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Efficient solving of quantified inequality constraints over the real numbers
 ACM Transactions on Computational Logic
"... Let a quantified inequality constraint over the reals be a formula in the firstorder predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In t ..."
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Cited by 29 (8 self)
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Let a quantified inequality constraint over the reals be a formula in the firstorder predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques. 1
Continuous FirstOrder Constraint Satisfaction
 ARTIFICIAL INTELLIGENCE, AUTOMATED REASONING, AND SYMBOLIC COMPUTATION, NUMBER 2385 IN LNCS
, 2002
"... This paper shows how to use constraint programming techniques for solving firstorder constraints over the reals (i.e., formulas in the firstorder predicate language over the structure of the real numbers). More specifically, based on a narrowing operator that implements an arbitrary notion of con ..."
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Cited by 27 (12 self)
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This paper shows how to use constraint programming techniques for solving firstorder constraints over the reals (i.e., formulas in the firstorder predicate language over the structure of the real numbers). More specifically, based on a narrowing operator that implements an arbitrary notion of consistency for atomic constraints over the reals (e.g., boxconsistency), the paper provides a narrowing operator for firstorder constraints that implements a corresponding notion of firstorder consistency, and a solver based on such a narrowing operator. As a consequence, this solver can take over various favorable properties from the field of constraint programming.
Quantified Constraints under Perturbation
 ARTICLE SUBMITTED TO JOURNAL OF SYMBOLIC COMPUTATION
"... Quantified constraints (i.e., firstorder formulae over the real numbers) are often exposed to perturbations: Constants that come from measurements usually are only known up to certain precision, and numerical methods only compute with approximations of real numbers. In this paper we study the be ..."
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Cited by 17 (12 self)
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Quantified constraints (i.e., firstorder formulae over the real numbers) are often exposed to perturbations: Constants that come from measurements usually are only known up to certain precision, and numerical methods only compute with approximations of real numbers. In this paper we study the behavior of quantified constraints under perturbation by showing that one can formulate the problem of solving quantified constraints as a nested parametric optimization problem followed by one sign computation. Using the fact that minima and maxima are stable under perturbation, but the sign of a real number is stable only for nonzero inputs, we derive practically useful conditions for the stability of quantified constraints under perturbation.
A Branch and Prune Algorithm for the Approximation of NonLinear AESolution Sets
 In SAC ’06: Proceedings of the 2006 ACM symposium on Applied computing
"... Nonlinear AEsolution sets are a special case of parametric systems of equations where universally quantified parameters appear first. They allow to model many practical situations. A new branch and prune algorithm dedicated to the approximation of nonlinear AEsolution sets is proposed. It is b ..."
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Cited by 12 (6 self)
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Nonlinear AEsolution sets are a special case of parametric systems of equations where universally quantified parameters appear first. They allow to model many practical situations. A new branch and prune algorithm dedicated to the approximation of nonlinear AEsolution sets is proposed. It is based on a new generalized interval (intervals whose bounds are not constrained to be ordered) parametric HansenSengupta operator. In spite of some restrictions on the form of the AEsolution set which can be approximated, it allows to solve problems which were before out of reach of previous numerical methods. Some promising experimentations are presented.
Search Heuristics for Box Decomposition Methods
 Journal of Global Optimization
, 2001
"... In this paper we study search heuristics for box decomposition methods that solve problems such as global optimization, minimax optimization, or quantified constraint solving. For this we unify these methods as nested branchandbound algorithms, and develop a model that is more convenient for study ..."
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Cited by 6 (5 self)
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In this paper we study search heuristics for box decomposition methods that solve problems such as global optimization, minimax optimization, or quantified constraint solving. For this we unify these methods as nested branchandbound algorithms, and develop a model that is more convenient for studying heuristics for these algorithms than the traditional models from Artificial Intelligence. We use the result to prove various theorems about heuristics and apply the outcome to the box decomposition methods under consideration. We support the findings with timings for the method of quantified constraint solving developed by the author.
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 2 (0 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.
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 2 (1 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.
Robust pole clustering of parametric uncertain systems using interval methods
"... In this paper a new methodology to solve the pole clustering problem for parametric uncertain systems is introduced: The problem of clustering the closed loop poles into prescribed Dregions in the complex plane is stated as a quantified constraint problem that represents bounded uncertain parame ..."
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Cited by 2 (0 self)
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In this paper a new methodology to solve the pole clustering problem for parametric uncertain systems is introduced: The problem of clustering the closed loop poles into prescribed Dregions in the complex plane is stated as a quantified constraint problem that represents bounded uncertain parameters by intervals; and an engineeringoriented approach based on interval methods is developed to solve this quantified constraint problem. The result is a new, robust, reliable and design oriented method to deal with parametric uncertain systems. The
An Efficient Algorithm for a Sharp Approximation of Universally Quantified Inequalities
 In Proceedings of ACM SAC 2008
, 2008
"... 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 her ..."
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Cited by 1 (0 self)
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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.