# 数学代写|组合优化代写Combinatorial optimization代考|ORIE6334

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## 数学代写|组合优化代写Combinatorial optimization代考|Interior Point Algorithm

Although three algorithms have been presented in previous sections for LP, they all are running not in polynomial-time. The reason is that in general, the number of extreme points (i.e., vertices) of feasible domain is exponential. In this section, we present a polynomial-time algorithm which moves from a feasible point to another feasible point in the interior of the feasible domain. Hence, it is called the interior point algorithm.
First, we assume that the LP is in the following form:
\begin{aligned} \min & c x \ \text { subject to } & A x=b \ & x \geq 0 \end{aligned}
where without loss of generality, assume

• $A$ is $m \times n$ matrix with full rank $m$, i.e., $\left(A A^T\right)^{-1}$ exists,
• the feasible domain is bounded,
• the minimum value of objective function is zero, and
• an initial feasible solution $x^0$ is available.

Actually, from LP (P) and its dual (D’) in Sect. 6.6, we can obtain LP as follows:
\begin{aligned} \min & w-z=y b-c x \ \text { subject to } & y A \geq c \ & A x=b \ & x \geq 0 . \end{aligned}
This LP has zero as the objective function value of optimal solution. Modify it into standard form. Then we will obtain an LP satisfying our assumptions.

In order to keep moving in the interior of feasible domain, we need to replace our linear objective function by a nonlinear one, called the potential function,
$$f(x)=q \log (c x)-\sum_{i=1}^n \log x_i$$
which contains a barrier terms $\sum_{i=1}^n \log x_i$ to keep moving away from boundaries. Moreover, for simplicity of notation, we assume the base of $\log$ is 2 in this section.

## 数学代写|组合优化代写Combinatorial optimization代考|Polyhedral Techniques

A polyhedron is a set of all points bounded by a system of linear inequalities and linear equalities. For example, the feasible domain of each LP is a polyhedron. Since LP is polynomial-time solvable, we may use LP as a tool for solving other combinatorial optimization problems in the following way: Find a polyhedron $P$ such that every vertex of $P$ is a feasible solution of considered combinatorial optimization problem, so that the problem is transformed into an LP. This method is called the polyhedral technique. In this section, we introduce this technique through a few examples.
The first example is the maximum weight bipartite matching.
Problem 6.8.1 (Maximum Weight Bipartite Matching) Given a bipartite graph $\left(V_1, V_2, E\right)$ with nonnegative edge weight $w: E \rightarrow R_{+}$, find a matching with maximum total edge weight.
The polyhedron of bipartite matching is defined as follows:
Definition 6.8.2 (Polyhedron of Bipartite Matching) For each matching $M$, define $\chi_M \in{0,1}^{|\mathrm{E}|}$ by
$$\chi_M(e)= \begin{cases}1 & \text { if } e \in M, \ 0 & \text { otherwise }\end{cases}$$
Define the polyhedron of bipartite match $P_{b m a t c h}$ to be the convex hull of $\chi_M$ for $M$ over all matchings that is
$$P_{\text {bmatch }}=\left{\sum_{M \in \mathcal{M}} \alpha_M \chi_M \mid \alpha_M \geq 0, \sum_{M \in \mathcal{M}} \alpha_M=1\right}$$
where $\mathcal{M}$ the set of all matchings.
Note that a bounded polyhedron is also called a polytope. Thus, the convex hull of a finite number of vectors must be a bounded region. Therefore, $P_{b m a t c h}$ is also called the polytope of bipartite matching.
Let $\delta(v)$ denote the set of edges incident to vertex $v$.

# 组合优化代考

## 数学代写|组合优化代写Combinatorial optimization代考|Initial Feasible Basis

(1) 成本函数值 $w$ 降为 0 ，所有的人工变量都从可行的基 础上去除。在这种情况下，最终可行基可以作为原始 LP 中的初始可行基。
(2)成本函数达到负最大值。在这种情况下，原LP无可行 解。
(3) 成本函数值 $w$ 降为 0 ；然而，在可行的基础上有一个 人为变量 $y_i$ 让 $b_i$ 和 $a_{i j}$ 表示最后时刻的约束系数。在这种 情况下，我们必须有 $y_i=b_i=0$ ；否则， $w=e y>0$ 。请注意，存在一个变量 $x_j$ 使得 $a_{i j} \neq 0$ 元素来移动 $y_i$ 从可行的基础上移出并移入 $x_j$ ，保留成本 函数值 0 。当所有的人工变量都从可行的基础上移出 后，这种情况就简化为情况 (1)。

## 数学代写|组合优化代写Combinatorial optimization代考|Primal-Dual Algorithm

$(P): \quad \max z=c x$ subject to $A x \quad=b x \geq$

$(D): \quad \min w=y b$ subject to $y A \geq c$,

$$y a_j>c_j \Rightarrow x_j=0,$$
or
$$x_j>0 \Rightarrow y a_j=c_j$$

J(y)=Vleft{i Imid y a_j=c_jlright $} y$ 是最优的当且仅当存在一 个原始可行解 $x$ 满足与 $y$ 的互补松他条件，即以下 LP 具 有最优值:
$$(R P): \quad \max -\sum_{i=1}^m u_i \text { subject to } \quad \sum_{j \in J(y)} a_{i j} x_j$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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