A constraint hierarchies approach to geometric constraints on sketches

International audienceWe propose an approach that uses preferences on the constraints in order to deal with over-constrained geometric constraint problems. This approach employs constraint hierarchies, a paradigm that has close relations with the traditional graph-based approaches used in geometric constraint solving. We also remark that any geometric constraint problem defined by imposing relations on a sketch becomes overconstrained as soon as the sketch is imposed as a weak constraint representing the designers intents. As a result our method appears very appropriate in CAD/CAM tools

Splitting heuristics for disjunctive numerical constraints

International audienceRatschan has recently proposed a general framework for first-order formulas whose atoms are numerical constraints. It extends the notion of consistency to logical terms, but little is done with respect to the splitting operation. In this paper, we explore the potential of splitting heuristics that exploit the logical structure of disjunctive numerical constraint problems in order to simplify the problem along the search. First experiments on CNF formulas show that interesting solving time gains can be achieved by choosing the right splitting points

Algorithmes pour la détection de rigidités dans les CSP géométriques

National audienceLe théorème de Laman permet de caractériser la rigidité des syst emes a barres en 2D. La rigidité structurelle est basée sur une généralisation de ce théorème. Elle est généralement considérée comme une bonne heuristique pour identifier des sous-parties rigides dans les CSP géométriques (GCSP), mais peut en réalité se tromper sur des sous-syst emes très simples car elle ne tient pas compte des propriétés géométriques vérifiées par les objets. Hoffmann et al. ont proposé en 1997 des algorithmes a base de flots s'appuyant sur la caractérisation par rigidité structurelle pour répondre aux principales questions liées au concept de rigidité : déterminer si un GCSP est rigide, identifier ses composantes rigide, sur-rigide et sous-rigide, en minimiser la taille, etc. La rigidité structurellé etendue, une nouvelle caractérisation de la rigidité, a été proposée par Jermann et al. en 2002. Elle permet de prendre en compte les pro-priétés géométriques du GCSP etudié et s'av ere ainsi plus fiable. Dans le présent article, nous présentons des algorithmes qui répondent aux principales questions liées a la notion de rigidité en utilisant cette nouvelle caractérisation. Plus précisément, nous montrons que deux modifications de la fonction de distribution de flot utilisée dans les algorithmes de Hoffmann et al. permettent l'obtention d'une famille d'algorithmes basés sur la rigidité structurellé etendue. Nous démontrons la correction et la complétude des nouveaux algorithmes et etudions leur complexité en pire cas

Constraint Based Computation of Periodic Orbits of Chaotic Dynamical Systems

International audienceThe chaos theory emerged at the end of the 19th century, and it has given birth to a deep mathematical theory in the 20th century, with a strong practical impact (e.g., weather forecast, turbulence analysis). Periodic orbits play a key role in understanding chaotic systems. Their rigorous computation provides some insights on the chaotic behavior of the system and it enables computer assisted proofs of chaos related properties (e.g., topological entropy). In this paper, we show that the (numerical) constraint programming framework provides a very convenient and efficient method for computing periodic orbits of chaotic dynamical systems: Indeed, the flexibility of CP modeling allows considering various models as well as including additional constraints (e.g., symmetry breaking constraints). Furthermore, the richness of the different solving techniques (tunable local propagators, search strategies, etc.) leads to highly efficient computations. These strengths of the CP framework are illustrated by experimental results on classical chaotic systems from the literature

Search Strategies for an Anytime Usage of the Branch and Prune Algorithm

International audienceWhen applied to numerical CSPs, the branch and prune algorithm (BPA) computes a sharp covering of the solution set. The BPA is therefore impractical when the solution set is large, typically when it has a dimension larger than four or five which is often met in underconstrained problems. The purpose of this paper is to present a new search tree exploration strategy for BPA that hybridizes depth-first and breadth-first searches. This search strategy allows the BPA discovering potential solutions in different areas of the search space in early stages of the exploration, hence allowing an anytime usage of the BPA. The merits of the proposed search strategy are experimentally evaluated

