This paper investigates the mitigation of limit cycle oscillations (LCOs) using optimally designed nonlinear energy sinks (NESs). The study is based on a NES attached to a two degree-of-freedom airfoil with nonlinear stiffness in pitch and heave. Depending on the nonlinear properties of the airfoil, it can exhibit super-or sub-critical LCOs. The design of the NES is based on a stochastic optimization approach, which accounts for various sources of uncertainties, including initial and loading conditions. The optimization approach is also tailored to handle potential discontinuities in the response of the airfoil. The non-smoothness of the response can be either due to sub-critical LCOs or to the activation of the NES. Several stochastic optimization problems with the objective of maximizing the mean reduction of LCOs are presented.