AI 中文总结
该研究针对非凸QCQPs，提出受MIR启发的联合范围不等式，通过投影-提升方法构建，在几何实验中可大幅缩减投影松弛面积。
AI 中文摘要
我们通过受混合整数舍入（MIR）不等式启发的“投影-提升”方法研究非凸二次约束二次规划（QCQPs）的割平面。针对扩展QCQP公式的两个基础有效不等式，我们将相关的两行松弛投影到二维集合中，分析两个基础不等式中二次函数的联合范围。对于非凸联合范围，我们给出投影集合的闭式凸包描述；对于凸联合范围，我们给出其半定表示。这产生了一类新的联合范围不等式，可被提升回扩展QCQP公式。MIR不等式可处理“混合”项：连续变量或整数变量的分数线性组合。类似地，我们提出更灵活的正割混合联合范围不等式，能更好地揭示和利用非凸联合范围。该方法保持稀疏性，因为每个提升不等式的支撑由基础不等式的支撑控制。在初步几何实验中，联合范围不等式使通过重构-线性化技术构造的投影松弛的面积大幅减少。
英文摘要
We study cutting planes for nonconvex quadratically constrained quadratic programs (QCQPs) through a project-then-lift approach inspired by mixed-integer rounding (MIR) inequalities. Given two base valid inequalities for the extended QCQP formulation, we project the associated two-row relaxation into a two-dimensional set and analyze the joint range of quadratic functions in two base inequalities. For the nonconvex joint range, we give a closed-form convex hull description of the projected set; for the convex joint range, we give its semidefinite representation. This yields a new family of joint-range inequalities, which can be lifted back to the extended QCQP formulation. MIR inequalities can handle ``mixed'' terms: continuous variables or fractional linear combinations of integer variables. Similarly, we propose more flexible secant mixed-joint-range inequalities, which better expose and exploit the nonconvex joint range. The proposed approach preserves sparsity, since the support of each lifted inequality is controlled by that of the base inequalities. In preliminary geometric experiments, the joint-range inequalities yield substantial area reduction of the projected relaxation constructed via reformulation-linearization-technique.
