The objective of the current work is to define a design optimization methodology in aerodynamics, in which all numerical components are based on a unique geometrical representation, consistent with Computer-Aided Design (CAD) standards. In particular, the design is parameterized by Non-Uniform Rational B-Splines (NURBS), the computational domain is automatically constructed using rational Bézier elements extracted from NURBS boundaries without any approximation and the resolution of the flow equations relies on an adaptive Discontinuous Galerkin (DG) method based on rational representations. A Bayesian framework is used to optimize NURBS control points, in a single-or multi-objective, constrained, global optimization framework. The resulting methodology is therefore fully CAD-consistent, high-order in space and time, includes local adaption and shock capturing capabilities, and exhibits high parallelization performance. The proposed methods are described in details and their properties are established. Finally, two design optimization problems are provided as illustrations: the shape optimization of an airfoil in transonic regime, for drag reduction with lift constraint, and the multi-objective optimization of the control law of a morphing airfoil in subsonic regime, regarding the time-averaged lift, the minimum instantaneous lift and the energy consumption.