This paper presents a set membership method (named Interval Analysis Localization (IAL)) to deal with the global localization problem of mobile robots. By using a LIDAR (LIght Detection And Ranging) range sensor, the odometry and a discrete map of an indoor environment, a robot has to determine its pose (position and orientation) in the map without any knowledge of its initial pose. In a bounded error context, the IAL algorithm searches a set of boxes (interval vector), with a cardinality as small as possible that includes the robot’s pose. The localization process is based on constraint propagation and interval analysis tools, such as bisection and relaxed intersection. The proposed method is validated using real data recorded during the CAROTTE challenge, organized by the French ANR (National Research Agency) and the French DGA (General Delegation of Armament). IAL is then compared with the well-known Monte Carlo Localization showing weaknesses and strengths of both algorithms. As it is shown in this paper with the IAL algorithm, interval analysis can be an efficient tool to solve the global localization problem.