A fresh CP look at mixed-binary QPs: new formulations and relaxations

• Published in: Mathematical programming Vol. 166; no. 1-2; pp. 159 - 184
• Main Authors: Bomze, Immanuel M., Cheng, Jianqiang, Dickinson, Peter J. C., Lisser, Abdel
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2017; Springer Nature B.V; Springer Verlag
• Subjects: Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Computer Science; Cones; Formulations; Full Length Paper; Hierarchies; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Matrix methods; Numerical Analysis; Operations Research; Optimization and Control; Theoretical; 90C20; Nonconvex optimization; 90C30; Quadratic optimization; Copositivity; Completely positive; Reformulations; Nonlinear optimization; 90C26; Conic optimization
• DOI: 10.1007/s10107-017-1109-8
• ISSN: 0025-5610

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.