  Bell nonlocality is an intrinsic feature of quantum mechanics, which can be
certified via the violation of Bell inequalities. It is therefore a fundamental
question to certify Bell nonlocality from experimental data. Here, we present
an optimization scheme to improve nonlocality certification by exploring
flexible mappings between Bell inequalities and Hamiltonians corresponding to
the Bell operators. We show that several Hamiltonian models can be mapped to
new inequalities with improved classical bounds than the original one, enabling
a more robust detection of nonlocality. From the other direction, we
investigate the mapping from fixed Bell inequalities to Hamiltonians, aiming to
maximize quantum violations while considering experimental imperfections. As a
practical demonstration, we apply this method to an XXZ-like honeycomb-lattice
model utilizing over 70 superconducting qubits. The successful application of
this technique, as well as combining the two directions to form an optimization
loop, may open new avenues for developing more practical and noise-resilient
nonlocality certification techniques and enable broader experimental
explorations.
