数学代写|组合优化代写Combinatorial optimization代考|APM6664

2022年10月13日

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数学代写|组合优化代写Combinatorial optimization代考|Khachiyan’s Theorem

In this section we shall prove Khachiyan’s theorem: the ELLIPSOID METHOD can be applied to LINEAR PROGRAMMING in order to obtain a polynomial-time algorithm. Let us first prove that it suffices to have an algorithm for checking feasibility of linear inequality systems:

Proposition 4.16. Suppose there is a polynomial-time algorithm for the following problem: “Given a matrix $A \in \mathbb{Q}^{m \times n}$ and a vector $b \in \mathbb{Q}^m$, decide if ${x: A x \leq b}$ is empty.” Then there is a polynomial-time algorithm for LINEAR PROGRAMMING which finds an optimum basic solution if there exists one.

Proof: Let an LP $\max {c x: A x \leq b}$ be given. We first check if the primal and dual LPs are both feasible. If at least one of them is infeasible, we are done by Theorem 3.27. Otherwise, by Corollary 3.21, it is sufficient to find an element of ${(x, y): A x \leq b, y A=c, y \geq 0, c x=y b}$.

We show (by induction on $k$ ) that a solution of a feasible system of $k$ inequalities and $l$ equalities can be found by $k$ calls to the subroutine checking emptiness of polyhedra plus additional polynomial-time work. For $k=0$ a solution can be found easily by GAUSSIAN ELIMINATION (Corollary 4.11).

Now let $k>0$. Let $a x \leq \beta$ be an inequality of the system. By a call to the subroutine we check whether the system becomes infeasible by replacing $a x \leq \beta$ by $a x=\beta$. If so, the inequality is redundant and can be removed (cf. Proposition $3.8$ ). If not, we replace it by the equality. In both cases we reduced the number of inequalities by one, so we are done by induction.

If there exists an optimum basic solution, the above procedure generates one, because the final equality system contains a maximal feasible subsystem of $A x=b$.
Before we can apply the ElLIPSOID METHOD, we have to take care that the polyhedron is bounded and full-dimensional:

Proposition 4.17. (Khachiyan [1979], Gács and Lovász [1981]) Let $A \in \mathbb{Q}^{m \times n}$ and $b \in \mathbb{Q}^m$. The system $A x \leq b$ has a solution if and only if the system
$$A x \leq b+\epsilon \mathbb{\mathbb { 1 }}, \quad-R \mathbb{1} \leq x \leq R \mathbb{1}$$
has a solution, where 1 is the all-one vector, $\frac{1}{\epsilon}=2 n 2^{4(\operatorname{size}(A)+\operatorname{size}(b))}$ and $R=$ $1+2^{4(\operatorname{size}(A)+\operatorname{size}(b))}$

If $A x \leq b$ has a solution, then volume $\left(\left{x \in \mathbb{R}^n: A x \leq b+\epsilon \mathbb{1},-R \mathbb{1} \leq x \leq\right.\right.$ $R \mathbb{1}}) \geq\left(\frac{2 \epsilon}{n 2^{\operatorname{size}(A)}}\right)^n$.

数学代写|组合优化代写Combinatorial optimization代考|Separation and Optimization

The above method (in particular Proposition 4.16) requires that the polyhedron be given explicitly by a list of inequalities. However, a closer look shows that this is not really necessary. It is sufficient to have a subroutine which – given a vector $x-$ decides if $x \in P$ or otherwise returns a separating hyperplane, i.e. a vector $a$ such that $a x>\max {a y: y \in P}$. We shall prove this for full-dimensional polytopes; for the general (more complicated) case we refer to Grötschel, Lovász and Schrijver [1988] (or Padberg [1995]). The results in this section are due to Grötschel, Lovász and Schrijver [1981] and independently to Karp and Papadimitriou [1982] and Padberg and Rao [1981].

