  The problem of computing the optimum of a function on a finite set is an
important problem in mathematics and computer science. Many combinatorial
problems such as MAX-SAT and MAXCUT can be recognized as optimization problems
on the hypercube $C_2^n = \{-1,1\}^n$ consisting of $2^n$ points. It has been
noticed that if a finite set is equipped with an abelian group structure, then
one can efficiently certify nonnegative functions on it by Fourier sum of
squares (FSOS). Motivated by these works, this paper is devoted to developing a
framework to find a lower bound of a function on a finite abelian group
efficiently. We implement our algorithm by the SDP solver SDPNAL+ for computing
the verified lower bound of $f$ on $G$ and test it on the MAX-2SAT and
  MAX-3SAT benchmark problems from the Max-SAT competitions in 2009 and 2016.
Beyond that, we also test our algorithm on random functions on $C_3^n$. These
experiments demonstrate the advantage of our algorithm over previously known
methods.
