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数学代写|线性规划代写Linear Programming代考|MATH340 Negative Transpose Property

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

In our discussion of duality in Chapter 5 , 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} & \text { 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}

数学代写|线性规划代写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}

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

$$\text { maximize } c^T x \quad \text { subject to } A x \leq b x \geq 0$$

$$\operatorname{minimize} b^T y \quad \text { subject to } A^T y \geq c y \geq 0 .$$

数学代写|线性规划代写Linear Programming代考|Sensitivity Analysis

$$\zeta=\zeta^{-} z_{\mathcal{N}}^T x_{\mathcal{N}} x_{\mathcal{B}} \quad=x_{\mathcal{B}}^*-B^{-1} N x_{\mathcal{N}}$$

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。