With the results of this section one can solve certain linear programs in polynomial time although the polytope has an exponential number of facets. Many examples will be discussed later in this book; see e.g. Corollary $12.22$ or Theorem 20.34. By considering the dual LP one can also deal with linear programs with a huge number of variables.

Let $P \subseteq \mathbb{R}^n$ be a full-dimensional polytope, or more generally, a fulldimensional bounded convex set. We assume that we know the dimension $n$ and two balls $B\left(x_0, r\right)$ and $B\left(x_0, R\right)$ such that $B\left(x_0, r\right) \subseteq P \subseteq B\left(x_0, R\right)$. But we do not assume that we know a linear inequality system defining $P$. In fact, this would not make sense if we want to solve linear programs with an exponential number of constraints in polynomial time, or even optimize linear objective functions over nonlinearly constrained convex sets.

Below we shall prove that, under some reasonable assumptions, we can optimize a linear function over a polyhedron $P$ in polynomial time (independent of the number of constraints) if we have a so-called separation oracle: a subroutine for the following problem: Note that such a vector $d$ exists if $P$ is a rational polyhedron or a compact convex set (cf. Exercise 21 of Chapter 3). Given a convex set $P$ by such a separation oracle, we look for an oracle algorithm using this as a black box. In an oracle algorithm we may ask the oracle at any time and we get a correct answer in one step. We can regard this concept as a subroutine whose running time we do not take into account. (In Chapter 15 we shall give a formal definition.)

组合优化代考

数学代写|组合优化代写组合优化代考|哈奇扬定理

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$$A x \leq b+\epsilon \mathbb{\mathbb { 1 }}, \quad-R \mathbb{1} \leq x \leq R \mathbb{1}$$

数学代写|组合优化代写combinatoroptimization代考|Separation and optimization

.拆分优化 以上方法(特别是命题4.16)要求多面体由一系列不等式显式给出。然而，仔细一看就会发现这并不是真的必要。有一个子程序就足够了——给定一个向量$x-$，它决定$x \in P$或以其他方式返回一个分离的超平面，即一个向量$a$，使得$a x>\max {a y: y \in P}$。我们将对全维多面体证明这一点;对于一般的(更复杂的)情况，我们参考Grötschel, Lovász和Schrijver[1988](或Padberg[1995])。本节的结果来自Grötschel, Lovász和Schrijver[1981]，独立来自Karp和Papadimitriou[1982]和Padberg和Rao [1981] 根据本节的结果，我们可以在多项式时间内求解某些线性规划，尽管多边形有指数数量的面。本书后面将讨论许多例子;参见e.g.推论$12.22$或定理20.34。通过考虑双LP，我们也可以处理具有大量变量的线性规划 设$P \subseteq \mathbb{R}^n$是一个全维多边形，或者更一般地说，一个全维有界凸集。我们假设我们知道尺寸$n$和两个球$B\left(x_0, r\right)$和$B\left(x_0, R\right)$，这样$B\left(x_0, r\right) \subseteq P \subseteq B\left(x_0, R\right)$。但我们不假设我们知道一个定义$P$的线性不等式系统。事实上，如果我们想要在多项式时间内解决具有指数数量约束的线性程序，或者甚至优化非线性约束凸集上的线性目标函数，这是没有意义的 下面我们将证明，在一些合理的假设下，我们可以在多项式时间内(独立于约束的数量)优化多面体$P$上的线性函数，如果我们有一个所谓的分离oracle:以下问题的子例程:请注意，如果$P$是一个有理多面体或一个紧凸集，则存在这样一个向量$d$(参见第3章练习21)。给定这样一个分离oracle的凸集$P$，我们寻找一个使用它作为黑箱的oracle算法。在oracle算法中，我们可以在任何时候向oracle提问，一步就能得到正确的答案。我们可以把这个概念看作一个子例程，我们不考虑它的运行时间。(在第十五章我们将给出一个正式的定义)

有限元方法代写

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MATLAB代写

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