A multiple-objective optimization is implemented for a double row of staggered film holes on the suction surface of a turbine vane. The optimization aims to maximize the film cooling performance, which is assessed using the cooling effectiveness, while minimizing the corresponding aerodynamic loss, which is measured with a mass-averaged total pressure coefficient. Three geometric variables defining the hole shape are optimized: the conical expansion angle, compound angle and length to diameter ratio of the non-diffused portion of the hole. The optimization employs a non-dominated sorting genetic algorithm coupled with an artificial neural network to generate the Pareto front. Reynolds-averaged Navier-Stokes simulations are employed to construct the neural network and investigate the aerodynamic and thermal optimum solutions. The optimum designs exhibit improved performance in comparison to the reference design. The optimization methodology allowed investigation into the impact of varying the geometric variables on the cooling effectiveness and the aerodynamic loss.