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

2023年2月3日

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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

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