Interval contractor

In mathematics, an interval contractor (or contractor for short)^[1] associated to a set ${\displaystyle X}$ is an operator ${\displaystyle C}$ which associates to a hyperrectangle ${\displaystyle [x]}$ in ${\displaystyle {\mathbf {R}}^{n}}$ another box ${\displaystyle C([x])}$ of ${\displaystyle {\mathbf {R}}^{n}}$ such that the two following properties are always satisfied:

• ${\displaystyle C([x])\subset [x]}$ (contractance property)

• ${\displaystyle C([x])\cap X=[x]\cap X}$ (completeness property)

A contractor associated to a constraint (such as an equation or an inequality) is a contractor associated to the set ${\displaystyle X}$ of all ${\displaystyle x}$ which satisfy the constraint. Contractors make it possible to improve the efficiency of branch-and-bound algorithms classically used in interval analysis.