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# 数学代考|The Simplex Method Is Not Polynomial-Time 运筹学代写

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## 运筹学代写

When the simplex method is used to solve a linear program in standard form with coefficient matrix $\mathbf{A} \in E^{m \times n}, \mathbf{b} \in E^{m}$ and $\mathbf{c} \in E^{n}$, the number of pivot steps to solve the problem starting from a basic feasible solution is typically a small multiple of $m$ : usually between $2 m$ and $3 m$. In fact, Dantzig observed that for problems with $m \leq 50$ and $n \leq 200$ the number of iterations is ordinarily less than $1.5 m$.

At one time researchers believed-and attempted to prove-that the simplex algorithm (or some variant thereof) always requires a number of iterations that is bounded by a polynomial expression in the problem size. That was until Victor Klee and George Minty exhibited a class of linear programs each of which requires an
$5.2$ * The Simplex Method Is Not Polynomial-Time
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One form of the Klee-Minty example is
$\operatorname{maximize} \sum_{j=1}^{n} 10^{n-j} x_{j}$
subject to $2 \sum_{j=1}^{i-1} 10^{i-j} x_{j}+x_{i} \leq 100^{i-1} i=1, \ldots, n$
$x_{j} \geq 0 \quad j=1, \ldots, n .$
The problem above is easily cast as a linear program in standard form.
A specific case is that for $n=3$, giving
$$\text { maximize } 100 x_{1}+10 x_{2}+x_{3}$$

One form of the Klee-Minty example is
\begin{aligned} &\text { maximize } \sum_{j=1}^{n} 10^{n-j} x_{j} \ &\text { subject to } 2 \sum_{j=1}^{i-1} 10^{i-j} x_{j}+x_{i} \leq 100^{i-1} i=1, \ldots, n \ &x_{j} \geq 0 \quad j=1, \ldots, n \end{aligned}
The problem above is easily cast as a linear program in standard form.
A specific case is that for $n=3$, giving
maximize $100 x_{1}+10 x_{2}+x_{3}$
$200 x_{1}+20 x_{2}+x_{3} \leq 10,000$
$x_{1} \geqslant 0, x_{2} \geqslant 0, x_{3} \geqslant 0 .$
In this case, we have three constraints and three variables (along with their nonnegativity constraints). After adding slack variables, the problem is in standard form. The system has $m=3$ equations and $n=6$ nonnegative variables. It can be verified that it takes $2^{3}-1=7$ pivot steps to solve the problem with the simplex method when at each step the pivot column is chosen to be the one with the largest (because this a maximization problem) reduced cost. (See Exercise 1.)

The general problem of the class (1) takes $2^{n}-1$ pivot steps and this is in fact the number of vertices minus one (which is the starting vertex). To get an idea of how bad this can be, consider the case where $n=50$. We have $2^{50}-1 \approx 10^{15}$. In a year with 365 days, there are approximately $3 \times 10^{7} \mathrm{~s}$. If a computer ran continuously, performing a million pivots of the simplex algorithm per second, it would take approximately
$$\frac{10^{15}}{3 \times 10^{7} \times 10^{6}} \approx 33 \text { years }$$
to solve a problem of this class using the greedy pivot selection rule.
Although it is not polynomial in the worst case, the simplex method remains one of major solvers for linear programming. In fact, the method has been recently proved to be (strongly) polynomial for solving the Markov Decision Process with any fixed discount rate.

$5.2$ * 单纯形法不是多项式时间
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Klee-Minty 示例的一种形式是
$\operatorname{最大化} \sum_{j=1}^{n} 10^{n-j} x_{j}$

$x_{j} \geq 0 \quad j=1, \ldots, n .$

$$\text { 最大化 } 100 x_{1}+10 x_{2}+x_{3}$$

Klee-Minty 示例的一种形式是
$$\开始{对齐} &\text { 最大化 } \sum_{j=1}^{n} 10^{n-j} x_{j} \ &\text { 服从 } 2 \sum_{j=1}^{i-1} 10^{ij} x_{j}+x_{i} \leq 100^{i-1} i=1, \ldots , n \ &x_{j} \geq 0 \quad j=1, \ldots, n \end{对齐}$$

$200 x_{1}+20 x_{2}+x_{3} \leq 10,000$
$x_{1} \geqslant 0, x_{2} \geqslant 0, x_{3} \geqslant 0 .$

$$\frac{10^{15}}{3 \times 10^{7} \times 10^{6}} \大约 33 \text { 年 }$$

## 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。