function. In [@olsson2009branch], they proposed a provably optimal algorithm. They employed convex underestimators with branch-and-bound methods to iteratively compute the global minimum of the cost function. Although this algorithm can guarantee global optimality, it is very time-consuming. Olsson [@olsson2008solving] reduced the computational complexity by applying the Lagrangian dual relaxation to approximate the cost function. Their experimental results showed that a single convex semidefinite program can well approximate the original problem. Recently, an improved result is obtained by a strengthened Lagrangian dual relaxation [@briales2017convex]. Although without theoretical guarantees, their experimental results show that this algorithm can achieve the global minimum. Although these algorithms [@olsson2008solving; @briales2017convex] have made progress in efforts to reduce the computational time, they are still not suitable to demanding real-time applications. Recently, Wientapper [@wientapper2018universal] provided an efficient algorithm based on Gröbner basis polynomial solver. They introduced the first order derivatives of the norm-one constraint of the quaternion into the first order optimality conditions of the cost function as [@kneip2014upnp], rather than apply the Lagrangian formulation for the norm-one constraint of the quaternion. This results in an efficient solution, however, their result is not optimal in theory. Furthermore, as the number of equations is greater than the number of 