Multi-Directional Search method
Torczon's multi-directional search is a simplex method which does not require derivatives.
I optimized the RAE2822 airfoil for drag reduction keeping the lift constant. The cost function is
J = Cd/Cd0 + 10e4 * max(0, 1-Cl/Cl0)
where Cd0=0.00917209414, Cl0=0.854904661 are the drag and lift for the RAE2822 airfoil. The shape deformation is parametrized by 20 Hicks-Henne functions, 10 on lower and 10 on upper surface. For the optimized airfoil the drag and lift are Cd=0.00353457467 and Cl=0.854904701 respectively.
This figure shows the solution for the RAE2822 airfoil:
and this shows the solution on the optimized airfoil:
Note the disappearance of the shock in the optimized airfoil. The airfoils are shown in the following figure: