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# 数学代考|The Analytic Center 运筹学代写

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## 运筹学代写

The new interior-point algorithms introduced by Karmarkar move by successive steps inside the feasible region. It is the interior of the feasible set rather than the vertices and edges that plays a dominant role in this type of algorithm. In fact, these algorithms purposely avoid the edges of the set, only eventually converging to one as a solution.

Our study of these algorithms begins in the next section, but it is useful at this point to introduce a concept that definitely focuses on the interior of a set, termed the set’s analytic center. As the name implies, the center is away from the edge. In addition, the study of the analytic center introduces a special structure, termed a barrier or potential that is fundamental to interior-point methods.
Consider a set $\mathcal{S}$ in a subset of $\mathcal{X}$ of $E^{n}$ defined by a group of inequalities as
$$\mathcal{S}=\left{\mathbf{x} \in \mathcal{X}: g_{j}(\mathbf{x}) \geqslant 0, j=1,2, \ldots, m\right}$$
5 Interior-Point Methods
138 and assume that the functions $g_{j}$ are continuous. $\mathcal{S}$ has a nonempty interior $\mathcal{S}=$ $\left{\mathbf{x} \in \mathcal{X}: g_{j}(\mathbf{x})>0\right.$, all $\left.j\right}$. Associated with this definition of the set is the potential function
$$\psi(\mathbf{x})=-\sum_{j=1}^{m} \log g_{j}(\mathbf{x})$$
defined on $\mathcal{S}$.
The analytic center of $\mathcal{S}$ is the vector (or set of vectors) that minimizes the potential; that is, the vector (or vectors) that solve
$$\min \psi(\mathbf{x})=\min \left{-\sum_{j=1}^{m} \log g_{j}(\mathbf{x}): \mathbf{x} \in \mathcal{X}, g_{j}(\mathbf{x})>0 \text { for each } j\right}$$
Example 1 (A Cube) Consider the set $\mathcal{S}$ defined by $x_{i} \geqslant 0$, $\left(1-x_{i}\right) \geqslant 0$, for $i=1,2, \ldots, n$. This is $\mathcal{S}=[0,1]^{n}$, the unit cube in $E^{n}$. The analytic center can be found by differentiation to be $x_{i}=1 / 2$, for all $i$. Hence, the analytic center is identical to what one would normally call the center of the unit cube.

In general, the analytic center depends on how the set is defined-on the particular inequalities used in the definition. For instance, the unit cube is also defined by the inequalities $x_{i} \geqslant 0$, $\left(1-x_{i}\right)^{d} \geqslant 0$ with odd $d>1$. In this case the solution is $x_{i}=1 /(d+1)$ for all $i$. For large $d$ this point is near the inner corner of the unit cube.

Also, the addition of redundant inequalities can change the location of the analytic center. For example, repeating a given inequality will change the center’s location.

There are several sets associated with linear programs for which the analytic center is of particular interest. One such set is the feasible region itself. Another is the set of optimal solutions. There are also sets associated with dual and primal-dual formulations. All of these are related in important ways.

Let us illustrate by considering the analytic center associated with a bounded polytope $\Omega$ in $E^{m}$ represented by $n(>m)$ linear inequalities; that is,
$$\Omega=\left{\mathbf{y} \in E^{m}: \mathbf{c}^{T}-\mathbf{y}^{T} \mathbf{A} \geqslant \mathbf{0}\right}$$
where $\mathbf{A} \in E^{m \times n}$ and $\mathbf{c} \in E^{n}$ are given and $\mathbf{A}$ has rank $m$. Denote the interior of $\Omega$ by
$$\stackrel{\Omega}{\Omega}=\left{\mathbf{y} \in E^{m}: \mathbf{c}^{T}-\mathbf{y}^{T} \mathbf{A}>\mathbf{0}\right}$$

Karmarkar 引入的新内点算法在可行区域内连续移动。在这类算法中起主导作用的是可行集的内部而不是顶点和边。事实上，这些算法故意避开集合的边缘，最终收敛到一个作为解决方案。

$$\mathcal{S}=\left{\mathbf{x} \in \mathcal{X}: g_{j}(\mathbf{x}) \geqslant 0, j=1,2, \ldots, m\right }$$
5 内点法
138 并假设函数 $g_{j}$ 是连续的。 $\mathcal{S}$ 有一个非空内部 $\mathcal{S}=$ $\left{\mathbf{x} \in \mathcal{X}: g_{j}(\mathbf{x})>0 \right.$，所有 $\left.j\right}$。与集合的这个定义相关的是势函数
$$\psi(\mathbf{x})=-\sum_{j=1}^{m} \log g_{j}(\mathbf{x})$$

$\mathcal{S}$ 的解析中心是使势最小化的向量（或向量集）；即求解的向量（或多个向量）
$$\min \psi(\mathbf{x})=\min \left{-\sum_{j=1}^{m} \log g_{j}(\mathbf{x}): \mathbf{x} \在 \mathcal{X} 中，g_{j}(\mathbf{x})>0 \text { 对于每个 } j\right}$$

$$\Omega=\left{\mathbf{y} \in E^{m}: \mathbf{c}^{T}-\mathbf{y}^{T} \mathbf{A} \geqslant \mathbf{0 }\正确的}$$

$$\stackrel{\Omega}{\Omega}=\left{\mathbf{y} \in E^{m}: \mathbf{c}^{T}-\mathbf{y}^{T} \mathbf{ A}>\mathbf{0}\right}$$

## 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。