论文标题
自偶有的多面体锥及其松弛矩阵
Self-dual polyhedral cones and their slack matrices
论文作者
论文摘要
我们分析了自偶有的多面体锥体，并证明了有关其松弛矩阵的几种特性。特别是，我们表明自以为是等同于存在阳性半菲尼斯（PSD）松弛的存在。除此之外，我们表明，如果基础锥是不可还原的，那么相应的PSD休闲剂不仅是双重的非负矩阵（DNN），而且是DNN矩阵锥的极端光线，它与以前未描述的极端射线家族相对应。更令人惊讶的是，我们表明，除非圆锥体是简单的，否则PSD松弛不仅不能完全是正矩阵，而且还位于完全阳性的半芬属矩阵的锥体外。最后，我们展示了如何使用半限定编程用给定的组合物来探测自偶锥的存在。我们的结果是针对多面体锥体给出的，但我们还讨论了对自极多极化的一些后果。
We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD) slack. Beyond that, we show that if the underlying cone is irreducible, then the corresponding PSD slacks are not only doubly nonnegative matrices (DNN) but are extreme rays of the cone of DNN matrices, which correspond to a family of extreme rays not previously described. More surprisingly, we show that, unless the cone is simplicial, PSD slacks not only fail to be completely positive matrices but they also lie outside the cone of completely positive semidefinite matrices. Finally, we show how one can use semidefinite programming to probe the existence of self-dual cones with given combinatorics. Our results are given for polyhedral cones but we also discuss some consequences for negatively self-polar polytopes.
