Optimization Letters

, Volume 12, Issue 3, pp 499–518 | Cite as

A semidefinite programming method for integer convex quadratic minimization

  • Jaehyun Park
  • Stephen Boyd
Original Paper


We consider the NP-hard problem of minimizing a convex quadratic function over the integer lattice \({\mathbf{Z}}^n\). We present a simple semidefinite programming (SDP) relaxation for obtaining a nontrivial lower bound on the optimal value of the problem. By interpreting the solution to the SDP relaxation probabilistically, we obtain a randomized algorithm for finding good suboptimal solutions, and thus an upper bound on the optimal value. The effectiveness of the method is shown for numerical problem instances of various sizes.


Convex optimization Integer quadratic programming Mixed-integer programming Semidefinite relaxation Branch-and-bound 



We thank three anonymous referees for providing helpful comments and constructive remarks.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Stanford UniversityStanfordUSA
  2. 2.Stanford UniversityStanfordUSA

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