Journal of Combinatorial Optimization

, Volume 28, Issue 2, pp 376–399 | Cite as

A Branch and Bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation

  • Alain Billionnet
  • Sourour Elloumi
  • Amélie LambertEmail author


Let \((MQP)\) be a general mixed-integer quadratic program that consists of minimizing a quadratic function \(f(x) = x^TQx +c^Tx\) subject to linear constraints. Our approach to solve \((MQP)\) is first to consider an equivalent general mixed-integer quadratic problem. This equivalent problem has additional variables \(y_{ij}\), additional quadratic constraints \(y_{ij}=x_ix_j\), a convex objective function, and a set of valid inequalities. Contrarily to the reformulation proposed in Billionnet et al. (Math Program 131(1):381–401, 2012), the equivalent problem cannot be directly solved by a standard solver. Here, we propose a new Branch and Bound process based on the relaxation of the non-convex constraints \(y_{ij}=x_ix_j\) to solve \((MQP)\). Computational experiences are carried out on pure- and mixed-integer quadratic instances. The results show that the solution time of most of the considered instances with up to 60 variables is improved by our Branch and Bound algorithm in comparison with the approach of Billionnet et al. (2012) and with the general mixed-integer nonlinear solver BARON (Sahinidis and Tawarmalani, Global optimization of mixed-integer nonlinear programs, user’s manual, 2010).


General mixed-integer quadratic programming Branch and Bound Quadratic convex relaxation Experiments 


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Alain Billionnet
    • 1
  • Sourour Elloumi
    • 1
  • Amélie Lambert
    • 2
    Email author
  1. 1.CEDRIC-ENSIIEEvry CedexFrance
  2. 2.CEDRIC-CNAMParis Cedex 03France

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