发表机构
Bowdoin College; Illinois State University; Bocconi University（博多因学院; 伊利诺伊州立大学; 博科尼大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对单纯形法的退化问题，提出一种转轴规则，可规避退化并追踪影子边方向，仅需线性数量退化转轴，还改进了0/1多面体上线性规划的退化转轴数量上界。
AI 中文摘要
单纯形法是求解线性规划问题最广泛使用的方法之一。该算法从可行域的一个顶点解出发，通过一系列基交换（称为转轴操作）推进，每次交换对应沿多面体的一条改进边移动至更优顶点。影响单纯形法效率的核心挑战是退化问题，它会导致转轴操作序列冗长且无法改变当前顶点解。本文证明，存在一种转轴规则，当在某个相容基上初始化时，该规则能规避退化问题并追踪任意给定的影子边方向，仅需线性数量的退化单纯形转轴操作。作为该结果的副产品，我们得到了求解定义在0/1多面体上的线性规划所需退化单纯形转轴操作数量的改进上界。
英文摘要
The Simplex method is among the most widely used approaches for solving linear programs. Starting at a vertex solution of the feasible region, the algorithm proceeds through a sequence of basis exchanges (known as pivots), each corresponding to a move along an improving edge of the polyhedron toward a better vertex. A central challenge affecting the efficiency of the Simplex method is degeneracy, which can cause long sequences of pivot operations that fail to change the current vertex solution. In this paper, we prove the existence of a pivot rule which is able to escape degeneracy and follow any given shadow edge-direction when initialized at some compatible basis, with a linear number of degenerate Simplex pivots. As a byproduct of our result, we obtain an improved bound on the number of degenerate Simplex pivots needed to solve linear programs defined on 0/1 polytopes.
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