Mathematical Programming

, Volume 158, Issue 1–2, pp 235–266 | Cite as

Exact quadratic convex reformulations of mixed-integer quadratically constrained problems

  • Alain Billionnet
  • Sourour Elloumi
  • Amélie Lambert
Full Length Paper Series A


We propose a solution approach for the general problem (QP) of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints. The resolution is based on the reformulation of the original problem (QP) into an equivalent quadratic problem whose continuous relaxation is convex, so that it can be effectively solved by a branch-and-bound algorithm based on quadratic convex relaxation. We concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible. Furthermore, we extend our method to the case of mixed-integer quadratic problems with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. Finally, we illustrate the different results of the article by small examples and we present some computational experiments on pure-integer and mixed-integer instances of (QP). Most of the considered instances with up to 53 variables can be solved by our approach combined with the use of Cplex.


Integer quadratic programming Equivalent convex reformulation Semidefinite programming Branch-and-bound algorithm 

Mathematics Subject Classification

90C11 90C20 


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  • Alain Billionnet
    • 1
  • Sourour Elloumi
    • 1
  • Amélie Lambert
    • 2
  1. 1.CEDRIC-ENSIIEEvry cedexFrance
  2. 2.CEDRIC-CnamParis cedex 03France

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