New compact engine architectures such as pressure gain combustion require ad-hoc turbomachinery to ensure an adequate range of operation with high performance. A critical factor for supersonic turbines is to ensure the starting of the flow passages, which limits the flow turning and airfoil thickness. Radial outflow turbines inherently increase the cross section along the flow path, which holds great potential for high turning of supersonic flow with a low stage number and guarantees a compact design. First the preliminary design space is described. Afterwards a differential evolution multi-objective optimization with 12 geometrical design parameters is deducted. With the design tool AutoBlade 10.1, 768 geometries were generated and hub, shroud, and blade camber line were designed by means of Bezier curves. Outlet radius, passage height, and axial location of the outlet were design variables as well. Structured meshes with around 3.7 million cells per passage were generated. Steady three dimensional Reynolds averaged Navier Stokes (RANS) simulations, enclosed by the k-omega SST turbulence model were solved by the commercial solver CFD++. The geometry was optimized towards low entropy and high power output. To prove the functionality of the new turbine concept and optimization, a full wheel unsteady RANS simulation of the optimized geometry exposed to a nozzled rotating detonation combustor (RDC) has been performed and the advantageous flow patterns of the optimization were also observed during transient operation.