主元规则与单调路径的多面体几何
The Polyhedral Geometry of Pivot Rules and Monotone Paths
摘要：
受单纯形法性能分析的启发，我们研究了线性规划主元规则族的行为。我们引入规范化权重枢轴规则，这是基本的原因如下：首先，它们是无记忆的，在这个意义上，枢轴是由本地信息编码的树形图。其次，许多最常用的透视规则属于该类，我们表明这个子类对于理解所有透视规则的复杂性至关重要。最后，归一化权重枢轴规则可以以自然连续的方式参数化。我们证明了两个多面体的存在，枢纽规则多面体和cubotopes，捕获的行为规范化的重量枢纽规则多面体和线性规划。我们解释他们的脸结构的多树状。我们计算相干树形图的数量的上界，也就是我们的多面体的顶点。除了优化，我们的结构提供了新的视角，经典的几何组合。我们引入了一个归一化权重主元规则，我们称之为最大斜率主元规则，它推广了阴影顶点主元规则。相应的主元规则多面体和主元规则多面体改进了Billera-Sturmfels的单调路径多面体。此外，我们的多面体的特殊情况下产生置换面体，associahedra，和multiplihedra。为了最大程度地改进主元规则，我们绘制了扫描多面体和多拟阵的连接。
英文摘要：
Motivated by the analysis of the performance of the simplex method we study the behavior of families of pivot rules of linear programs. We introduce normalized-weight pivot rules which are fundamental for the following reasons: First, they are memory-less, in the sense that the pivots are governed by local information encoded by an arborescence. Second, many of the most used pivot rules belong to that class, and we show this subclass is critical for understanding the complexity of all pivot rules. Finally, normalized-weight pivot rules can be parametrized in a natural continuous manner. We show the existence of two polytopes, the pivot rule polytopes and the neighbotopes, that capture the behavior of normalized-weight pivot rules on polytopes and linear programs. We explain their face structure in terms of multi-arborescences. We compute upper bounds on the number of coherent arborescences, that is, vertices of our polytopes. Beyond optimization, our constructions provide new perspectives on classical geometric combinatorics. We introduce a normalized-weight pivot rule, we call the max-slope pivot rule which generalizes the shadow-vertex pivot rule. The corresponding pivot rule polytopes and neighbotopes refine monotone path polytopes of Billera--Sturmfels. Moreover special cases of our polytopes yield permutahedra, associahedra, and multiplihedra. For the greatest improvement pivot rules we draw connections to sweep polytopes and polymatroids.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
