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.

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..

Here are some examples of work carried out MAIAA :

- A curved path model without uncertainty, based on B-splines allows a formulation of the problem of conflict resolution in the form of a continuous optimization problem. How defined the objective function and constraint functions also allows to calculate the gradients. Three different optimization methods are compared on this problem: genetic algorithms, an interior point method and an optimization method without derivatives.
- A MINLP formulation (Mixed-integer non-linear programming) is proposed for the problem of conflict resolution by speed adjustments. Branch and bound space with convex stress relaxation provides a global optimum for instances of small size (less than 6 aircraft). For instances of medium size (10 aircraft), we propose a local approach with a heuristic based on a decomposition into sub-problems.

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

- Reformulation of models
- Improvement heuristic for modularity maximization criterion in graphs

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.