Using polarity, we give an outer polyhedral approximation for the epigraph of set functions. For a submodular function, we prove that the corresponding polar relaxation is exact; hence, it is equivalent to the Lovász extension. The polar approach provides an alternative proof for the convex hull description of the epigraph of a submodular function. Computational experiments show that the inequalities from outer approximations can be effective as cutting planes for solving submodular as well as non-submodular set function minimization problems.

中文翻译：

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亚模函数最小化和极性

使用极性，我们给出了集合函数的题词的外部多面体近似。对于亚模函数，我们证明相应的极弛豫是精确的；因此，它等效于Lovász扩展。极地方法为子模函数的硬标的凸包描述提供了另一种证明。计算实验表明，来自外部逼近的不等式可以有效地用作解决次模块和非次模块集合函数最小化问题的切平面。