  Certification of quantum nonlocality plays a central role in practical
applications like device-independent quantum cryptography and random number
generation protocols. These applications entail the challenging problem of
certifying quantum nonlocality, something that is hard to achieve when the
target quantum state is weakly entangled, or when the source of errors is high,
e.g. when photons propagate through the atmosphere or a long optical fiber.
Here, we introduce a technique to find a Bell inequality with the largest
possible gap between the quantum prediction and the classical local hidden
variable limit for a given set of measurement frequencies. Our method
represents an efficient strategy to certify quantum nonlocal correlations from
experimental data without requiring extra measurements, in the sense that there
is no Bell inequality with a larger gap than the one provided. Furthermore, we
also reduce the photodetector efficiency required to close the detection
loophole. We illustrate our technique by improving the detection of quantum
nonlocality from experimental data obtained with weakly entangled photons.
