On the connection of facially exposed and nice cones
On the connection of facially exposed and nice cones
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论面部暴露与漂亮锥体的连接
DOI：
10.1016/j.jmaa.2012.10.033
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发表时间：
2012
影响因子：
1.3
通讯作者：
G. Pataki
中科院分区：
文献类型：
--
作者：
G. Pataki
如果集合K∗+F⊥对K的所有F个面都是闭合的，那么有限维欧氏空间中的一个闭合凸锥K就被称为nice，其中K∗是K的对偶锥，而F⊥是F的线性张成的正交补。nice的性质在Borwein和Wolkowicz的面约简算法中起着重要作用，关于nice锥的对偶的线性像是否闭合的问题也有一个简单的答案。我们证明了好锥体的几个特征，并显示了与面部暴露的强烈联系。我们证明了一个好的圆锥体必须正面暴露；相反，面部暴露加上附加条件意味着友善。我们推测好看的和面部暴露的锥体实际上是一样的，并给出了支持证据。
A closed convex cone K in a finite dimensional Euclidean space is called nice if the set K∗+F⊥is closed for all F faces of K, where K∗is the dual cone of K, and F⊥is the orthogonal complement of the linear span of F. The niceness property plays a role in the facial reduction algorithm of Borwein and Wolkowicz, and the question of whether the linear image of the dual of a nice cone is closed also has a simple answer. We prove several characterizations of nice cones and show a strong connection with facial exposedness. We prove that a nice cone must be facially exposed; conversely, facial exposedness with an added condition implies niceness. We conjecture that nice, and facially exposed cones are actually the same, and give supporting evidence.