREM 5.2. If the primal problem has an optimal solution,
$$x^=\left(x_1^, x_2^, \ldots, x_n^\right)$$
then the dual also has an optimal solution,
$$y^=\left(y_1^, y_2^, \ldots, y_m^\right)$$
such that
$$\sum_j c_j x_j^=\sum_i b_i y_i^ .$$
Carefully written proofs, while attractive for their tightness, sometimes obfuscate the main idea. In such cases, it is better to illustrate the idea with a simple example. Anyone who has taken a course in linear algebra probably already appreciates such a statement. In any case, it is true here as we explain the strong duality theorem.

The main idea that we wish to illustrate here is that, as the simplex method solves the primal problem, it also implicitly solves the dual problem, and it does so in such a way that $(5.2)$ holds.

To see what we mean, let us return to the example discussed in Section 5.1. We start by introducing variables $w_i, i=1,2$, for the primal slacks and $z_j, j=1,2,3$, for the dual slacks. Since the inequality constraints in the dual problem are greaterthan constraints, each dual slack is defined as a left-hand side minus the corresponding right-hand side. For example,
$$z_1=y_1+3 y_2-4$$

## 数学代写|线性规划代写Linear Programming代考|The Weak Duality Theorem

$$\sum_j c_j x_j \leq \sum_i b_i y_i .$$

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

$$\sum_j c_j x_j^=\sum_i b_i y_i^,$$

$$\sum_j c_j x_j \leq \sum_i b_i y_i^=\sum_j c_j x_j^$$

## 数学代写|线性规划代写Linear Programming代考|The Strong Duality Theorem

$$x^=\left(x_1^, x_2^, \ldots, x_n^\right)$$

$$y^=\left(y_1^, y_2^, \ldots, y_m^\right)$$

$$\sum_j c_j x_j^=\sum_i b_i y_i^ .$$

$$z_1=y_1+3 y_2-4$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|线性规划代写Linear Programming代考|The Perturbation/Lexicographic Method

As we have seen, there is not just one algorithm called the simplex method. Instead, the simplex method is a whole family of related algorithms from which we can pick a specific instance by specifying what we have been referring to as pivoting rules. We have also seen that, using the most natural pivoting rules, the simplex method can fail to converge to an optimal solution by occasionally cycling indefinitely through a sequence of degenerate pivots associated with a nonoptimal solution.

So this raises a natural question: are there pivoting rules for which the simplex method will definitely either reach an optimal solution or prove that no such solution exists? The answer to this question is yes, and we shall present two choices of such pivoting rules.

The first method is based on the observation that degeneracy is sort of an accident. That is, a dictionary is degenerate if one or more of the $\bar{b}_i$ ‘s vanish. Our examples have generally used small integers for the data, and in this case it doesn’t seem too surprising that sometimes cancellations occur and we end up with a degenerate dictionary. But each right-hand side could in fact be any real number, and in the world of real numbers the occurrence of any specific number, such as zero, seems to be quite unlikely. So how about perturbing a given problem by adding small random perturbations independently to each of the right-hand sides? If these perturbations are small enough, we can think of them as insignificant and hence not really changing the problem. If they are chosen independently, then the probability of an exact cancellation is zero.
Such random perturbation schemes are used in some implementations, but what we have in mind as we discuss perturbation methods is something a little bit different. Instead of using independent identically distributed random perturbations, let us consider using a fixed perturbation for each constraint, with the perturbation getting much smaller on each succeeding constraint. Indeed, we introduce a small positive number $\epsilon_1$ for the first constraint and then a much smaller positive number $\epsilon_2$ for the second constraint, etc. We write this as
$$0<\epsilon_m \ll \cdots \ll \epsilon_2 \ll \epsilon_1 \ll \text { all other data. }$$

## 数学代写|线性规划代写Linear Programming代考|Bland’s Rule

The second pivoting rule we shall consider is called Bland’s rule. It stipulates that both the entering and the leaving variable be selected from their respective sets of choices by choosing the variable $x_k$ with the smallest index $k$.

THEOREM 3.3. The simplex method always terminates provided that both the entering and the leaving variable are chosen according to Bland’s rule.

The proof may look rather involved, but the reader who spends the time to understand it will find the underlying elegance most rewarding.

