A Semidefinite Programming Method for Integer Convex Quadratic Minimization

We consider the NP-hard problem of minimizing a convex quadratic function over the integer lattice Z. We present a semidefinite programming (SDP) method 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. The effectiveness of the method is shown for numerical problem instances of various sizes. Finally, we introduce several extensions to the idea, including how to reduce the search space of existing branch-and-bound type enumeration algorithms for solving the problem globally, by using the tighter lower and upper bounds.