  We present a novel approach to non-convex optimization with certificates,
which handles smooth functions on the hypercube or on the torus. Unlike
traditional methods that rely on algebraic properties, our algorithm exploits
the regularity of the target function intrinsic in the decay of its Fourier
spectrum. By defining a tractable family of models, we allow at the same time
to obtain precise certificates and to leverage the advanced and powerful
computational techniques developed to optimize neural networks. In this way the
scalability of our approach is naturally enhanced by parallel computing with
GPUs. Our approach, when applied to the case of polynomials of moderate
dimensions but with thousands of coefficients, outperforms the state-of-the-art
optimization methods with certificates, as the ones based on Lasserre's
hierarchy, addressing problems intractable for the competitors.
