Solving Standard Quadratic Programming by Cutting Planes

Standard quadratic programs are non-convex quadratic programs with the only constraint that variables must belong to a simplex. By a famous result of Motzkin and Straus, those problems are connected to the clique number of a graph. In this paper, we study cutting plane techniques to obtain strong bounds for standard quadratic programs. Our cuts are derived in the context of a Spatial Branch & Bound where linearization variables are introduced to represent products. Their validity is based on the result of Motzkin and Straus in that it depends on the clique number of certain graphs. We derive in particular cuts that correspond to an underlying complete bipartite graph structure. We study the relation between these cuts and the classical ones obtained by the first level of the reformulation-linearization technique. By studying this relation, we derive a new type of valid inequalities that generalize both types of cuts and are stronger. We present extensive computational results using the different cutting planes we propose in the context of the Spatial Branch & Bound implemented by the commercial solver CPLEX. We show that our cuts allow to obtain a significantly better bound than reformulation-linearization cuts and reduce computing times for global optimality. Finally, we show how to generalize the cuts to non-convex quadratic knapsack problems, i.e., to attack problems in which the feasible region is not restricted to be a simplex.