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Quadratic programming (QP) is a well-studied fundamental NP-hard optimizationproblem which optimizes a quadratic objective over a set of linear constraints.In this paper, we reformulate QPs as a mixed-integer linear problem (MILP).This is done via the reformulation of QP as a linear complementary problem, andthe use of binary variables and big-M constraints, to model the complementaryconstraints. To obtain such reformulation, we show how to impose bounds on thedual variables without eliminating all the (globally) optimal primal solutions;using some fundamental results on the solution of perturbed linear systems.  Reformulating non-convex QPs as MILPs provides an advantageous way to obtainglobal solutions as it allows the use of current state-of-the-art MILP solvers.To illustrate this, we compare the performance of our solution approach,labeled quadprogIP, with the current benchmark global QP solver quadprogBB, aswell as with BARON, one of the leading non-linear programming (NLP) solvers,and CPLEX's non-convex QP solver, on a large variety of QP test instances. Inpractice, quadprogIP is shown to typically outperform by orders of magnitudequadprogBB, BARON, and CPLEX on standard QPs. For general QPs, quadprogIPoutperforms quadprogBB, outperforms BARON in most instances, while CPLEXperforms the best on these instances. For box-constrained QPs, quadprogIP has acomparable performance to quadprogBB and BARON in small- to medium-scaleinstances, but is outperformed by these solvers on large-scale instances; whileCPLEX performs the best on box-constrained QP instances. Also, unlikequadprogBB, the solution approach proposed here is able to solve QP instanceswhose dual feasible set is unbounded. The MATLAB code, called quadprogIP, andthe instances used to perform these numerical experiments are publiclyavailable at https://github.com/xiawei918/quadprogIP.

Globally solving Non-Convex Quadratic Programs via Linear Integer Programming techniques