A fresh CP look at mixed-binary QPs: new formulations and relaxations
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Triggered by Burer’s seminal characterization from 2009, many copositive reformulations of mixed-binary QPs have been discussed by now. Most of them can be used as proper relaxations, if the intractable co(mpletely)positive cones are replaced by tractable approximations. While the widely used approximation hierarchies have the disadvantage to use positive-semidefinite (psd) matrices of orders which rapidly increase with the level of approximation, alternatives focus on the problem of keeping psd matrix orders small, with the aim to avoid memory problems in the interior point algorithms. This work continues this approach, proposing new reformulations and relaxations. Moreover, we provide a thorough comparison of the respective duals and establish a monotonicity relation among their duality gaps. We also identify sufficient conditions for strong duality/zero duality gap in some of these formulations and generalize some of our observations to general conic problems.
KeywordsCopositivity Completely positive Conic optimization Quadratic optimization Reformulations Nonlinear optimization Nonconvex optimization
Mathematics Subject Classification90C20 90C26 90C30
The authors would like to thank the anonymous handling Associate Editor and two anonymous referees for their very useful comments and suggestions which helped to improve our paper significantly. This research benefited from the support of the “FMJH Program Gaspard Monge in Optimization and Operations Research”, and from the support to this program by EDF.
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