In this paper, we investigate the maximum achievable sum-rate of the two-user Gaussian interference channel with Gaussian superposition coding and successive decoding. We first examine an approximate deterministic formulation of the problem, and introduce the complementarity conditions that capture the use of Gaussian coding and successive decoding. In the deterministic channel problem, we find the constrained sum-capacity and its achievable schemes with the minimum number of messages, first in symmetric channels, and then in general asymmetric channels. We show that the constrained sum-capacity oscillates as a function of the cross link gain parameters between the information theoretic sum-capacity and the sum-capacity with interference treated as noise. Furthermore, we show that if the number of messages of either of the two users is fewer than the minimum number required to achieve the constrained sum-capacity, the maximum achievable sum-rate drops to that with interference treated as noise. We provide two algorithms to translate the optimal schemes in the deterministic channel model to the Gaussian channel model. We also derive two upper bounds on the maximum achievable sum-rate of the Gaussian Han-Kobayashi schemes, which automatically upper bound the maximum achievable sum-rate using successive decoding of Gaussian codewords. Numerical evaluations show that, similar to the deterministic channel results, the maximum achievable sum-rate with successive decoding in the Gaussian channels oscillates between that with Han-Kobayashi schemes and that with single message schemes.