Significant progress has been accomplished during the past decades about geometric constraint solving, in particular thanks to its applications in industrial fields like CAD and robotics. In order to tackle problems of industrial size, many solving methods use, as a preprocessing, decomposition techniques that transform a large geometric constraint system into a set of smaller ones.In this paper, we propose a survey of the decomposition techniques for geometric constraint problems a . We classify them into four categories according to their modus operandi, establishing some similarities between methods that are traditionally separated. We summarize the advantages and limitations of the different approaches, and point out key issues for meeting industrial requirements such as generality and reliability.

Abstract:Interval Taylor has been proposed in the sixties by the interval analysis community for relaxing non-convex continuous 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 × n (square) systems of equations. This paper presents an interval Newton-like operator, called ❳✲◆❡✇t♦♥, 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 ❳✲◆❡✇t♦♥ operator can be rapidly encoded, and produces good speedups in constrained global optimization and non-convex constraint satisfaction. First experiments compare ❳✲◆❡✇t♦♥ with affine arithmetic.

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