The Boolean quadric polytope: some characteristics, facets and relatives
Mathematical Programming: Series A and B
Computational study of a family of mixed-integer quadratic programming problems
Mathematical Programming: Series A and B
Perspective cuts for a class of convex 0–1 mixed integer programs
Mathematical Programming: Series A and B
A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations
Mathematical Programming: Series A and B
Algorithm for cardinality-constrained quadratic optimization
Computational Optimization and Applications
On the copositive representation of binary and continuous nonconvex quadratic programs
Mathematical Programming: Series A and B
On Nonconvex Quadratic Programming with Box Constraints
SIAM Journal on Optimization
Computable representations for convex hulls of low-dimensional quadratic forms
Mathematical Programming: Series A and B - Series B - Special Issue: Combinatorial Optimization and Integer Programming
Perspective reformulations of mixed integer nonlinear programs with indicator variables
Mathematical Programming: Series A and B - Series B - Special Issue: Combinatorial Optimization and Integer Programming
Valid inequalities for the pooling problem with binary variables
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Extending the QCR method to general mixed-integer programs
Mathematical Programming: Series A and B
Fixed-Charge Transportation with Product Blending
Transportation Science
SDP diagonalizations and perspective cuts for a class of nonseparable MIQP
Operations Research Letters
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In this paper we study valid inequalities for a set that involves a continuous vector variable x∈[0,1]n, its associated quadratic form xxT, and binary indicators on whether or not x0. This structure appears when deriving strong relaxations for mixed integer quadratic programs (MIQPs). Valid inequalities for this set can be obtained by lifting inequalities for a related set without binary variables (QPB), that was studied by Burer and Letchford. After closing a theoretical gap about QPB, we characterize the strength of different classes of lifted QPB inequalities. We show that one class, lifted-posdiag-QPB inequalities, capture no new information from the binary indicators. However, we demonstrate the importance of the other class, called lifted-concave-QPB inequalities, in two ways. First, all lifted-concave-QPB inequalities define the relevant convex hull for the case of convex quadratic programming with indicators. Second, we show that all perspective constraints are a special case of lifted-concave-QPB inequalities, and we further show that adding the perspective constraints to a semidefinite programming relaxation of convex quadratic programs with binary indicators results in a problem whose bound is equivalent to the recent optimal diagonal splitting approach of Zheng et al.. Finally, we show the separation problem for lifted-concave-QPB inequalities is tractable if the number of binary variables involved in the inequality is small. Our study points out a direction to generalize perspective cuts to deal with non-separable nonconvex quadratic functions with indicators in global optimization. Several interesting questions arise from our results, which we detail in our concluding section.