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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 various definitions of the central path directly suggest corresponding strategies for solution of a linear program. We outline three general approaches here: the primal barrier or path-following method, the primal-dual path-following method, and the primal-dual potential reduction method, although the details of their implementation and analysis must be deferred to later chapters after study of general nonlinear methods. Table $5.1$ depicts these solution strategies and the simplex methods described in Chaps. 4 and 3 with respect to how they meet the three optimality conditions: Primal Feasibility, Dual Feasibility, and Zero-Duality during the iterative process.

For example, the primal simplex method keeps improving a primal feasible solution, maintains the zero-duality gap (complementarity slackness condition) and moves toward dual feasibility; while the dual simplex method keeps improving a dual feasible solution, maintains the zero-duality gap (complementarity condition) and moves toward primal feasibility (see Sect. 3.3). The primal barrier method keeps improving a primal feasible solution and moves toward dual feasibility and primal and dual feasible solution pair and move toward complementarity.
algorithms complementarity; and the primal and dual feasible solution Primal Barrier Method A direct approach is to use
Primal Barrier Method A direct approach is to use the barrier construction and solve the problem $$\text { minimize } \mathbf{c}^{T} \mathbf{x}-\mu \sum_{j=1}^{n} \log x_{j}$$
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for a very small value of $\mu$. In fact, if we desire to reduce the duality gap to $\varepsilon$ it is only necessary to solve the problem for $\mu=\varepsilon / n$. Unfortunately, when $\mu$ is small, conditions are nearly singular. This makes it difficult to directly solve the problem for small $\mu$.

An overall strategy, therefore, is to start with a moderately large $\mu$ (say $\mu=$ 100) and solve that problem approximately. The corresponding solution is a point approximately on the primal central path, but it is likely to be quite distant from the point corresponding to the limit of $\mu \rightarrow 0$. However this solution point at $\mu=100$ can be used as the starting point for the problem with a slightly smaller $\mu$, for this point is likely to be close to the solution of the new problem. The value of $\mu$ might be reduced at each stage by a specific factor, giving $\mu_{k+1}=\gamma \mu_{k}$, where $\gamma$ is a fixed positive parameter less than one and $k$ is the stage count.

If the strategy is begun with a value $\mu_{0}$, then at the $k$ th stage we have $\mu_{k}=\gamma^{k} \mu_{0}$. Hence to reduce $\mu_{k} / \mu_{0}$ to below $\varepsilon$, requires
$$k=\frac{\log \varepsilon}{\log \gamma}$$
stages.

Primal Barrier Method 一种直接的方法是使用屏障构造并解决问题 $$\text { minimum } \mathbf{c}^{T} \mathbf{x}-\mu \sum_{j=1}^{n } \log x_{j}$$
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$$k=\frac{\log \varepsilon}{\log \gamma}$$

## 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。