Multiuser downlink beamforming for sum-rate maximization has been intensively studied in the literature assuming that the achievable data rates of the mobile stations (MSs) are continuous and strictly increasing functions of the received signal-to-interference-plus-noise ratios (SINRs). However, in practical cellular networks that employ adaptive modulation and coding, the data rates of the MSs are determined by the specific modulation and coding schemes and thus attain discrete values. We consider in this paper discrete rate adaptation and downlink beamforming (RAB), where the discrete rate assignment is jointly optimized along with the beamformer design to achieve the maximum sum-rate with minimum total transmitted power of the base station. User admission control is embedded in the discrete rate assignment procedure. We address the RAB problem using a mixed integer second-order cone program (MI-SOCP) approach, proposing a standard big-M MI-SOCP formulation that supports the branch-and-cut (BnC) method. To reduce the complexity of the BnC algorithm, we further develop an improved extended MI-SOCP formulation. We analytically show that the extended formulation generally admits strictly tighter continuous relaxations (and thus less computational complexity) than that of the big-M formulation. Efficient strategies are proposed to customize the standard BnC method for the RAB problem. For applications in large-scale networks, we develop low-complexity SOCP based inflation and deflation procedures to find suboptimal solutions of the RAB problem. Simulations show that the inflation and deflation procedures yield sum-rates that are very close to that of the optimal solutions.