We present the BMOO algorithm for multi-objective optimization in the presence of non-linear and expensive-to-evaluate constraints and an application to the design of a commercial aircraft environment control system (ECS). The BMOO algorithm implements a Bayesian approach to this optimization problem. The emphasis is on conducting the optimization using a limited number of system simulations and, as a particularity, the algorithm is run on a non-hypercubic design domain and implements hidden constraints handling capabilities. The ECS is composed of two cross-flow heat exchangers, a centrifugal compressor and a radial turbine, the geometries of which are simultaneously optimized to achieve minimal weight and entropy generation of the system as a whole, while respecting strict specifications. While both objectives impact the overall performance of the aircraft, they are shown to be antagonistic and a set of trade-off design solutions is identified.