Supersonic aircraft are challenging to optimally design due to the widely varying constraints and flight conditions they experience. Additionally, a large number of disciplinary subsystems must be considered due to highly complex design requirements. One subsystem that has a major effect on overall performance is the engine. In this work, we construct a supersonic mixed-flow variable cycle engine and perform multipoint gradient-based optimization using this model. We see that the operational variables allow the optimizer to tailor performance at each individual flight condition, leading to better overall performance. To simulate airframe integration constraints, we run successive optimizations with increasingly restrictive inlet areas and see decreases in engine performance. This work is part of a larger effort to incorporate engine design into aero-thermal-mission optimization of a supersonic aircraft.