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

In the previous sections we considered integral polyhedra. For general polyhedra $P$ we have $P \supset P_I$. If we want to solve an integer linear program $\max \left{c x: x \in P_I\right}$, it is a natural idea to cut off certain parts of $P$ such that the resulting set is again a polyhedron $P^{\prime}$ and we have $P \supset P^{\prime} \supset P_I$. Hopefully $\max \left{c x: x \in P^{\prime}\right}$ is attained by an integral vector; otherwise we can repeat this cutting-off procedure for $P^{\prime}$ in order to obtain $P^{\prime \prime}$ and so on. This is the basic idea behind the cutting plane method, first proposed for a special problem (the TSP) by Dantzig, Fulkerson and Johnson [1954].

Gomory $[1958$, 1963] found an algorithm which solves general integer programs with the cutting plane method. In this section we restrict ourselves to the theoretical background. Gomory’s algorithm does not run in polynomial time and has little practical relevance in its original form. However, the general idea of cutting plane methods is used very often and successfully in practice. We shall discuss this in Section 21.6. The following presentation is mainly based on Schrijver [1986].
Definition 5.29. Let $P \subseteq \mathbb{R}^n$ be a convex set. Then we define
$$P^{\prime}:=\bigcap_{P \subseteq H} H_I,$$
where the intersection ranges over all rational affine half-spaces $H={x: c x \leq \delta}$ containing $P$. We set $P^{(0)}:=P$ and $P^{(i+1)}:=\left(P^{(i)}\right)^{\prime} . P^{(i)}$ is called the $i$-th Gomory-Chvátal-truncation of $P$.

If $P$ is a rational polyhedron, then $P^{\prime}$ is also a rational polyhedron, as we will show below. Therefore $P \supseteq P^{\prime} \supseteq P^{(2)} \supseteq \cdots \supseteq P_I$ and $P_I=\left(P^{\prime}\right)_I$. We remark that Dadush, Dey, and Vielma [2011] proved that if $P$ is any compact convex set, then $P^{\prime}$ is a rational polytope.
Proposition 5.30. For any rational polyhedron $P={x: A x \leq b}$,
$$P^{\prime}={x: u A x \leq\lfloor u b\rfloor \text { for all } u \geq 0 \text { with } u A \text { integral }} .$$
Proof: We first make two observations. For any rational affine half-space $H=$ ${x: c x \leq \delta}$ with $c$ integral we obviously have
$$H^{\prime}=H_I \subseteq{x: c x \leq\lfloor\delta\rfloor} .$$
If in addition the components of $c$ are relatively prime (i.e., their greatest common divisor is 1), we claim that
$$H^{\prime}=H_I={x: c x \leq\lfloor\delta\rfloor} .$$

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

Suppose we have an integer linear program $\max \left{c x: A x \leq b, A^{\prime} x \leq b^{\prime}, x\right.$ integral $}$ that becomes substantially easier to solve when the constraints $A^{\prime} x \leq b^{\prime}$ are omitted. We write $Q:=\left{x \in \mathbb{Z}^n: A x \leq b\right}$ and assume that we can optimize linear objective functions over $Q$ (for example if $\operatorname{conv}(Q)={x: A x \leq b}$ ). Lagrangean relaxation is a technique to get rid of some troublesome constraints (in our case $A^{\prime} x \leq b^{\prime}$ ). Instead of explicitly enforcing the constraints we modify the objective function in order to punish infeasible solutions. More precisely, instead of optimizing
$$\max \left{c^{\top} x: A^{\prime} x \leq b^{\prime}, x \in Q\right}$$
we consider, for any vector $\lambda \geq 0$,

$$L R(\lambda):=\max \left{c^{\top} x+\lambda^{\top}\left(b^{\prime}-A^{\prime} x\right): x \in Q\right} .$$
For each $\lambda \geq 0, L R(\lambda)$ is an upper bound for (5.7) which is relatively easy to compute. (5.8) is called the Lagrangean relaxation of (5.7), and the components of $\lambda$ are called Lagrange multipliers.

Lagrangean relaxation is a useful technique in nonlinear programming; but here we restrict ourselves to (integer) linear programming.

Of course one is interested in as good an upper bound as possible. Observe that $\lambda \mapsto L R(\lambda)$ is a convex function. The following procedure (called subgradient optimization) can be used to minimize $L R(\lambda)$ :

Start with an arbitrary vector $\lambda^{(0)} \geq 0$. In iteration $i$, given $\lambda^{(i)}$, find a vector $x^{(i)}$ maximizing $c^{\top} x+\left(\lambda^{(i)}\right)^{\top}\left(b^{\prime}-A^{\prime} x\right)$ over $Q$ (i.e. compute $L R\left(\lambda^{(i)}\right)$ ). Note that $L R(\lambda)-L R\left(\lambda^{(i)}\right) \geq\left(\lambda-\lambda^{(i)}\right)^{\top}\left(b^{\prime}-A^{\prime} x^{(i)}\right)$ for all $\lambda$, i.e. $b^{\prime}-A^{\prime} x^{(i)}$ is a subgradient of $L R$ at $\lambda^{(i)}$. Set $\lambda^{(i+1)}:=\max \left{0, \lambda^{(i)}-t_i\left(b^{\prime}-A^{\prime} x^{(i)}\right)\right}$ for some $t_l>0$. Polyak [1967] showed that if $\lim {t \rightarrow \infty} t_l=0$ and $\sum{i=0}^{\infty} t_i=\infty$, then $\lim _{i \rightarrow \infty} L R\left(\lambda^{(i)}\right)=\min {L R(\lambda): \lambda \geq 0}$. For more results on the convergence of subgradient optimization, see Goffin [1977] and Anstreicher and Wolsey [2009].

# 组合优化代考

## 数学代写|组合优化代写组合优化代考|切割平面

.切割平面 .切割平面

Gomory $[1958$, 1963]发现了一个用切割平面法求解一般整数程序的算法。在这一节中，我们只讨论理论背景。Gomory的算法不是在多项式时间内运行的，它的原始形式几乎没有实际意义。然而，切割平面法的一般思想在实践中得到了广泛的应用并取得了成功。我们将在第21.6节对此进行讨论。下面的介绍主要基于Schrijver[1986]。

$$P^{\prime}:=\bigcap_{P \subseteq H} H_I,$$
，其中交集范围是所有包含$P$的有理仿射半空间$H={x: c x \leq \delta}$。我们将$P^{(0)}:=P$和$P^{(i+1)}:=\left(P^{(i)}\right)^{\prime} . P^{(i)}$称为$P$的$i$ -th Gomory-Chvátal-truncation。

$$P^{\prime}={x: u A x \leq\lfloor u b\rfloor \text { for all } u \geq 0 \text { with } u A \text { integral }} .$$

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