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Numerical Constraint Satisfaction Problems with Nonisolated Solutions
 In: Global Optimization and Constraint Satisfaction. Volume LNCS 2861., SpringerVerlag
, 2003
"... Abstract. In recent years, interval constraintbased solvers have shown their ability to efficiently solve complex instances of nonlinear numerical CSPs. However, most of the working systems are designed to deliver pointwise solutions with an arbitrary accuracy. This works generally well for syste ..."
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Cited by 11 (7 self)
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Abstract. In recent years, interval constraintbased solvers have shown their ability to efficiently solve complex instances of nonlinear numerical CSPs. However, most of the working systems are designed to deliver pointwise solutions with an arbitrary accuracy. This works generally well for systems with isolated solutions but less well when there is a continuum of feasible points (e.g. underconstrained problems, problems with inequalities). In many practical applications, such large sets of solutions express equally relevant alternatives which need to be identified as completely as possible. In this paper, we address the issue of constructing concise inner and outer approximations of the complete solution set for nonlinear CSPs. We propose a technique which combines the extreme vertex representation of orthogonal polyhedra [1–3], as defined in computational geometry, with adapted splitting strategies [4] to construct the approximations as unions of interval boxes. This allows for compacting the explicit representation of the complete solution set and improves efficiency. 1
Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data
"... Constraint Programming (CP) has proved an e ective paradigm to model and solve di cult combinatorial satisfaction and optimisation problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertai ..."
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Cited by 4 (2 self)
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Constraint Programming (CP) has proved an e ective paradigm to model and solve di cult combinatorial satisfaction and optimisation problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertainty are less suited to uncertainty arising due to incomplete and erroneous data, because they do not build reliable models and solutions guaranteed to address the user's genuine problem as she perceives it. Other elds such as reliable computation o er combinations of models and associated methods to handle these types of uncertain data, but lack an expressive framework characterising the resolution methodology independently of the model. We present a unifying framework that extends the CP formalism in both model and solutions, to tackle illde ned combinatorial problems with incomplete or erroneous data. The certainty closure framework brings together modelling and solving methodologies from di erent elds into the CP paradigm to provide reliable and e cient approches for uncertain constraint problems. We demonstrate the applicability of the framework on a case study in network diagnosis. We de ne resolution forms that give generic templates, and their associated operational semantics, to derive practical solution methods for reliable solutions.
Efficient Handling of Universally Quantified Inequalities
, 2008
"... This paper introduces a new framework for solving quantified constraint satisfaction problems (QCSP) defined by universally quantified inequalities on continuous domains. This class of QCSPs has numerous applications in engineering and technology. We introduce a generic branch and prune algorithm to ..."
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Cited by 4 (1 self)
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This paper introduces a new framework for solving quantified constraint satisfaction problems (QCSP) defined by universally quantified inequalities on continuous domains. This class of QCSPs has numerous applications in engineering and technology. We introduce a generic branch and prune algorithm to tackle these continuous CSPs with parametric constraints, where the pruning and the solution identification processes are dedicated to universally quantified inequalities. Special rules are proposed to handle the parameter domains of the constraints. The originality of our framework lies in the fact that it solves the QCSP as a nonquantified CSP where the quantifiers are handled locally, at the level of each constraint. Experiments show that our algorithm outperforms the state of the art methods based on constraint techniques.
An efficient algorithm for a sharp approximation of universally quantified inequalities
 In Proceedings of ACM SAC
, 2008
"... ABSTRACT This paper introduces a new algorithm for solving a subclass of quantified constraint satisfaction problems (QCSP) where existential quantifiers precede universally quantified inequalities on continuous domains. This class of QCSPs has numerous applications in engineering and design. We pr ..."
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Cited by 1 (0 self)
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ABSTRACT This paper introduces a new algorithm for solving a subclass of quantified constraint satisfaction problems (QCSP) where existential quantifiers precede universally quantified inequalities on continuous domains. This class of QCSPs has numerous applications in engineering and design. We propose here a new generic branch and prune algorithm for solving such continuous QCSPs. Standard pruning operators and solution identification operators are specialized for universally quantified inequalities. Special rules are also proposed for handling the parameters of the constraints. First experimentation show that our algorithm outperforms the state of the art methods.
Clustering for Disconnected Solution Sets of Numerical CSPs
 Recent Advances in Constraints: International Workshop on Constraint Solving and Constraint Logic Programming, CSCLP 2003
"... Abstract. This paper considers the issue of preprocessing the output of intervalbased solvers for further exploitations when solving numerical CSPs with continuum of solutions. Most intervalbased solvers cover the solution sets of such problems with a large collection of boxes. This makes it diffi ..."
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Abstract. This paper considers the issue of preprocessing the output of intervalbased solvers for further exploitations when solving numerical CSPs with continuum of solutions. Most intervalbased solvers cover the solution sets of such problems with a large collection of boxes. This makes it difficult to exploit their results for other purposes than simple querying. For many practical problems, it is highly desirable to run more complex queries on the representations of the solution set. We propose to use clustering techniques to reduce the number of boxes produced by intervalbased solvers, while providing some main characteristics of the solution set. Four new algorithms based on clustering are proposed. 1
Clustering the Search Tree for Numerical Constraints
"... Abstract. During search, most intervalbased solvers perform splitting when no effective reduction can be made. Some connected regions may be undesirably split into small boxes. This makes the number of branches growing quickly, then impacts the overall performance. Regrouping connected boxes reduce ..."
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Abstract. During search, most intervalbased solvers perform splitting when no effective reduction can be made. Some connected regions may be undesirably split into small boxes. This makes the number of branches growing quickly, then impacts the overall performance. Regrouping connected boxes reduces the number of branches of the search tree, thus improves the overall performance. We propose to use clustering techniques based on the connectedness of boxes for the regrouping. This paper introduces new clustering algorithms, that fit in the context of search, and shows experimental results on numerical constraints. 1
Generalized Interval Projection: A New Technique fo r Consistent Domain Extension, in "IJCAI
, 2007
"... This paper deals with systems of parametric equations over the reals, in the framework of interval constraint programming. As parameters vary within intervals, the solution set of a problem may have a non null volume. In these cases, an inner box (i.e., a box included in the solution set) instead of ..."
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This paper deals with systems of parametric equations over the reals, in the framework of interval constraint programming. As parameters vary within intervals, the solution set of a problem may have a non null volume. In these cases, an inner box (i.e., a box included in the solution set) instead of a single punctual solution is of particular interest, because it gives greater freedom for choosing a solution. Our approach is able to build an inner box for the problem starting with a single point solution, by consistently extending the domain of every variable. The key point is a new method called generalized projection. The requirements are that each parameter must occur only once in the system, variable domains must be bounded, and each variable must occur only once in each constraint. Our extension is based on an extended algebraic structure of intervals called generalized intervals, where improper intervals are allowed (e.g. [1,0]). 1