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

A relaxation approach to UBPPs based on equivalent DC penalized factorized matrix programs