SIAM Journal on Optimization, Volume 31, Issue 3, Page 1827-1849, January 2021.
Symmetries of a set are linear transformations that map the set to itself. A fundamental domain of a symmetric set is a subset that contains at least one representative from each of the symmetric equivalence classes (orbits) in the set. This paper contributes a novel polynomial algorithm for constructing minimal polytopic fundamental domains of polytopic sets. Our algorithm is applicable for generic linear symmetries of the set and has linear complexity in the number of facets and dimension of the symmetric polytope, e.g., the feasible region of an optimization problem. In addition, this paper contributes a novel polynomial algorithm for mapping an element of the polytope to its representative in the fundamental domain. The algorithms are demonstrated in four examples---two illustrative and two practical. In the first practical example, we show that a minimal fundamental domain of the hypercube under the symmetric group is the set of points with sorted elements. In the second practical example, we show how the construction algorithm can be applied to the max-cut problem.

中文翻译：

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对称优化的基本领域：构造和搜索

SIAM 优化杂志，第 31 卷，第 3 期，第 1827-1849 页，2021 年 1 月。
集合的对称性是将集合映射到自身的线性变换。对称集合的基本域是一个子集，它包含来自集合中每个对称等价类（轨道）的至少一个代表。本文提供了一种新颖的多项式算法，用于构建多面体集的最小多面体基本域。我们的算法适用于集合的一般线性对称，并且在对称多面体的面数和维度上具有线性复杂性，例如优化问题的可行区域。此外，本文提供了一种新颖的多项式算法，用于将多面体的元素映射到其在基本域中的代表。这些算法在四个示例中进行了演示——两个说明性的，两个实用性的。在第一个实际示例中，我们证明了对称群下超立方体的最小基本域是具有排序元素的点集。在第二个实际示例中，我们展示了如何将构造算法应用于最大切割问题。