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# 数学代考|The Ellipsoid Method 运筹学代写

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## 运筹学代写

The basic ideas of the ellipsoid method stem from research done in the $1960 \mathrm{~s}$ and 1970 s mainly in the Soviet Union (as it was then called) by others who preceded Khachiyan. In essence, the idea is to enclose the region of interest in ever smaller ellipsoids.

The significant contribution of Khachiyan was to demonstrate that under certain assumptions, the ellipsoid method constitutes a polynomially bounded algorithm for linear programming.

The version of the method discussed here is really aimed at finding a point of a polyhedral set $\Omega$ given by a system of linear inequalities.
$$\Omega=\left{\mathbf{y} \in E^{m}: \mathbf{y}^{T} \mathbf{a}{j} \leq c{j}, j=1, \ldots n .\right}$$
Finding a point of $\Omega$ can be thought of as equivalent to solving a linear programming problem.
Two important assumptions are made regarding this problem:
(A1) There is a vector $\mathbf{y}{0} \in E^{m}$ and a scalar $R>0$ such that the closed ball $S\left(\mathbf{y}{0}, R\right)$ with center $\mathbf{y}{0}$ and radius $R$, that is $$\left{\mathbf{y} \in E^{m}:\left|\mathbf{y}-\mathbf{y}{0}\right| \leq R\right}$$
contains $\Omega$. If $\Omega$ is nonempty, there is a scalar $r>0$ such that $\Omega$ contains a ball of the form $S(\mathbf{y}, r)$ with center at some $\mathbf{y} \in \Omega$ and radius $r$. (This assumption implies that if $\Omega$ is nonempty, then it has a nonempty interior and its volume is at least $\operatorname{vol}(S(\mathbf{0}, r))$. $)^{2}$
Definition An ellipsoid in $E^{m}$ is a set of the form
$$E=\left{\mathbf{y} \in E^{m}:(\mathbf{y}-\mathbf{z})^{T} \mathbf{Q}(\mathbf{y}-\mathbf{z}) \leq 1\right},$$
where $\mathbf{z} \in E^{m}$ is a given point (called the center) and $\mathbf{Q}$ is a positive definite matrix (see Sect. A.4 of Appendix A) of dimension $m \times m$. This ellipsoid is denoted $E(\mathbf{z}, \mathbf{Q})$.
The unit sphere $S(\mathbf{0}, 1)$ centered at the origin $\mathbf{0}$ is a special ellipsoid with $\mathbf{Q}=\mathbf{I}$, the identity matrix.

The axes of a general ellipsoid are the eigenvectors of $\mathbf{Q}$ and the lengths of the axes are $\lambda_{1}^{-1 / 2}, \lambda_{2}^{-1 / 2}, \ldots, \lambda_{m}^{-1 / 2}$, where the $\lambda_{i}$ ‘s are the corresponding eigenvalues. It can be shown that the volume of an ellipsoid is
$$\operatorname{vol}(E)=\operatorname{vol}(S(\mathbf{0}, 1)) \Pi_{i=1}^{m} \lambda_{i}^{-1 / 2}=\operatorname{vol}(S(\mathbf{0}, 1)) \operatorname{det}\left(\mathbf{Q}^{-1 / 2}\right) .$$

Khachiyan 的重要贡献是证明在某些假设下，椭球方法构成了线性规划的多项式有界算法。

$$\Omega=\left{\mathbf{y} \in E^{m}: \mathbf{y}^{T} \mathbf{a}{j} \leq c{j}, j=1, \ ldots n .\right}$$

(A1) 有一个向量 $\mathbf{y}{0} \in E^{m}$ 和一个标量 $R>0$ 使得封闭球 $S\left(\mathbf{y}{ 0}，R\right)$，中心为 $\mathbf{y}{0}$，半径为 $R$，即 $$\left{\mathbf{y} \in E^{m}:\left|\mathbf{y}-\mathbf{y}{0}\right| \leq R\右}$$

$$E=\left{\mathbf{y} \in E^{m}:(\mathbf{y}-\mathbf{z})^{T} \mathbf{Q}(\mathbf{y}-\mathbf {z}) \leq 1\right},$$

$$\operatorname{vol}(E)=\operatorname{vol}(S(\mathbf{0}, 1)) \Pi_{i=1}^{m} \lambda_{i}^{-1 / 2}= \operatorname{vol}(S(\mathbf{0}, 1)) \operatorname{det}\left(\mathbf{Q}^{-1 / 2}\right) 。$$

## 什么是运筹学代写

• 确定需要解决的问题。
• 围绕问题构建一个类似于现实世界和变量的模型。
• 使用模型得出问题的解决方案。
• 在模型上测试每个解决方案并分析其成功。
• 实施解决实际问题的方法。

## 运筹学代写的三个特点

• 优化——运筹学的目的是在给定的条件下达到某一机器或者模型的最佳性能。优化还涉及比较不同选项和缩小潜在最佳选项的范围。
• 模拟—— 这涉及构建模型，以便在应用解决方案刀具体的复杂大规模问题之前之前尝试和测试简单模型的解决方案。
• 概率和统计——这包括使用数学算法和数据挖掘来发现有用的信息和潜在的风险，做出有效的预测并测试可能的解决方法。