This paper presents a methodology for optimizing an aeronautical propeller to minimize power consumption. A multi-objective approach using blade element momentum (BEM) theory and evolutionary algorithms is employed to optimize propeller design by minimizing power consumption during takeoff and top-of-climb. Three different evolutionary algorithms generated a Pareto front, from which the optimal propeller design is selected. The selected propeller design is evaluated under optimal operational conditions for a specific mission. In this context, two operational approaches for the optimized propellers during flight missions are evaluated. The first approach considers the possibility of only three values for the propeller rotation, while the second allows continuous changes in the rotational speed and pitch angle values, known as the multi-rotational-speed approach. In the second approach, a modal analysis of the propeller is performed using rotating beam theory. The natural frequencies of vibration, constrained by the Campbell diagram, enable an operational analysis and ensure structural integrity by preventing resonance between propeller blades and the rotational procedures. The multi-rotational approach is conducted with and without frequency constraints, resulting in general flight energy reductions of 1.40% and 1.47%, respectively. However, substantial power savings are achieved, namely up to $10\%$ during critical flight states, which can have a significant impact on future engine design and operability. The main contributions of the research lie in analyzing the multi-rotational approach with vibrational constraints of the optimized propeller. This research advances sustainable aviation practices by focusing on reducing power consumption while maintaining performance.