Quadratic 0–1 optimization using separable underestimators

Binary programs with a quadratic objective function are NP-hard in general, even if the linear optimization problem over the same feasible set is tractable. In this paper, we address such problems by computing quadratic global underestimators of the objective function that are separable but not necessarily convex. Exploiting the binarity constraint on the variables, a minimizer of the separable underestimator over the feasible set can be computed by solving an appropriate linear minimization problem over the same feasible set. Embedding the resulting lower bounds into a branch-and-bound framework, we obtain an exact algorithm for the original quadratic binary program. The main practical challenge is the fast computation of an appropriate underestimator, which in our approach reduces to solving a series of semidefinite programs. We exploit the special structure of the resulting problems and propose a tailored coordinate-descent method for their solution. Our extensive experimental results on various quadratic combinatorial optimization problems show that our approach outperforms both Cplex and the related QCR method. For the quadratic shortest path problem, we thus provide the fastest exact approach currently available.