Proof. It suffices to show that such a variant of the simplex method never cycles. We prove this by assuming that cycling does occur and then showing that this assumption leads to a contradiction. So let’s assume that cycling does occur. Without loss of generality, we may assume that it happens from the beginning. Let $D_0, D_1, \ldots, D_{k-1}$ denote the dictionaries through which the method cycles. That is, the simplex method produces the following sequence of dictionaries:
$$D_0, D_1, \ldots, D_{k-1}, D_0, D_1, \ldots$$
We say that a variable is fickle if it is in some basis and not in some other basis. Let $x_t$ be the fickle variable having the largest index and let $D$ denote a dictionary in $D_0, D_1, \ldots, D_{k-1}$ in which $x_t$ leaves the basis. Again, without loss of generality we may assume that $D=D_0$. Let $x_s$ denote the corresponding entering variable. Suppose that $D$ is recorded as follows:
\begin{aligned} \zeta & =v+\sum_{j \in \mathcal{N}} c_j x_j \ x_i & =b_i-\sum_{j \in \mathcal{N}} a_{i j} x_j \quad i \in \mathcal{B} . \end{aligned}

## 数学代写|线性规划代写Linear Programming代考|The Perturbation/Lexicographic Method

$$0<\epsilon_m \ll \cdots \ll \epsilon_2 \ll \epsilon_1 \ll \text { all other data. }$$

## 数学代写|线性规划代写Linear Programming代考|Bland’s Rule

$$D_0, D_1, \ldots, D_{k-1}, D_0, D_1, \ldots$$

\begin{aligned} \zeta & =v+\sum_{j \in \mathcal{N}} c_j x_j \ x_i & =b_i-\sum_{j \in \mathcal{N}} a_{i j} x_j \quad i \in \mathcal{B} . \end{aligned}

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|线性规划代写Linear Programming代考|Comptroller as Pessimist

In another office at the production facility sits an executive called the comptroller. The comptroller’s problem (among others) is to assign a value to the raw materials on hand. These values are needed for accounting and planning purposes to determine the cost of inventory. There are rules about how these values can be set. The most important such rule (and the only one relevant to our discussion) is the following:
The company must be willing to sell the raw materials should an outside firm offer to buy them at a price consistent with these values.
Let $w_i$ denote the assigned unit value of the $i$ th raw material, $i=1,2, \ldots, m$. That is, these are the numbers that the comptroller must determine. The lost opportunity cost of having $b_i$ units of raw material $i$ on hand is $b_i w_i$, and so the total lost opportunity cost is
$$\sum_{i=1}^m b_i w_i .$$
The comptroller’s goal is to minimize this lost opportunity cost (to make the financial statements look as good as possible). But again, there are constraints. First of all, each assigned unit value $w_i$ must be no less than the prevailing unit market value $\rho_i$, since if it were less an outsider would buy the company’s raw material at a price lower than $\rho_i$, contradicting the assumption that $\rho_i$ is the prevailing market price. That is,
$$w_i \geq \rho_i, \quad i=1,2, \ldots, m .$$
Similarly,
$$\sum_{i=1}^m w_i a_{i j} \geq \sigma_j, \quad j=1,2, \ldots, n$$

## 数学代写|线性规划代写Linear Programming代考|The Linear Programming Problem

In the two examples given above, there have been variables whose values are to be decided in some optimal fashion. These variables are referred to as decision variables. They are usually written as
$$x_j, \quad j=1,2, \ldots, n .$$
In linear programming, the objective is always to maximize or to minimize some linear function of these decision variables
$$\zeta=c_1 x_1+c_2 x_2+\cdots+c_n x_n .$$
This function is called the objective function. It often seems that real-world problems are most naturally formulated as minimizations (since real-world planners always seem to be pessimists), but when discussing mathematics it is usually nicer to work with maximization problems. Of course, converting from one to the other is trivial both from the modeler’s viewpoint (either minimize cost or maximize profit) and from the analyst’s viewpoint (either maximize $\zeta$ or minimize $-\zeta$ ). Since this book is primarily about the mathematics of linear programming, we shall usually consider our aim one of maximizing the objective function.

