Integer quadratic quasi-polyhedra
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
On valid inequalities for quadratic programming with continuous variables and binary indicators
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
GloMIQO: Global mixed-integer quadratic optimizer
Journal of Global Optimization
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Nonconvex quadratic programming with box constraints is a fundamental $\mathcal{NP}$-hard global optimization problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterize their extreme points and vertices, show their invariance under certain affine transformations, and show that various linear inequalities induce facets. We also show that the sets are closely related to the Boolean quadric polytope, a fundamental polytope in the field of polyhedral combinatorics. Finally, we give a classification of valid inequalities and show that this yields a finite recursive procedure to check the validity of any proposed inequality.