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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This paper presents a new optimization metaheuristic called ID Walk (Intensification/Diversification Walk) that offers advantages for combining simplicity with effectiveness. In addition to the number S of moves, ID Walk uses only one parameter Max which is the maximum number of candidate neighbors studied in every move. This candidate list strategy manages the Max candidates so as to obtain a good tradeoff between intensification and diversification. A procedure has also been designed to tune the parameters automatically. We made experiments on several hard combinatorial optimization problems, and ID Walk compares favorably with correspondingly simple instances of leading metaheuristics, notably tabu search, simulated annealing and Metropolis. Thus, among algorithmic variants that are designed to be easy to program and implement, ID Walk has the potential to become an interesting alternative to such recognized approaches. Our automatic tuning tool has also allowed us to compare several variants of ID Walk and tabu search to analyze which devices (parameters) have the greatest impact on the computation time. A surprising result shows that the specific diversification mechanism embedded in ID Walk is very significant, which motivates examination of additional instances in this new class of "dynamic" candidate list strategies.

It is acknowledged that the symbolic form of the equations is crucial for interval-based solving techniques to efficiently handle systems of equations over the reals. However, only a few automatic transformations of the system have been proposed so far. Vu, Schichl, Sam-Haroud, Neumaier have exploited common subexpressions by transforming the equation system into a unique directed acyclic graph. They claim that the impact of common subexpressions elimination on the gain in CPU time would be only due to a reduction in the number of operations.This paper brings two main contributions. First, we prove theoretically and experimentally that, due to interval arithmetics, exploiting certain common subexpressions might also bring additional filtering/contraction during propagation. Second, based on a better exploitation of n-ary plus and times operators, we propose a new algorithm I-CSE that identifies and exploits all the "useful" common subexpressions. We show on a sample of benchmarks that I-CSE detects more useful common subexpressions than traditional approaches and leads generally to significant gains in performance, of sometimes several orders of magnitude.