In addition to the objective function, the examples also had constraints. Some of these constraints were really simple, such as the requirement that some decision variable be nonnegative. Others were more involved. But in all cases the constraints consisted of either an equality or an inequality associated with some linear combination of the decision variables:
$$a_1 x_1+a_2 x_2+\cdots+a_n x_n\left{\begin{array}{l} \leq \ = \ \geq \end{array}\right} b .$$

## 数学代写|线性规划代写Linear Programming代考|Comptroller as Pessimist

$$\sum_{i=1}^m b_i w_i .$$

$$w_i \geq \rho_i, \quad i=1,2, \ldots, m .$$

$$\sum_{i=1}^m w_i a_{i j} \geq \sigma_j, \quad j=1,2, \ldots, n$$

## 数学代写|线性规划代写Linear Programming代考|The Linear Programming Problem

$$x_j, \quad j=1,2, \ldots, n .$$

$$\zeta=c_1 x_1+c_2 x_2+\cdots+c_n x_n .$$

$$a_1 x_1+a_2 x_2+\cdots+a_n x_n\left{\begin{array}{l} \leq \ = \ \geq \end{array}\right} b .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|线性规划代写Linear Programming代考|Concavity and Curve Sketching

We have seen how the first derivative tells us where a function is increasing, where it is decreasing, and whether a local maximum or local minimum occurs at a critical point. In this section we see that the second derivative gives us information about how the graph of a differentiable function bends or turns. With this knowledge about the first and second derivatives, coupled with our previous understanding of symmetry and asymptotic behavior studied in Sections 1.1 and 2.6, we can now draw an accurate graph of a function. By organizing all of these ideas into a coherent procedure, we give a method for sketching graphs and revealing visually the key features of functions. Identifying and knowing the locations of these features is of major importance in mathematics and its applications to science and engineering, especially in the graphical analysis and interpretation of data. When the domain of a function is not a finite closed interval, sketching a graph helps to determine whether absolute maxima or absolute minima exist and, if they do exist, where they are located.
Concavity
As you can see in Figure 4.24 , the curve $y=x^3$ rises as $x$ increases, but the portions defined on the intervals $(-\infty, 0)$ and $(0, \infty)$ turn in different ways. As we approach the origin from the left along the curve, the curve turns to our right and falls below its tangents. The slopes of the tangents are decreasing on the interval $(-\infty, 0)$. As we move away from the origin along the curve to the right, the curve turns to our left and rises above its tangents. The slopes of the tangents are increasing on the interval $(0, \infty)$. This turning or bending behavior defines the concavity of the curve.

DEFINITION The graph of a differentiable function $y=f(x)$ is
(a) concave up on an open interval $I$ if $f^{\prime}$ is increasing on $I$;
(b) concave down on an open interval $I$ if $f^{\prime}$ is decreasing on $I$.
A function whose graph is concave up is also often called convex.
If $y=f(x)$ has a second derivative, we can apply Corollary 3 of the Mean Value Theorem to the first derivative function. We conclude that $f^{\prime}$ increases if $f^{\prime \prime}>0$ on $I$, and decreases if $f^{\prime \prime}<0$.
The Second Derivative Test for Concavity
Let $y=f(x)$ be twice-differentiable on an interval $I$.

1. If $f^{\prime \prime}>0$ on $I$, the graph of $f$ over $I$ is concave up.
2. If $f^{\prime \prime}<0$ on $I$, the graph of $f$ over $I$ is concave down.
If $y=f(x)$ is twice-differentiable, we will use the notations $f^{\prime \prime}$ and $y^{\prime \prime}$ interchangeably when denoting the second derivative.

