The multiplier method for constrained optimization problems is considered geometrically with the concept of supporting hypersurfaces. This geometric interpretation simplifies the comparison of the multiplier method with other existing methods and provides a basis from which to develop new algorithms. Moreover, though it is proved that the algorithm of the multiplier method can be derived from different viewpoints, the convergence of the multiplier method as a successive approximation algorithm is verified in this paper. Finally, the multiplier method is applied to optimal control problems with terminal state constraints, and some numerical experiments are presented.