In mathematics, set inversion is the problem of characterizing the preimage X of a set Y by a function f, i.e., X = f  −1(Y ) = . It can also be viewed as the problem of describing the solution set of the quantified constraint "Y(f (x))", where Y( y) is a constraint, e.g. an inequality, describing the set Y.
In most applications, f is a function from Rn to Rp and the set Y is a box of Rp (i.e. a Cartesian product of p intervals of R).
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. For each box [''x''], we perform the following tests:
To check the two first tests, we need an interval extension (or an inclusion function) [''f'' ] for f. Classified boxes are stored into subpavings, i.e., union of non-overlapping boxes. The algorithm can be made more efficient by replacing the inclusion tests by contractors.
The set X = f  −1([4,9]) where f (x1, x2) = x + x is represented on the figure.
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. Since [−1,1]2 + [2,{{sqrt|5}}]2 = [0,1] + [4,5] = [4,6] is inside [4,9], we conclude that the whole box [−1,1] × [2,{{sqrt|5}}] is inside 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.[7]