## 数学代写|线性规划代写Linear Programming代考|Points of Inflection

The curve $y=3+\sin x$ in Example 2 changes concavity at the point $(\pi, 3)$. Since the first derivative $y^{\prime}=\cos x$ exists for all $x$, we see that the curve has a tangent line of slope -1 at the point $(\pi, 3)$. This point is called a point of inflection of the curve. Notice from Figure 4.26 that the graph crosses its tangent line at this point and that the second derivative $y^{\prime \prime}=-\sin x$ has value 0 when $x=\pi$. In general, we have the following definition.
DEFINITION A point $(c, f(c))$ where the graph of a function has a tangent line and where the concavity changes is a point of inflection.
We observed that the second derivative of $f(x)=3+\sin x$ is equal to zero at the inflection point $(\pi, 3)$. Generally, if the second derivative exists at a point of inflection $(c, f(c))$, then $f^{\prime \prime}(c)=0$. This follows immediately from the Intermediate Value Theorem whenever $f^{\prime \prime}$ is continuous over an interval containing $x=c$ because the second derivative changes sign moving across this interval. Even if the continuity assumption is dropped, it is still true that $f^{\prime \prime}(c)=0$, provided the second derivative exists (although a more advanced argument is required in this noncontinuous case). Since a tangent line must exist at the point of inflection, either the first derivative $f^{\prime}(c)$ exists (is finite) or the graph has a vertical tangent at the point. At a vertical tangent neither the first nor second derivative exists. In summary, one of two things can happen at a point of inflection.
At a point of inflection $(c, f(c))$, either $f^{\prime \prime}(c)=0$ or $f^{\prime \prime}(c)$ fails to exist.

## 代写|线性规划代写Linear Programming代考|Concavity and Curve Sketching

(a)如果$f^{\prime}$在$I$上增加，则在开放区间$I$上凹上;
(b)如果$f^{\prime}$在$I$上减小，则在开放区间$I$上向下凹。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## avatest™帮您通过考试

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## 数学代写|线性规划代写Linear Programming代考|Extreme Values of Functions on Closed Intervals

This section shows how to locate and identify extreme (maximum or minimum) values of a function from its derivative. Once we can do this, we can solve a variety of optimization problems (see Section 4.5). The domains of the functions we consider are intervals or unions of separate intervals.
DEFINITIONS Let $f$ be a function with domain $D$. Then $f$ has an absolute maximum value on $D$ at a point $c$ if
$$f(x) \leq f(c) \quad \text { for all } x \text { in } D$$
and an absolute minimum value on $D$ at $c$ if
$$f(x) \geq f(c) \quad \text { for all } x \text { in } D \text {. }$$
Maximum and minimum values are called extreme values of the function $f$. Absolute maxima or minima are also referred to as global maxima or minima.

For example, on the closed interval $[-\pi / 2, \pi / 2]$ the function $f(x)=\cos x$ takes on an absolute maximum value of 1 (once) and an absolute minimum value of 0 (twice). On the same interval, the function $g(x)=\sin x$ takes on a maximum value of 1 and a minimum value of -1 (Figure 4.1).

Functions defined by the same equation or formula can have different extrema (maximum or minimum values), depending on the domain. A function might not have a maximum or minimum if the domain is unbounded or fails to contain an endpoint. We see this in the following example.

## 数学代写|线性规划代写Linear Programming代考|Local (Relative) Extreme Values

Figure 4.5 shows a graph with five points where a function has extreme values on its domain $[a, b]$. The function’s absolute minimum occurs at $a$ even though at $e$ the function’s value is smaller than at any other point nearby. The curve rises to the left and falls to the right around $c$, making $f(c)$ a maximum locally. The function attains its absolute maximum at $d$. We now define what we mean by local extrema.
DEFINITIONS A function $f$ has a local maximum value at a point $c$ within its domain $D$ if $f(x) \leq f(c)$ for all $x \in D$ lying in some open interval containing $c$.
A function $f$ has a local minimum value at a point $c$ within its domain $D$ if $f(x) \geq f(c)$ for all $x \in D$ lying in some open interval containing $c$.
If the domain of $f$ is the closed interval $[a, b]$, then $f$ has a local maximum at the endpoint $x=a$ if $f(x) \leq f(a)$ for all $x$ in some half-open interval $[a, a+\delta), \delta>0$. Likewise, $f$ has a local maximum at an interior point $x=c$ if $f(x) \leq f(c)$ for all $x$ in some open interval $(c-\delta, c+\delta), \delta>0$, and a local maximum at the endpoint $x=b$ if $f(x) \leq f(b)$ for all $x$ in some half-open interval $(b-\delta, b], \delta>0$. The inequalities are reversed for local minimum values. In Figure 4.5, the function $f$ has local maxima at $c$ and $d$ and local minima at $a, e$, and $b$. Local extrema are also called relative extrema. Some functions can have infinitely many local extrema, even over a finite interval. One example is the function $f(x)=\sin (1 / x)$ on the interval $(0,1]$. (We graphed this function in Figure 2.40.)