Intelligent Splitting for Disjunctive Numerical CSPs

International audienceDisjunctions in numerical CSPs appear in applications such as Design, Biology or Control. Generalized solving techniques have been proposed to handle these disjunctions in a Branch&Prune fashion. However, they focus essentially on the pruning operation. In this paper, we present experimental evidences that significant performance gains can be expected by exploiting the disjunctions in the branching operation

Interval-Based Projection Method for Under-Constrained Numerical Systems

International audienceThis paper presents an interval-based method that follows the branch-and-prune scheme to compute a verified paving of a projection of the solution set of an under-constrained system. Benefits of this algorithm include anytime solving process, homogeneous verification of inner boxes, and applicability to generic problems, allowing any number of (possibly nonlinear) equality and inequality constraints. We present three key improvements of the algorithm dedicated to projection problems: (i) The verification process is enhanced in order to prove faster larger boxes in the projection space. (ii) Computational effort is saved by pruning redundant portions of the solution set that would project identically. (iii) A dedicated branching strategy allows reducing the number of treated boxes. Experimental results indicate that various applications can be modeled as projection problems and can be solved efficiently by the proposed method

On Continuation Methods for Non-Linear Bi-Objective Optimization: Certified Interval-Based Approach

The global optimization of constrained Non-Linear Bi-Objective Optimization problems (MO) aims at covering their Pareto-optimal front which is in general a manifold in R^2. Continuation methods can help in this context as they can follow a continuous component of this front once an initial point on it is provided. They constitute somehow a generalization of the classical scalarizing framework which transforms the bi-objective problem into a parametric mono-objective problem. Recent works have shown that they can play a key role in global algorithms dedicated to bi-objective problems, e.g. population based algorithms, where they allow discovering large portions of locally Pareto optimal vectors, which turns out to strongly support diversification. In this paper, we provide a survey on continuation techniques in global optimization methods for MO, which allow discovering large portions of locally Pareto-optimal solutions. We also propose a rigorous active set management strategy on top of a previously proposed certified continuation method based on interval analysis, and illustrate it on a challenging bi-objective problem

A Branch and Prune Algorithm for the Computation of Generalized Aspects of Parallel Robots

International audienceParallel robots enjoy enhanced mechanical characteristics that have to be contrasted with a more complicated design. In particular, they often have parallel singularities at some poses, and the robots may become uncontrollable, and could even be damaged, in such configurations. The computation of the connected components in the set of nonsingular reachable configurations, called generalized aspects, is therefore a key issue in their design. This paper introduces a new method, based on numerical constraint programming, to compute a certified enclosure of the generalized aspects. Though this method does not allow counting their number rigorously, it constructs inner approximations of the nonsingular workspace that allow commanding parallel robots safely. It also provides a lower-bound on the exact number of generalized aspects. It is moreover the first general method able to handle any parallel robot in theory, though its computational complexity currently restricts its usage to robots with three degrees of freedom. Finally, the contraint programming paradigm it relies on makes it possible to consider various additional constraints (e.g., collision avoidance), making it suitable for practical considerations

Franco-Japanese Research Collaboration on Constraint Programming

International audienceConstraint programming is an emergent technology that allows modeling and solving various problems in many areas such as artificial intelligence, computer programming, computer-aided design, computer graphics, and user interfaces. In this report, we provide recent activities of research collaboration on constraint programming conducted by the authors and other researchers in France and Japan. First, we outline our joint research projects on constraint programming, and then present the backgrounds, goals, and approaches of several research topics treated in the projects. Second, we describe the two Franco-Japanese Workshops on Constraint Programming (FJCP), which we organized in Japan in October 2004 and in France in November 2005. We conclude with future prospects for collaboration between French and Japanese researchers in this area