论文标题
二次不平等和隐藏超平面凸的聚集
Aggregations of quadratic inequalities and hidden hyperplane convexity
论文作者
论文摘要
我们研究了二次不平等现象描述的集合$ s $的凸面的属性。产生不平等现象有效的一种简单方法是采用$ s $的定义不平等现象的非负线性组合。我们称这种不平等的聚合。特殊的聚合自然包含$ s $的凸壳，我们为这种聚合提供了足够的条件来定义凸壳。我们介绍了隐藏的超平面凸度（HHC）的概念，该概念与二次图隐藏凸的经典概念有关。我们表明，如果与$ s $相关的二次地图满足HHC，则$ s $的凸壳由特殊聚合定义。据我们所知，该结果概括了有关定义凸壳的聚集的所有已知结果。使用这种足够的条件，我们能够识别以前未知类别的集合，其中聚集导致凸壳。我们表明，称为正定义线性组合的条件以及隐藏的透平平凸度是有限的许多聚合来定义凸船体的足够条件。以上所有结果均用于使用开放二次不平等的集合。对于封闭的二次不平等现象，我们证明了有关给出凸面船体的聚合的新结果，而没有对$ S $的拓扑假设。
We study properties of the convex hull of a set $S$ described by quadratic inequalities. A simple way of generating inequalities valid on $S$ is to take a nonnegative linear combinations of the defining inequalities of $S$. We call such inequalities aggregations. Special aggregations naturally contain the convex hull of $S$, and we give sufficient conditions for such aggregations to define the convex hull. We introduce the notion of hidden hyperplane convexity (HHC), which is related to the classical notion of hidden convexity of quadratic maps. We show that if the quadratic map associated with $S$ satisfies HHC, then the convex hull of $S$ is defined by special aggregations. To the best of our knowledge, this result generalizes all known results regarding aggregations defining convex hulls. Using this sufficient condition, we are able to recognize previously unknown classes of sets where aggregations lead to convex hull. We show that the condition known as positive definite linear combination together with hidden hyerplane convexity is a sufficient condition for finitely many aggregations to define the convex hull. All the above results are for sets defined using open quadratic inequalities. For closed quadratic inequalities, we prove a new result regarding aggregations giving the convex hull, without topological assumptions on $S$.
