  The reduced density matrix is variationally optimized for the two-dimensional
Hubbard model. Exploiting all symmetries present in the system, we have been
able to study $6\times6$ lattices at various fillings and different values for
the on-site repulsion, using the highly accurate but computationally expensive
three-index conditions. To reduce the computational cost we study the
performance of imposing the three-index constraints on local clusters of
$2\times2$ and $3\times3$ sites. We subsequently derive new constraints which
extend these cluster constraints to incorporate the open-system nature of a
cluster on a larger lattice. The feasibility of implementing these new
constraints is demonstrated by performing a proof-of-principle calculation on
the $6\times6$ lattice. It is shown that a large portion of the three-index
result can be recovered using these extended cluster constraints, at a fraction
of the computational cost.
