# Relaxed intersection

The relaxed intersection of m sets corresponds to the classical intersection between sets except that it is allowed to relax few sets in order to avoid an empty intersection. This notion can be used to solve Constraints Satisfaction Problems that are inconsistent by relaxing a small number of constraints. When a bounded-error approach is considered for parameter estimation, the relaxed intersection makes it possible to be robust with respect to some outliers.

## Definition

The q-relaxed intersection of the m subsets ${\displaystyle X_{1},\dots ,X_{m}}$ of ${\displaystyle R^{n}}$, denoted by ${\displaystyle X^{\{q\}}=\bigcap ^{\{q\}}X_{i}}$ is the set of all ${\displaystyle x\in R^{n}}$ which belong to all ${\displaystyle X_{i}}$ 's, except ${\displaystyle q}$ at most. This definition is illustrated by Figure 1.

Figure 1. q-intersection of 6 sets for q=2 (red), q=3 (green), q= 4 (blue) , q= 5 (yellow).

Characterizing the q-relaxed intersection is a thus a set inversion problem. [1]

We have

## Relaxed intersection of intervals

The relaxed intersection of intervals is not necessary an interval. We thus take the interval hull of the result. If ${\displaystyle X_{i}}$'s are intervals, the relaxed intersection can be computed with a complexity of m.log(m) by using the Marzullo's algorithm. It suffices to sort all lower and upper bounds of the m intervals to represent the function ${\displaystyle \lambda }$. Then, we easily get the set

which corresponds to a union of intervals. We then return the smallest interval which contains this union.

Figure 2 shows the function ${\displaystyle \lambda (x)}$ associated to the previous example.

Figure 2. Set-membership function associated to the 6 intervals.

## Relaxed intersection of boxes

To compute the q-relaxed intersection of m boxes of ${\displaystyle R^{n}}$, we project all m boxes with respect to the n axes. For each of the n groups of m intervals, we compute the q-relaxed intersection. We return Cartesian product of the n resulting intervals. [2] Figure 3 provides an illustration of the 4-relaxed intersection of 6 boxes. Each point of the red box belongs to 4 of the 6 boxes.

Figure 3. The red box corresponds to the 4-relaxed intersection of the 6 boxes

## Relaxed union

The q-relaxed union of ${\displaystyle X_{1},\dots ,X_{m}}$ is defined by

Note that when q=0, the relaxed union/intersection corresponds to the classical union/intersection. More precisely, we have

and

## De Morgan's law

If ${\displaystyle {\overline {X}}}$ denotes the complementary set of ${\displaystyle X_{i}}$, we have

As a consequence

## Relaxation of contractors

is a contractor for ${\displaystyle X^{\{q\}}}$ and

is a contractor for ${\displaystyle {\overline {X}}^{\{q\}}}$, where

are contractors for

Combined with a branch-and-bound algorithm such as SIVIA (Set Inversion Via Interval Analysis), the q-relaxed intersection of m subsets of ${\displaystyle R^{n}}$ can be computed.

## Application to bounded-error estimation

The q-relaxed intersection can be used for robust localization ,[3] for robust localization [4] or for tracking [5]

Robust observers can also be implemented using the relaxed intersections to be robust with respect to outliers. [6]

We propose here a simple example [7] to illustrate the method. Consider a model the ith model output of which is given by

where ${\displaystyle p\in R^{2}}$. Assume that we have

where ${\displaystyle t_{i}}$ and ${\displaystyle [y_{i}]}$ are given by the following list

The sets ${\displaystyle \lambda ^{-1}(q)}$ for different ${\displaystyle q}$ are depicted on Figure 4.

Figure 4. Set of all parameter vectors consistent with exactly 6-q data bars (painted red), for q=1,2,3,4,5.

## References

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