The aerodynamic optimization community has recently started an effort to develop benchmark problems suitable for exercising aerodynamic optimization methods in a constrained design space. In the first round, four problems have been developed, two involving two-dimensional airfoils and the other two three-dimensional wings. In this paper, we address the two-dimensional problems which involve optimization of the NACA 0012 in inviscid transonic flow, as well as optimization of the RAE 2822 in viscous transonic flow. We solve the problems using a computationally efficient physics-based surrogate approach exploiting space mapping. Our results indicate that by shifting the computational burden to fast low-fidelity models, significant performance improvements can be achieved at the cost of a few evaluations of the expensive computational fluid dynamic models. In our approach, a commercial package FLUENT is used as the high-fidelity fluid flow solver with a hyperbolic C-mesh, whereas the versatile viscous-inviscid solver MSES is utilized as the low-fidelity model. The PARSEC parameterization method is used to describe the airfoil shapes with up to 10 design variables.