  Global polynomial optimization is an important tool across applied
mathematics, with many applications in operations research, engineering, and
physical sciences. In various settings, the polynomials depend on external
parameters that may be random. We discuss a stochastic sum-of-squares (S-SOS)
algorithm based on the sum-of squares hierarchy that constructs a series of
semidefinite programs to jointly find strict lower bounds on the global minimum
and extract candidates for parameterized global minimizers. We prove
quantitative convergence of the hierarchy as the degree increases and use it to
solve unconstrained and constrained polynomial optimization problems
parameterized by random variables. By employing $n$-body priors from condensed
matter physics to induce sparsity, we can use S-SOS to produce solutions and
uncertainty intervals for sensor network localization problems containing up to
40 variables and semidefinite matrix sizes surpassing $800 \times 800$.
