# 数学代写|组合优化代写Combinatorial optimization代考|COMP567

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## 数学代写|组合优化代写Combinatorial optimization代考|Gaussian Elimination

The most important algorithm in linear algebra is the so-called Gaussian elimination. It has been applied by Gauss but was known much earlier (see Schrijver [1986] for historical notes). Gaussian elimination is used to determine the rank of a matrix, to compute the determinant and to solve a system of linear equations. It occurs very often as a subroutine in linear programming algorithms; e.g. in (1) of the SIMPLEX ALGORITHM.

Given a matrix $A \in \mathbb{Q}^{m \times n}$, our algorithm for Gaussian Elimination works with an extended matrix $Z=(B C) \in \mathbb{Q}^{m \times(n+m)}$; initially $B=A$ and $C=I$. The algorithm transforms $B$ to the form $\left(\begin{array}{ll}I \ 0 & R \ 0\end{array}\right)$ by the following elementary operations: permuting rows and columns, adding a multiple of one row to another row, and (in the final step) multiplying rows by nonzero constants. At each iteration $C$ is modified accordingly, such that the property $C \tilde{A}=B$ is maintained throughout where $\tilde{A}$ results from $A$ by permuting rows and columns.

The first part of the algorithm, consisting of (2) and (3), transforms $B$ to an upper triangular matrix. Consider for example the matrix $Z$ after two iterations; it has the form

If $z_{33} \neq 0$, then the next step just consists of subtracting $\frac{z_{i 3}}{z_{33}}$ times the third row from the $i$-th row, for $i=4, \ldots, m$. If $z_{33}=0$ we first exchange the third row and/or the third column with another one. Note that if we exchange two rows, we have to exchange also the two corresponding columns of $C$ in order to maintain the property $C \tilde{A}=B$. To have $\tilde{A}$ available at each point we store the permutations of the rows and columns in variables $\operatorname{row}(i), i=1, \ldots, m$ and $\operatorname{col}(j), j=1, \ldots, n$. Then $\tilde{A}=\left(A_{\text {row }}(i), \operatorname{col}(j)\right)_{i \in{1, \ldots, m}, j \in{1, \ldots, n}}$.

## 数学代写|组合优化代写Combinatorial optimization代考|The Ellipsoid Method

In this section we describe the so-called ellipsoid method, developed by Iudin and Nemirovskii [1976] and Shor [1977] for nonlinear optimization. Khachiyan [1979] observed that it can be modified in order to solve LPs in polynomial time. Most of our presentation is based on (Grötschel, Lovász and Schrijver [1981]), (Bland, Goldfarb and Todd [1981]) and the book of Grötschel, Lovász and Schrijver [1988], which is also recommended for further study.

The idea of the ellipsoid method is very roughly the following. We look for either a feasible or an optimum solution of an LP. We start with an ellipsoid which we know a priori to contain the solutions (e.g. a large ball). At each iteration $k$, we check if the center $x_k$ of the current ellipsoid is a feasible solution. Otherwise, we take a hyperplane containing $x_k$ such that all the solutions lie on one side of this hyperplane. Now we have a half-ellipsoid which contains all solutions. We take the smallest ellipsoid completely containing this half-ellipsoid and continue.

Definition 4.12. An ellipsoid is a set $E(A, x)=\left{z \in \mathbb{R}^n:(z-x)^{\top} A^{-1}(z-x) \leq\right.$ 1 ) for some symmetric positive definite $n \times n$-matrix $A$.

Note that $B(x, r):=E\left(r^2 I, x\right)$ (with $I$ being the $n \times n$ unit matrix) is the $n$-dimensional Euclidean ball with center $x$ and radius $r$.
The volume of an ellipsoid $E(A, x)$ is known to be
$$\text { volume }(E(A, x))=\sqrt{\operatorname{det} A} \text { volume }(B(0,1))$$
(see Exercise 7). Given an ellipsoid $E(A, x)$ and a hyperplane ${z: a z=a x}$, the smallest ellipsoid $E\left(A^{\prime}, x^{\prime}\right)$ containing the half-ellipsoid $E^{\prime}={z \in E(A, x)$ : $a z \geq a x}$ is called the Löwner-John ellipsoid of $E^{\prime}$ (see Figure 4.1). It can be computed by the following formulas:
\begin{aligned} A^{\prime} &=\frac{n^2}{n^2-1}\left(A-\frac{2}{n+1} b b^{\top}\right), \ x^{\prime} &=x+\frac{1}{n+1} b, \ b &=\frac{1}{\sqrt{a^{\top} A a}} A a . \end{aligned}

# 组合优化代考

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## 数学代写|组合优化代写combinatoroptimization代考|The Ellipsoid Method

. The Ellipsoid Method . The Ellipsoid Method

$B(x, r):=E\left(r^2 I, x\right)$ (与 $I$ 成为 $n \times n$ 单位矩阵)是 $n$有中心的-维欧几里得球 $x$ 半径 $r$.

$$\text { volume }(E(A, x))=\sqrt{\operatorname{det} A} \text { volume }(B(0,1))$$
(见练习7) $E(A, x)$ 还有一个超平面 ${z: a z=a x}$，最小的椭球体 $E\left(A^{\prime}, x^{\prime}\right)$ 包含半椭球体的 $E^{\prime}={z \in E(A, x)$ : $a z \geq a x}$ 叫做Löwner-John椭球 $E^{\prime}$ (见图4.1)。其计算公式如下:
\begin{aligned} A^{\prime} &=\frac{n^2}{n^2-1}\left(A-\frac{2}{n+1} b b^{\top}\right), \ x^{\prime} &=x+\frac{1}{n+1} b, \ b &=\frac{1}{\sqrt{a^{\top} A a}} A a . \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。

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