This paper describes a novel optimization-based approach to conflict resolution in air traffic control, based on geometric programming. The main advantage of the approach is that Geometric Programs (GPs) can also capture various metering directives issued by the traffic flow management level, in contrast to most recent methods focusing purely on aircraft separation issues. GPs can also account for some of the nonlinearities present in the formulations of conflict resolution problems, while incurring only a small penalty in computation time with respect to the fastest linear programming based approaches. Additional integer variables can be introduced to improve the quality of the obtained solutions and handle combinatorial choices, resulting in Mixed-Integer Geometric Programs (MIGPs). We present GPs and MIGPs to solve a variety of joint metering and separation scenarios, e.g. including miles-in-trail and minutes-in-trail restrictions through airspace fixes and boundaries. Simulation results demonstrate the efficiency of the approach.