网络多面体上双线性项的凸化
Convexification of bilinear terms over network polytopes
AI总结：
针对网络多面体上双线性约束 McCormick 松弛对偶界差的问题，提出基于网络特殊树和森林结构的凸包定义面不等式构造方法，并通过计算实验验证其有效性。
摘要：
众所周知，针对双线性约束 $z=xy$ 的 McCormick 松弛在 $x$ 和 $y$ 的盒子域上给出了凸包。在网络应用中，当双线性变量的域由网络多面体描述时，McCormick 松弛（也称为线性化）无法提供凸包，且常导致较弱的对偶界。我们研究了包含双线性约束 $z_{i,j}=x_iy_j$ 的集合的凸包，其中 $x_i$ 表示网络多面体中的弧流变量，$y_j$ 位于单纯形中。对于单纯形包含单个 $y$ 变量的情形，我们提出了一种系统化程序，在变量的原始空间中获取上述集合的凸包，并证明凸包的所有定义面不等式均可通过识别底层网络中的特殊树结构显式获得。对于单纯形包含多个 $y$ 变量的推广情形，我们设计了一种构造性程序，获取底层双线性集合凸包的一类重要定义面不等式，该类不等式由底层网络中的特殊森林结构刻画。我们展示了计算实验以评估所提方法的有效性。
英文摘要：
It is well-known that the McCormick relaxation for the bilinear constraint $z=xy$ gives the convex hull over the box domains for $x$ and $y$. In network applications where the domain of bilinear variables is described by a network polytope, the McCormick relaxation, also referred to as linearization, fails to provide the convex hull and often leads to poor dual bounds. We study the convex hull of the set containing bilinear constraints $z_{i,j}=x_iy_j$ where $x_i$ represents the arc-flow variable in a network polytope, and $y_j$ is in a simplex. For the case where the simplex contains a single $y$ variable, we introduce a systematic procedure to obtain the convex hull of the above set in the original space of variables, and show that all facet-defining inequalities of the convex hull can be obtained explicitly through identifying a special tree structure in the underlying network. For the generalization where the simplex contains multiple $y$ variables, we design a constructive procedure to obtain an important class of facet-defining inequalities for the convex hull of the underlying bilinear set that is characterized by a special forest structure in the underlying network. Computational experiments are presented to evaluate the effectiveness of the proposed methods.
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