An absolute maximum is also a local maximum. Being the largest value overall, it is also the largest value in its immediate neighborhood. Hence, a list of all local maxima will automatically include the absolute maximum if there is one. Similarly, a list of all local minima will include the absolute minimum if there is one.

## 数学代写|线性规划代写Linear Programming代考|Extreme Values of Functions on Closed Intervals

$$f(x) \leq f(c) \quad \text { for all } x \text { in } D$$

$$f(x) \geq f(c) \quad \text { for all } x \text { in } D \text {. }$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|线性规划代写Linear Programming代考|Derivatives of Trigonometric Functions

Many phenomena of nature are approximately periodic (electromagnetic fields, heart rhythms, tides, weather). The derivatives of sines and cosines play a key role in describing periodic changes. This section shows how to differentiate the six basic trigonometric functions.
Derivative of the Sine Function
To calculate the derivative of $f(x)=\sin x$, for $x$ measured in radians, we combine the limits in Example 5a and Theorem 7 in Section 2.4 with the angle sum identity for the sine function:
$$\sin (x+h)=\sin x \cos h+\cos x \sin h$$
If $f(x)=\sin x$, then
\begin{aligned} & f^{\prime}(x)=\lim {h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim {h \rightarrow 0} \frac{\sin (x+h)-\sin x}{h} \quad \text { Derivative definition } \ & =\lim {h \rightarrow 0} \frac{(\sin x \cos h+\cos x \sin h)-\sin x}{h} \quad \text { Identity for } \sin (x+h) \ & =\lim {h \rightarrow 0} \frac{\sin x(\cos h-1)+\cos x \sin h}{h} \ & =\lim {h \rightarrow 0}\left(\sin x \cdot \frac{\cos h-1}{h}\right)+\lim {h \rightarrow 0}\left(\cos x \cdot \frac{\sin h}{h}\right) \ & =\sin x \cdot \underbrace{\lim {h \rightarrow 0} \frac{\cos h-1}{h}}{\text {limit } 0}+\cos x \cdot \underbrace{\lim {h \rightarrow 0} \frac{\sin h}{h}}{\text {limit } 1} \ & =\sin x \cdot 0+\cos x \cdot 1=\cos x . \ & \end{aligned}
Example 5a and
Theorem 7, Section 2.4
The derivative of the sine function is the cosine function:
$$\frac{d}{d x}(\sin x)=\cos x$$

## 数学代写|线性规划代写Linear Programming代考|Derivative of the Cosine Function

With the help of the angle sum formula for the cosine function,
$$\cos (x+h)=\cos x \cos h-\sin x \sin h,$$
we can compute the limit of the difference quotient:
$$\begin{array}{rlrl} \frac{d}{d x}(\cos x) & =\lim {h \rightarrow 0} \frac{\cos (x+h)-\cos x}{h} & & \text { Derivative definition } \ & =\lim {h \rightarrow 0} \frac{(\cos x \cos h-\sin x \sin h)-\cos x}{h} & & \begin{array}{l} \text { Cosine angle } \ \text { sum identity } \end{array} \ & =\lim {h \rightarrow 0} \frac{\cos x(\cos h-1)-\sin x \sin h}{h} & \ & =\lim {h \rightarrow 0}\left(\cos x \cdot \frac{\cos h-1}{h}\right)-\lim {h \rightarrow 0}\left(\sin x \cdot \frac{\sin h}{h}\right) & & \ & =\cos x \cdot \lim {h \rightarrow 0} \frac{\cos h-1}{h}-\sin x \cdot \lim _{h \rightarrow 0} \frac{\sin h}{h} & & \ & =\cos x \cdot 0-\sin x \cdot 1 & & \begin{array}{l} \text { Example 5a } \ \text { and Theorem 7, } \end{array} \ & =-\sin x . & & \text { Section 2.4 } \end{array}$$
Derivative definition
Example 5a and Theorem 7,
Section 2.4
The derivative of the cosine function is the negative of the sine function:
$$\frac{d}{d x}(\cos x)=-\sin x$$

