发表机构
DAMO Academy, Alibaba Group (U.S.) Inc.（达摩院，阿里巴巴集团（美国）公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对多胞体最大体积内接椭球问题，提出基于极性的迭代过程，以极性MinCE对数体积为势函数证明线性收敛，并分析非精确求解的容差与算法选择。
AI 中文摘要
我们通过基于极性的几何迭代研究多胞体的最大体积内接椭球（MaxIE）问题。给定一个内点，该方法形成平移后的极性多胞体，计算其最小体积覆盖椭球（MinCE），然后将该覆盖椭球极化为新的内接椭球。该过程由Khachiyan和Todd于1993年提出。先前的工作研究了基本的极性恒等式，并给出了精确过程的渐近收敛论证，但未提供体积比的速率。独立于该渐近论证，我们基于极性MinCE的对数体积作为势函数开发了一种新的分析。我们证明了其凸性并给出了显式梯度公式，并建立了沿每条轨迹的势间隙的全局线性收缩。收缩因子是存在性的且依赖于实例。这既给出了独立的收敛性证明，也给出了有限的体积比迭代界。我们进一步分析了非精确极性过程，其中每个MinCE子问题仅近似求解，并推导了产生规定体积近似的充分预言机容差。将该外层分析与现有的MinCE算法相结合，基于路径跟踪牛顿法、重心坐标下降法和远离步Frank-Wolfe方法给出了条件算术估计。数值实验将这些预言机选择与两个MaxIE基线进行了比较，并说明性能取决于将MinCE求解器与实例几何相匹配。
英文摘要
We study the maximum-volume inscribed ellipsoid (MaxIE) problem for a polytope through a geometric iteration based on polarity. Given an interior point, the method forms the shifted polar polytope, computes its minimum-volume covering ellipsoid (MinCE), and then polarizes this covering ellipsoid back to obtain a new inscribed ellipsoid. This procedure was suggested by Khachiyan and Todd in 1993. Prior work studied basic polarity identities and gave an asymptotic-convergence argument for the exact process, but did not furnish a volume-ratio rate. Independently of that asymptotic argument, we develop a new analysis based on the log-volume of the polar MinCE as a potential function. We prove its convexity with an explicit gradient formula and establish global linear contraction of the potential gap along each trajectory. The contraction factor is existential and instance-dependent. This gives both a separate convergence proof and a finite volume-ratio iteration bound. We further analyze an inexact polarity process in which each MinCE subproblem is solved only approximately, and derive sufficient oracle tolerances for producing a prescribed volume approximation. Combining this outer analysis with existing algorithms for MinCE gives conditional arithmetic estimates based on the path-following Newton method, the barycentric coordinate descent, and the away-step Frank-Wolfe method. Numerical experiments compare these oracle choices with two MaxIE baselines and illustrate that performance depends on matching the MinCE solver to the instance geometry.
