  Two dimensional tensor networks such as projected entangled pairs states
(PEPS) are generally hard to contract. This is arguably the main reason why
variational tensor network methods in 2D are still not as successful as in 1D.
However, this is not necessarily the case if the tensor network represents a
gapped ground state of a local Hamiltonian; such states are subject to many
constraints and contain much more structure. In this paper we introduce a new
approach for approximating the expectation value of a local observable in
ground states of local Hamiltonians that are represented as PEPS
tensor-networks. Instead of contracting the full tensor-network, we try to
estimate the expectation value using only a local patch of the tensor-network
around the observable. Surprisingly, we demonstrate that this is often easier
to do when the system is frustrated. In such case, the spanning vectors of the
local patch are subject to non-trivial constraints that can be utilized via a
semi-definite program to calculate rigorous lower- and upper-bounds on the
expectation value. We test our approach in 1D systems, where we show how the
expectation value can be calculated up to at least 3 or 4 digits of precision,
even when the patch radius is smaller than the correlation length.
