Aggregations of Quadratic Inequalities and Hidden Hyperplane Convexity
研究了由二次不等式描述的集合的凸包性质，提出隐藏超平面凸性概念，并给出聚合不等式定义凸包的充分条件，推广了已有结果。
.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 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 intersection of 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 the positive definite linear combination for every triple of inequalities, together with HHC, is sufficient for finitely many aggregations to define the convex hull, answering a question raised in [S. S. Dey, G. Munoz, and F. Serrano, On Obtaining the Convex Hull of Quadratic Inequalities via Aggregations, arXiv:2106.12629, 2021]. 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 \(S\), which were needed in [S. Modaresi and J. P. Vielma, Math. Program., 164 (2017), pp. 383–409; S. S. Dey, G. Munoz, and F. Serrano, On Obtaining the Convex Hull of Quadratic Inequalities via Aggregations, arXiv:2106.12629, 2021].Keywordsquadratic programmingpolynomial optimizationconvex geometryMSC codes90C20