Globally Solving Nonconvex Quadratic Programs via Linear Integer Programming Techniques

摘要

We reformulate a (indefinite) quadratic program (QP) as a mixed-integer linear programming (MILP) problem by first reformulating a QP as a linear complementary problem, and then using binary variables and big-M constraints to model its complementary constraints. To obtain such reformulation, we use fundamental results on the solution of perturbed linear systems to impose bounds on the QP's dual variables without eliminating any of its (globally) optimal primal solutions. Reformulating a nonconvex QP as a MILP problem allows the use of current state-of-the-art MILP solvers to find its global optimal solution. To illustrate this, we compare the performance of this MILP-based solution approach, labeled quadprog IP, with quadprogBB, BARON, and CPLEX. In practice, quadprog IP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. Also, unlike quadprogBB, quadproglP is able to solve QP instances in which the dual feasible set is unbounded. The MATLAB code quadproglP and the instances used to perform the reported numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP.