Interval contractor

such that the two following properties are always satisfied: A contractor associated to a constraint (such as an equation or an inequality) is a contractor associated to the set

Contractors make it possible to improve the efficiency of branch-and-bound algorithms classically used in interval analysis.

, where [A] is the interval hull of the set A, i.e., the smallest box enclosing A.

where {x} denotes the degenerated box enclosing x as a single point.

The contractor C is convergent if for all sequences [x](k) of boxes containing x, we have

Figure 1 represents the set X painted grey and some boxes, some of them 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.

[2] The intersection, the union, the composition and the repetition are defined as follows.

There exist different ways to build contractors associated to equations and inequalities, say, f(x) in [y].

Most of them are based on interval arithmetic.

One of the most efficient and most simple is the forward/backward contractor (also called as HC4-revise).

[3][4] The principle is to evaluate f(x) using interval arithmetic (this is the forward step).

A backward evaluation of f(x) is then performed in order to contract the intervals for the xi (this is the backward step).

We can evaluate the function f(x) 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.

is obtained by evaluating the forward and the backward constraints using interval analysis.