利用Sum-of-Squares规划最大化半代数集的切片体积
Maximizing Slice-Volumes of Semialgebraic Sets using Sum-of-Squares Programming
AI总结：
针对半代数集仿射切片体积最大化问题，提出将其转化为无穷维线性规划，借助moment-Sum-of-Squares半定规划层级求解收敛上界，可通过拓扑结构、对称性及斯托克斯冗余约束优化效率，并在示例集上完成验证。
摘要：
本文提出一种最大化给定半代数集的仿射切片体积的算法。受半代数集体积计算相关前期工作启发，该切片体积任务被表述为连续函数域上的无穷维线性规划。研究通过规模递增的半定规划moment-Sum-of-Squares层级，计算得到最大切片体积的一系列收敛上界。该方案的计算复杂度可通过利用拓扑结构（在2、3、4、8维下）与对称性来降低，还可引入基于斯托克斯定理的冗余约束来加速数值收敛。研究在多个示例集合上开展了切片体积计算的验证演示。
英文摘要：
This paper presents an algorithm to maximize the volume of an affine slice through a given semialgebraic set. This slice-volume task is formulated as an infinite-dimensional linear program in continuous functions, inspired by prior work in volume computation of semialgebraic sets. A convergent sequence of upper-bounds to the maximal slice volume are computed using the moment-Sum-of-Squares hierarchy of semidefinite programs in increasing size. The computational complexity of this scheme can be reduced by utilizing topological structure (in dimensions 2, 3, 4, 8) and symmetry. This numerical convergence can be accelerated through the introduction of redundant Stokes-based constraints. Demonstrations of slice-volume calculation are performed on example sets.
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