  Given a renormalization scheme, we show how to formulate a tractable convex
relaxation of the set of feasible local density matrices of a many-body quantum
system. The relaxation is obtained by introducing a hierarchy of constraints
between the reduced states of ever-growing sets of lattice sites. The
coarse-graining maps of the underlying renormalization procedure serve to
eliminate a vast number of those constraints, such that the remaining ones can
be enforced with reasonable computational means. This can be used to obtain
rigorous lower bounds on the ground state energy of arbitrary local
Hamiltonians, by performing a linear optimization over the resulting convex
relaxation of reduced quantum states. The quality of the bounds crucially
depends on the particular renormalization scheme, which must be tailored to the
target Hamiltonian. We apply our method to 1D translation-invariant spin
models, obtaining energy bounds comparable to those attained by optimizing over
locally translation-invariant states of $n\gtrsim 100$ spins. Beyond this
demonstration, the general method can be applied to a wide range of other
problems, such as spin systems in higher spatial dimensions, electronic
structure problems, and various other many-body optimization problems, such as
entanglement and nonlocality detection.
