Mathematics provide a strong theoretical framework and a wide range of methods to deal with different optimization problems that may be encountered in the management of air traffic.

The standard form of an optimization problem usually provides as follows:


where f is the objective-function (to be minimized in the formulation), g the functions which model the inequality constraints, and h the functions which model the equality constraints.

Different classes of optimization methods

The choice of the optimization method depends on the problem addressed and choice modeling (discrete variables, continuous or random modeling uncertainties or not, etc.). It is also conditioned by the nature and properties of the objective function and constraints: linearity, convexity, presence or absence of constraints, the existence of an analytical formulation of functions and derivatives. Finally, we will aim to find a local optimum in a certain neighborhood, or search for a global optimum of the objective function.

Depending on the case, so we will talk about local or global optimization, continuous or discrete (combinatorial or integer) or mixed (combining continuous and integer variables), with or without constraints, deterministic or stochastic, convex or non-convex.

For their implementation, optimization methods most often use iterative algorithms. From a starting point chosen in the search space, or more points for the population algorithms, we try to iteratively improve the standard of the objective function.

These algorithms are themselves either deterministic or stochastic. In the latter case, they use a random walk guided by the heuristic. This is called meta-heuristics: evolutionary algorithms, simulated annealing, differential evolution, particle swarm, etc..

The mathematical optimization applied to the air traffic management

Here are some examples of work carried out MAIAA :

Research on optimization methods

MAIAA researchers also lead a research on mathematical optimization methods, including the following topics :

See also the page optimization and parallelism in Research / Computer science menu for ongoing research on the combination of meta-heuristic methods and intervals by a cooperative mechanism parallelism.