This paper investigates a concept of an electrically supercharged internal combustion engine powertrain. A supercharger consists of an electric motor and a compressor. It draws its power from an electric energy buffer (e.g., a battery) and helps the engine during short-duration high-power demands. Both the engine and the buffer are sized to reduce the sum of the vehicle operational (fuel) and component (engine and buffer) costs. For this purpose, a convex, driving cycle-based vehicle model is derived, enabling the formulation of an underlying optimization problem as a second order cone program. Such a program can be efficiently solved using dedicated numerical tools (for a given gear selection strategy), which provides not only the optimal engine/buffer sizes but also the optimal vehicle control and state trajectories (e.g., compressor power and buffer energy). Finally, the results obtained from a representative, numerical case study are discussed in detail.