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# 数学代考|The Central Path 运筹学代写

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## 运筹学代写

The concept underlying interior-point methods for linear programming is to use nonlinear programming techniques of analysis and methodology. The analysis is often based on differentiation of the functions defining the problem. Traditional linear programming does not require these techniques since the defining functions are linear. Duality in general nonlinear programs is typically manifested through Lagrange multipliers (which are called dual variables in linear programming). The analysis and algorithms of the remaining sections of the chapter use these nonlinear techniques. These techniques are discussed systematically in later chapters, so rather detail in their application to linear programming. It is expected that most readers are already familiar with the basic method for minimizing a function by setting its derivative to zero, and for incorporating constraints by introducing Lagrange multipliers. These methods are discussed in detail in Chaps. 11-15.
The computational algorithms of nonlinear programming are typically iterative in nature, often characterized as search algorithms. At any step with a given point, a direction for search is established and then a move in that direction is made to are systematically presented throughout the text. In this chapter, we use versions of Newton’s method as the search algorithm, but we postpone a detailed study of the method until later chapters.
(5.5) Not only have nonlinear methods improved linear programming, but interiorpoint methods for linear programming have been extended to provide new approaches to nonlinear programming. This chapter is intended to show how this merger of linear and nonlinear programming produces elegant and effective programming. Study of them here, even without all the detailed analysis, should provide good intuitive background for the more general manifestations. Consider a primal linear program in standard form
\begin{aligned} \text { (LP) } \operatorname{minimize} & \mathbf{c}^{T} \mathbf{x} \ \text { subject to } & \mathbf{A x}=\mathbf{b}, \mathbf{x} \geqslant \mathbf{0} . \end{aligned}
We denote the feasible region of this program by $\mathcal{F}{p}$. We assume that $\dot{\mathcal{F}}{p}={\mathbf{x}$ : $\mathbf{A x}=\mathbf{b}, \mathbf{x}>\mathbf{0}}$ is nonempty and the optimal solution set of the problem is bounded. Associated with this problem, we define for $\mu \geqslant 0$ the barrier problem
\begin{aligned} &\text { (BP) minimize } \mathbf{c}^{T} \mathbf{x}-\mu \sum_{j=1}^{n} \log x_{j} \ &\text { subject to } \mathbf{A x}=\mathbf{b}, \mathbf{x}>\mathbf{0} \end{aligned}
5 Interior-Point Methods
142 It is clear that $\mu=0$ corresponds to the original problem $(5.5)$. As $\mu \rightarrow \infty$, the solution approaches the analytic center of the feasible region (when it is bounded), since the barrier term swamps out $\mathbf{c}^{T} \mathbf{x}$ in the objective. As $\mu$ is varied continuously toward 0 , there is a path $\mathbf{x}(\mu)$ defined by the solution to (BP). This path $\mathbf{x}(\mu)$ is termed the primal central path. As $\mu \rightarrow 0$ this path converges to the analytic center of the op of (LP).
A strategy for solving (LP) is to solve (BP) for smaller and smaller values of $\mu$ and thereby approach a solution to (LP). This is indeed the basic idea of interior- point methods. point methods.
At any $\mu>0$, under the assumptions that we have made for problem (5.5), the necessary and sufficient conditions for a unique and bounded solution are obtained form the Lagrangian (see Chap. 11)
$$\mathbf{c}^{T} \mathbf{x}-\mu \sum_{j=1}^{n} \log x_{j}-\mathbf{y}^{T}(\mathbf{A} \mathbf{x}-\mathbf{b})$$
The derivatives with respect to the $x_{j}$ ‘s are set to zero, leading to the conditions
$$c_{j}-\mu / x_{j}-\mathbf{y}^{T} \mathbf{a}{j}=0, \text { for each } j$$ form the Lagrangia The derivatives or equivalently where as before $\mathbf{a}{j}$ is the $j$ th column of $\mathbf{A}, \mathbf{1}$ is the vector of 1’s, and $\mathbf{X}$ is the diagonal matrix whose diagonal entries are the components of $\mathbf{x}>\mathbf{0}$. Setting $s_{j}=\mu / x_{j}$ the complete set of conditions can be rewritten
\begin{aligned} \mathbf{x} \circ \mathrm{s} &=\boldsymbol{\mu} \mathbf{1} \ \mathbf{A x} &=\mathbf{b} \end{aligned}

(5.5) 非线性方法不仅改进了线性规划，而且线性规划的内点方法也得到了扩展，为非线性规划提供了新的方法。本章旨在展示线性和非线性编程的这种结合如何产生优雅而有效的编程。在这里对它们进行研究，即使没有所有详细的分析，也应该为更普遍的表现提供良好的直观背景。考虑标准形式的原始线性程序
\begin{aligned} \text { (LP) } \operatorname{minimize} & \mathbf{c}^{T} \mathbf{x} \ \text { 服从 } & \mathbf{A x}=\ mathbf{b}, \mathbf{x} \geqslant \mathbf{0} 。 \end{对齐}

$$\开始{对齐} &\text { (BP) 最小化 } \mathbf{c}^{T} \mathbf{x}-\mu \sum_{j=1}^{n} \log x_{j} \ &\text { 服从 } \mathbf{A x}=\mathbf{b}, \mathbf{x}>\mathbf{0} \end{对齐}$$
5 内点法
142 很明显，$\mu=0$ 对应于原始问题 $(5.5)$。由于 $\mu \rightarrow \infty$，解接近可行域的解析中心（当它有界时），因为障碍项淹没了 $\mathbf{c}^{T} \mathbf{x}$目标。由于 $\mu$ 不断地向 0 变化，因此 (BP) 的解定义了一条路径 $\mathbf{x}(\mu)$。这条路径 $\mathbf{x}(\mu)$ 被称为原始中心路径。由于 $\mu \rightarrow 0$，这条路径收敛到 (LP) 的运算的解析中心。

$$\mathbf{c}^{T} \mathbf{x}-\mu \sum_{j=1}^{n} \log x_{j}-\mathbf{y}^{T}(\mathbf{A} \mathbf{x}-\mathbf{b})$$

$$c_{j}-\mu / x_{j}-\mathbf{y}^{T} \mathbf{a}{j}=0, \text { for each } j$$ 形成 Lagrangia 导数或等价物 其中 $\mathbf{a}{j}$ 是 $\mathbf{A} 的第$j$列，\mathbf{1}$ 是 1 的向量，$\mathbf{X}$ 是对角矩阵，其对角元素是 $\mathbf{x}>\mathbf{0}$ 的分量。设置 $s_{j}=\mu / x_{j}$ 可以重写完整的条件集
$$\开始{对齐} \mathbf{x} \circ \mathrm{s} &=\boldsymbol{\mu} \mathbf{1} \ \mathbf{A x} &=\mathbf{b} \end{对齐}$$

## 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。