Scramjet engines are considered as one of the most promising hypersonic airbreathing propulsion techniques for efficient and economic access to space, with the axisymmetric full Busemann intakes known for their high intake efficiency for inviscid flows. This paper presents the results and insights obtained from a single-point as well as multi-point multi-objective optimization studies of a Busemann based intake by means of a surrogate-assisted evolutionary algorithm coupled with high-fidelity computational fluid dynamics. Intake geometries are generated by applying geometric alterations to the full Busemann intake via leading-edge truncation, axial stunting and radial contraction, aiming to simultaneously minimize intake drag and maximize the compression efficiency at two different design conditions, i.e., Mach 7.7 at an altitude of 30 km and Mach 10 at 33.5 km on a constant dynamic pressure ascent trajectory. From the single-point optimization study it has been found that the non-dominated solutions obtained from minimizing drag and maximizing compression efficiency are approximately the same as those obtained from maximizing total pressure recovery and minimizing static pressure ratio. From the multi-point optimization study, the optimal solutions have been found to retain the classical advantages of the reference full Busemann intakes in terms of high compression efficiency while leveraging their limitations with shorter intakes with higher static pressure as well as adequately high mean exit temperature for both design conditions. Further, the non-dominated solutions have been found to form tendency of clusters based on their geometric parameters.