Many combinatorial optimization problems can be formulated as
the minimization of a 0–1 quadratic function subject to linear constraints. In
this paper, we are interested in the exact solution of this problem through a
two-phase general scheme. The first phase consists in reformulating the
initial problem either into a compact mixed integer linear program or into a
0–1 quadratic convex program. The second phase simply consists in
submitting the reformulated problem to a standard solver. The efficiency of
this scheme strongly depends on the quality of the reformulation obtained in
phase 1. We show that a good compact linear reformulation can be obtained by
solving a continuous linear relaxation of the initial problem. We also show
that a good quadratic convex reformulation can be obtained by solving a
semidefinite relaxation. In both cases, the obtained reformulation profits
from the quality of the underlying relaxation. Hence, the proposed scheme gets
around, in a sense, the difficulty to incorporate these costly relaxations in
a branch-and-bound algorithm.