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# 数学代考|Sensitivity and Complementary Slackness运筹学代写

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## 运筹学代写

The optimal values of the dual variables in a linear program can, as we have seen, be interpreted as prices. In this section this interpretation is explored in further detail.
Sensitivity
Suppose we denote the minimal value function of the right-hand-side data vector $\mathbf{b}$ in the linear program
\begin{aligned} z(\mathbf{b}):=& \text { minimize } \mathbf{c}^{T} \mathbf{x} \ & \text { subject to } \mathbf{A x}=\mathbf{b}, \mathbf{x} \geqslant \mathbf{0} \end{aligned}
(3.9)
$3.4$ Sensitivity and Complementary Slackness
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the optimal basis is $\mathbf{B}$ with corresponding solution $\left(\mathbf{x}{\mathbf{B}}, \mathbf{0}\right)$, where $\mathbf{x}{\mathbf{B}}=\mathbf{B}^{-1} \mathbf{b} . \mathbf{A}$ solution to the corresponding dual is $\mathbf{y}^{T}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1}$. Now, assuming nondegeneracy, small changes in the vector $\mathbf{b}$ will not cause the optimal basis to change. Thus for $\mathbf{b}+\Delta \mathbf{b}$ the optimal solution is $$\mathbf{x}=\left(\mathbf{x}{\mathbf{B}}+\boldsymbol{\Delta} \mathbf{x}{\mathbf{B}}, \mathbf{0}\right)$$ where $\boldsymbol{\Delta} \mathbf{x}{\mathbf{B}}=\mathbf{B}^{-1} \boldsymbol{\Delta} \mathbf{b}$. Thus the corresponding increment in the cost function is
This equation shows that $\mathbf{y}$ gives the sensitivity of the optimal cost with respect to small changes in the vector $\mathbf{b}$. In other words, if a new program were solved with $\mathbf{b}$ changed to $\mathbf{b}+\boldsymbol{\Delta} \mathbf{b}$, the change in the optimal value of the objective function would be $\mathbf{y}^{T} \boldsymbol{\Delta} \mathbf{b}$.
This interpretation of the dual vector $\mathbf{y}$ is intimately related to its interpretation as a vector of simplex multipliers. Since $y_{i}$ is the price of the unit vector $\mathbf{e}{i}$ when constructed from the basis $\mathbf{B}$, it directly measures the change in cost due to a change be considered as the marginal price of the component $b{i}$, since if $b_{i}$ is changed to $b_{i}+\Delta b_{i}$ the value of the optimal solution changes by $y_{i} \Delta b_{i}$.
If the linear program is interpreted as a diet problem, for instance, then $y_{i}$ is the maximum price per unit that the dietitian would be willing to pay for a small amount of the $i$ th nutrient, because decreasing the amount of nutrient that must be supplied by food will reduce the food bill by $\lambda_{i}$ dollars per unit. If, as another who must select levels $x_{1}, x_{2}, \ldots, x_{n}$ of $n$ production activities in order to meet certain required levels of output $b_{1}, b_{2}, \ldots, b_{m}$ while minimizing production costs, the $y_{i}$ ‘s are the marginal prices of the outputs. They show directly how much the theorem to summarize the observations.
Theorem The minimal value function $z$ (b) of linear program (3.9) is a convex function, and the optimal dual solution $\mathbf{y}^{}$ is a sub-gradient vector of the function at $\mathbf{b}$, written as $\nabla z(b)=\mathbf{y}^{}$.
Proof Let $\mathbf{x}^{1}$ and $\mathbf{x}^{2}$ be the two optimal solutions of (3.9) corresponding to two right-hand-side vectors $\mathbf{b}^{1}$ and $\mathbf{b}^{2}$, respectively. Then for any scalar $0 \leq \alpha \leq 1$, $\left(\alpha \mathbf{x}^{1}+(1-\alpha) \mathbf{x}^{2}\right)$ the minimal value
\begin{aligned} \left.\mathrm{b}^{2}\right) & \leq \mathbf{c}^{T}\left(\alpha \mathbf{x}^{1}+(1-\alpha) \mathbf{x}^{2}\right) \ &=\alpha \cdot \mathbf{c}^{T} \mathbf{x}^{1}+(1-\alpha) \cdot \mathbf{c}^{T} \mathbf{x}^{2} \ &=\alpha z\left(\mathbf{b}^{1}\right)+(1-\alpha) z\left(\mathbf{b}^{2}\right) \end{aligned}
which implies the first claim.
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3 Duality and Complementarity
Furthermore, let $\mathbf{y}^{1}$ be the optimal dual solution with $\mathbf{b}=\mathbf{b}^{1}$. Note that $\mathbf{y}^{1}$ remains feasible for the dual of the primal with $\mathbf{b}=\mathbf{b}^{2}$ because the dual feasible region is independent of change in b. Thus
$z\left(\mathbf{b}^{2}\right)-z\left(\mathbf{b}^{1}\right)=\mathbf{c}^{T} \mathbf{x}^{2}-\left(\mathbf{y}^{1}\right)^{T} \mathbf{b}^{1} \quad$ (the zero-duality gap theorem) $\geq\left(\mathbf{y}^{1}\right)^{T} \mathbf{b}^{2}-\left(\mathbf{y}^{1}\right)^{T} \mathbf{b}^{1} \quad$ (the weak duality lemma) $=\left(\mathbf{y}^{1}\right)^{T}\left(\mathbf{b}^{2}-\mathbf{b}^{1}\right)$,

\begin{aligned} z(\mathbf{b}):=& \text { 最小化 } \mathbf{c}^{T} \mathbf{x} \ & \text { 服从 } \mathbf{A x }=\mathbf{b}, \mathbf{x} \geqslant \mathbf{0} \end{对齐}
(3.9)
3.4 美元的灵敏度和互补的松弛度
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## 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。