半定可表示集的暴露面
Exposed faces of semidefinitely representable sets
摘要：
线性矩阵不等式（LMI）是一个条件，说明一个对称矩阵，其元素是仿射线性组合的变量是半正定的。由于对角线性矩阵不等式定义了多面体，因此线性矩阵不等式的解集被称为谱面体。谱面体的线性像称为半定可表示集。对谱面体和半定可表示集的兴趣部分来自于这样一个事实，即人们可以通过半定规划有效地优化它们上的线性函数，就像人们可以通过线性规划在多面体上做一样。众所周知，分光面体的每个面都是暴露的。这在一般的严格凸集的上下文中也是正确的。我们研究同样的问题，半定可表示集。Lasserre提出了一种矩矩阵方法来构造某些集合的半定表示。我们的主要结果是，这种方法只能工作，如果所有的考虑集的面都暴露。这个必要条件补充了Lasserre，Helton和Nie最近证明的充分条件。
英文摘要：
A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of the interest in spectrahedra and semidefinite representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, like one can do on polyhedra by linear programming. It is known that every face of a spectrahedron is exposed. This is also true in the general context of rigidly convex sets. We study the same question for semidefinite representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can only work if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton and Nie.
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