A feasible method for solving an SDP relaxation of the quadratic knapsack problem

In this paper, we consider an SDP relaxation of the quadratic knapsack problem (QKP). After using the Burer-Monteiro factorization, we get a non-convex optimization problem, whose feasible region is an algebraic variety. Although there might be non-regular points on the algebraic variety, we prove that the algebraic variety is a smooth manifold except for a trivial point for a generic input data. We also analyze the local geometric properties of non-regular points on this algebraic variety. In order to maintain the equivalence between the SDP problem and its non-convex formulation, we derive a new rank condition under which these two problems are equivalent. This new rank condition can be much weaker than the classical rank condition if the coefficient matrix has certain special structures. We also prove that under an appropriate rank condition, any second order stationary point of the non-convex problem is also a global optimal solution without any regularity assumption. This result is distinguished from previous results based on LICQ-like smoothness assumption. With all these theoretical properties, we design an algorithm that equip a manifold optimization method with a strategy to escape from non-optimal non-regular points. Our algorithm can also be used as a heuristic for solving the quadratic knapsack problem. Numerical experiments are conducted to verify the high efficiency and robustness of our algorithm as compared to other SDP solvers and a heuristic method based on dynamic programming. In particular, our algorithm is able to solve the SDP relaxation of a one-million dimensional QKP with a sparse cost matrix very accurately in about 20 minutes on a modest desktop computer.