## Section: New Results

### Interval analysis

#### A Contractor Based on Convex Interval Taylor

Participants : Gilles Trombettoni [correspondant] , Bertrand Neveu.

Interval Taylor has been proposed in the sixties by the interval
analysis community for relaxing continuous non-convex constraint
systems. However, it generally produces a non-convex relaxation of
the solution set. A simple way to build a convex polyhedral
relaxation is to select a *corner* of the studied domain/box as
expansion point of the interval Taylor form, instead of the usual
midpoint. The idea has been proposed by Neumaier to produce a sharp
range of a single function and by Lin and Stadtherr to handle $n\times n$ (square) systems of equations.

This paper presents an interval Newton-like operator, called `X-Newton` , that iteratively calls this interval convexification
based on an endpoint interval Taylor. This general-purpose
contractor uses no preconditioning and can handle any system of
equality and inequality constraints. It uses Hansen's variant to
compute the interval Taylor form and uses two opposite corners of
the domain for every constraint.

The `X-Newton` operator can be rapidly encoded, and produces
good speedups in constrained global optimization and constraint
satisfaction. First experiments compare `X-Newton` with affine
arithmetic[31] , [19] , [20]