Recently, a concave optimization approach has been proposed to solve the robust point matching (RPM) problem. This method is globally optimal, but it requires that each model point has a counterpart in the data point set. Unfortunately, such a requirement may not be satisfied in certain applications when there are outliers in both point sets. To address this problem, we relax this condition and reduce the objective function of RPM to a function with few nonlinear terms by eliminating the transformation variables. The resulting function, however, is no longer quadratic. We prove that it is still concave over the feasible region of point correspondence. The branch-and-bound (BnB) algorithm can then be used for optimization. To further improve the efficiency of the BnB algorithm whose bottleneck lies in the costly computation of the lower bound, we propose a new lower bounding scheme which has a k-cardinality linear assignment formulation and can be efficiently solved. Experimental results show that the proposed algorithm outperforms state-of-the-arts in its robustness to disturbances and point matching accuracy.