When f is nonlinear the set inversion problem can be solved[1] using interval analysis combined with a branch-and-bound algorithm.
[2] The main idea consists in building a paving of Rp made with non-overlapping boxes.
The algorithm can be made more efficient by replacing the inclusion tests by contractors.
For instance, since [−2,1]2 + [4,5]2 = [0,4] + [16,25] = [16,29] does not intersect the interval [4,9], we conclude that the box [−2,1] × [4,5] is outside X.
Set inversion is mainly used for path planning, for nonlinear parameter set estimation,[3][4] for localization[5][6] or for the characterization of stability domains of linear dynamical systems.