  This paper is concerned with the unconstrained binary polynomial program
(UBPP), which has a host of applications in many science and engineering
fields. By leveraging the global exact penalty for its DC constrained SDP
reformulation, we achieve an equivalent DC penalized SDP, and propose a
continuous relaxation approach by seeking the critical point of the
Burer-Monteiro factorization for a finite number of DC penalized SDPs with
increasing penalty factors. A globally convergent majorization-minimization
(MM) method with extrapolation is also developed to capture such critical
points. Under a mild condition, we show that the rank-one projection of the
output for the relaxation approach is an approximate feasible solution of the
UBPP and quantify the upper bound of its objective value from the optimal
value. Numerical comparisons with the SDP relaxation method armed with a
special random rounding technique and the DC relaxation approach based on the
solution of linear SDPs confirm the efficiency of the proposed relaxation
approach, which can solve the instance of \textbf{20000} variables in
\textbf{15} minutes and yield an upper bound to the optimal value and the known
best value with a relative error at most \textbf{1.824\%} and \textbf{2.870\%},
respectively.
