In mathematics, an interval contractor associated to a set X is an operatorC which associates to a box in Rn another box C of Rn such that the two following properties are always satisfied
A contractor associated to a constraint is a contractor associated to the set X of all x which satisfy the constraint. Contractors make it possible to improve the efficiency of branch-and-bound algorithms classically used in interval analysis.
Properties of contractors
A contractor C is monotonic if we have It is minimal if for all boxes , we have where is the interval hull of the set A, i.e., the smallest box enclosing A. The contractor C is thin if for all points x, where denotes the degenerated box enclosing x as a single point. The contractor C is idempotent if for all boxes , we have The contractor C is convergent if for all sequences of boxes containing x, we have
Illustration
Figure 1 represents the set X painted grey and some boxes. Some of them of degenerated, i.e., they correspond to singletons. Figure 2 represents these boxes after contraction. Note that no point of X has been removed by the contractor. The contractor is minimal for the cyan box but is pessimistic for the green one. All degenerated blue boxes are contracted to the empty box. The magenta box and the red box cannot be contracted.
Some operations can be performed on contractors to build more complex contractors. The intersection, the union, the composition and the repetition are defined as follows.
Building contractors
There exist different ways to build contractors associated to equations and inequalities, say, f in . Most of them are based on interval arithmetic. One of the most efficient and most simple is the forward/backward contractor. The principle is to evaluate f using interval arithmetic. The resulting interval is intersected with . A backward evaluation of f is then performed in order to contract the intervals for the xi. We now illustrate the principle on a simple example. Consider the constraint We can evaluate the function f by introducing the two intermediate variables a and b, as follows The two previous constraints are called forward constraints. We get the backward constraints by taking each forward constraint in the reverse order and isolating each variable on the right hand side. We get The resulting forward/backward contractor is obtained by evaluating the forward and the backward constraints using interval analysis.
