Aircraft conflict detection and resolution is currently attracting the interest of many air transportation service providers and is concerned with the following question: Given a set of airborne aircraft and their intended trajectories, what control strategy should be followed by the pilots and the air traffic service provider to prevent the aircraft from coming too close to each other? This paper addresses this problem by presenting a resolution methodology whereby each aircraft proposes its desired heading while a centralized air traffic control authority resolves any conflict arising between aircraft, while minimizing the deviation between desired and conflict-free heading for each aircraft. The resolution methodology relies on a combination of convex programming and randomized searches: It is shown that a version of the planar, multiaircraft conflict resolution problem, accounting for all possible crossing patterns among aircraft, might be recast as a nonconvex, quadratically constrained quadratic program. For this type of problem, there exist efficient numerical relaxations, based on semidefinite programming, that provide lower bounds on the best achievable objective. These relaxations also lead to a random search technique to compute feasible, locally optimal, and conflict-free strategies. This approach is demonstrated on numerical examples and discussed.