## 数学代写|线性规划代写Linear Programming代考|Derivatives of Trigonometric Functions

$$\sin (x+h)=\sin x \cos h+\cos x \sin h$$

\begin{aligned} & f^{\prime}(x)=\lim {h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim {h \rightarrow 0} \frac{\sin (x+h)-\sin x}{h} \quad \text { Derivative definition } \ & =\lim {h \rightarrow 0} \frac{(\sin x \cos h+\cos x \sin h)-\sin x}{h} \quad \text { Identity for } \sin (x+h) \ & =\lim {h \rightarrow 0} \frac{\sin x(\cos h-1)+\cos x \sin h}{h} \ & =\lim {h \rightarrow 0}\left(\sin x \cdot \frac{\cos h-1}{h}\right)+\lim {h \rightarrow 0}\left(\cos x \cdot \frac{\sin h}{h}\right) \ & =\sin x \cdot \underbrace{\lim {h \rightarrow 0} \frac{\cos h-1}{h}}{\text {limit } 0}+\cos x \cdot \underbrace{\lim {h \rightarrow 0} \frac{\sin h}{h}}{\text {limit } 1} \ & =\sin x \cdot 0+\cos x \cdot 1=\cos x . \ & \end{aligned}

$$\frac{d}{d x}(\sin x)=\cos x$$

## 数学代写|线性规划代写Linear Programming代考|Derivative of the Cosine Function

$$\cos (x+h)=\cos x \cos h-\sin x \sin h,$$

$$\begin{array}{rlrl} \frac{d}{d x}(\cos x) & =\lim {h \rightarrow 0} \frac{\cos (x+h)-\cos x}{h} & & \text { Derivative definition } \ & =\lim {h \rightarrow 0} \frac{(\cos x \cos h-\sin x \sin h)-\cos x}{h} & & \begin{array}{l} \text { Cosine angle } \ \text { sum identity } \end{array} \ & =\lim {h \rightarrow 0} \frac{\cos x(\cos h-1)-\sin x \sin h}{h} & \ & =\lim {h \rightarrow 0}\left(\cos x \cdot \frac{\cos h-1}{h}\right)-\lim {h \rightarrow 0}\left(\sin x \cdot \frac{\sin h}{h}\right) & & \ & =\cos x \cdot \lim {h \rightarrow 0} \frac{\cos h-1}{h}-\sin x \cdot \lim _{h \rightarrow 0} \frac{\sin h}{h} & & \ & =\cos x \cdot 0-\sin x \cdot 1 & & \begin{array}{l} \text { Example 5a } \ \text { and Theorem 7, } \end{array} \ & =-\sin x . & & \text { Section 2.4 } \end{array}$$

$$\frac{d}{d x}(\cos x)=-\sin x$$

## 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。

Posted on Categories:Linear Programming, 数学代写, 线性规划

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## 数学代写|线性规划代写Linear Programming代考|Two-Phase Simplex Method Without Artificial Variables

We will now present one version of the algorithm for determining the initial one a basic admissible solution that does not require the introduction of artificial variables. The main disadvantage of the previous method is the increase in the dimension of the linear problem. We consider the standard form of linear programming problem without restrictions on the sign of the coefficients $\beta_i$ in which the base is an unknown permissible solution. The existence of a permissible solution follows from the following lemma:

Lemma 2.6.1. Let $\beta_i=\alpha_{i 0}$ be a coefficient for some and. If $\alpha_{i j} \geq$ $0, j=1, \ldots, n$, then the set of admissible solutions is empty.

Proof. Given the conditions in the lemma, in the set of restriction, there is an equation in which the sum of the products is negative colors a negative number, which is impossible.

Let the set of solutions be linear programming is permissible and let $y$ be a basic impermissible solution with $q$ negative coordinates. Let (new numbering if necessary) achieved $y=\left(y_1, \ldots, y_m, 0, \ldots, 0\right)$ basic inadmissible solution such that $y_1, \ldots, y_q<0, q \leq m$, and $y_p \geq 0$ for $p>q$. The corresponding base is $K_1, \ldots, K_m$.



## 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。