  Standard variational methods tend to obtain upper bounds on the ground state
energy of quantum many-body systems. Here we study a complementary method that
determines lower bounds on the ground state energy in a systematic fashion,
scales polynomially in the system size and gives direct access to correlation
functions. This is achieved by relaxing the positivity constraint on the
density matrix and replacing it by positivity constraints on moment matrices,
thus yielding a semi-definite programme. Further, the number of free parameters
in the optimization problem can be reduced dramatically under the assumption of
translational invariance. A novel numerical approach, principally a combination
of a projected gradient algorithm with Dykstra's algorithm, for solving the
optimization problem in a memory-efficient manner is presented and a proof of
convergence for this iterative method is given. Numerical experiments that
determine lower bounds on the ground state energies for the Ising and
Heisenberg Hamiltonians confirm that the approach can be applied to large
systems, especially under the assumption of translational invariance.
