Detection and resolution of aircraft conflicts in en-route flights controls the separation (typically set at 5NM) between aircraft trajectories. It is crucial to ensure flight safety and remains a challenging problem in Air Traffic Management (ATM). Taking into account the increasing air traffic on the world scale and its impact on air traffic controllers' workload, a higher level of automation in ATM urgently needs to be introduced to deal with aircraft conflict avoidance. In the present work, we propose two novel optimization formulations, where the decision levers are both aircraft speed changes and heading angle changes. Both formulations exploit the removal of the infinite-dimensional feature of the separation constraint using a method introduced in (Cafieri and Durand, 2014), and rely on a linearization of angle-related nonlinear terms. The first formulation is based on Mixed-Integer Nonlinear Programming (MINLP). MINLP enables the simultaneous consideration of continuous variables (aircraft speeds, heading angles, etc.) as well as integer ones (in particular, binary, to model logical choices), and to model the inerently combinatorial pairwise nonlinear separation constraints. The second formulation we propose is a purely continuous optimization model, where we introduce an exact l1-penalty function, tailored to the problem at hand, to deal with the aircraft separation constraints. A variant of the introduced models that aims at increasing the aircraft separation distance, results in robust solutions for air traffic controllers.\\ Numerical results on a set of problem instances validate the approaches while highlighting the versatility of the proposed models.