Globally solving concave quadratic programs via doubly nonnegative relaxation
- We consider the problem of maximizing a convex quadratic function over a bounded
polyhedral set. We design a new framework based on SDP relaxations and cutting
plane methods for solving the associated reference value problem. The major novelty
is a new way to generate valid cuts through the doubly nonnegative (DNN) relaxation.
We establish various theoretical properties of the DNN relaxation, including its equiv-
alence with the Shor relaxation of an equivalent quadratically constrained problem,
the strong duality, and the generation of valid cuts from an approximate solution of
the DNN relaxation returned by an arbitrary SDP solver. Computational results on
both real and synthetic data demonstrate the efficiency of the proposed method and
its ability to solve high-dimensional problems with dense data. In particular, our new
algorithm successfully solves in 3 days the reference value problem arising from com-
putational biology for a dataset containing more than 300,000 instances of dimension
78. In contrast, CPLEX or Gurobi is estimated to require years of computational time
for the same dataset on the same computing platform.