Set inversion

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) = {xRn | f(x) ∈ 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, for example, 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:

  1. if f([x]) ⊂ Y we conclude that [x] ⊂ X;
  2. if f([x]) ∩ Y = ∅ we conclude that [x] ∩ X = ∅;
  3. Otherwise, the box [x] the box is bisected except if its width is smaller than a given precision.

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) = x2
+ x2
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 + [2,5]2 = [0,1] + [4,5] = [4,6] is inside [4,9], we conclude that the whole box [-1,1] × [2,5] 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] .


  1. Jaulin, L.; Walter, E. (1993). "Set inversion via interval analysis for nonlinear bounded-error estimation" (PDF). Automatica. 29 (4): 1053–1064. doi:10.1016/0005-1098(93)90106-4.
  2. Jaulin, L.; Kieffer, M.; Didrit, O.; Walter, E. (2001). Applied Interval Analysis. Berlin: Springer. ISBN 1-85233-219-0.
  3. Jaulin, L.; Godet, J.L; Walter, E.; Elliasmine, A.; Leduff, Y. (1997). "Light scattering data analysis via set inversion" (PDF). Journal of Physics A: Mathematical and General. 30: 7733–7738. Bibcode:1997JPhA...30.7733J. doi:10.1088/0305-4470/30/22/012.
  4. Braems, I.; Berthier, F.; Jaulin, L.; Kieffer, M.; Walter, E. (2001). "Guaranteed Estimation of Electrochemical Parameters by Set Inversion Using Interval Analysis" (PDF). Journal of Electroanalytical Chemistry. 495 (1).
  5. Colle, E.; Galerne, S. (2013). "Mobile robot localization by multiangulation using set inversion". Robotics and Autonomous Systems. 66 (1). doi:10.1016/j.robot.2012.09.006.
  6. Drevelle, V.; Bonnifait, Ph. (2011). "A set-membership approach for high integrity height-aided satellite positioning". GPS Solutions. 15 (4).
  7. Walter, E.; Jaulin, L. (1994). "Guaranteed characterization of stability domains via set inversion" (PDF). IEEE Trans. Autom. Control. 39 (4).
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