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Complete search in continuous global optimization and constraint satisfaction, Acta Numerica 13
, 2004
"... A chapter for ..."
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Novel Approaches to Numerical Software with Result Verification
 NUMERICAL SOFTWARE WITH RESULT VERIFICATION, INTERNATIONAL DAGSTUHL SEMINAR, DAGSTUHL
, 2003
"... Traditional design of numerical software with result verification is based on the assumption that we know the algorithm ¦¨§� © ©���� £��������� � that transforms input © ©�� into �� � £��������� � ©���� the output, and we £��������� � know the intervals of possible values of the inputs. Many real ..."
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Cited by 26 (18 self)
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Traditional design of numerical software with result verification is based on the assumption that we know the algorithm ¦¨§� © ©���� £��������� � that transforms input © ©�� into �� � £��������� � ©���� the output, and we £��������� � know the intervals of possible values of the inputs. Many reallife problems go beyond this paradigm. In some cases, we do not have an algorithm ¦, we only know some relation (constraints) between ©� � and. In other cases, in addition to knowing the intervals, we may know some relations between; we may have some information about the probabilities of different values of © � , and we may know the exact values of some of the inputs (e.g., we may know that © £ ���¨�� �). In this paper, we describe the approaches for solving these reallife problems. In Section 2, we describe interval consistency techniques related to handling constraints; in Section 3, we describe techniques that take probabilistic information into consideration, and in Section 4, we overview techniques for processing exact real numbers.
Global optimization by continuous GRASP
 Optimization Letters
"... ABSTRACT. We introduce a novel global optimization method called Continuous GRASP (CGRASP) which extends Feo and Resende’s greedy randomized adaptive search procedure (GRASP) from the domain of discrete optimization to that of continuous global optimization. This stochastic local search method is s ..."
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Cited by 24 (9 self)
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ABSTRACT. We introduce a novel global optimization method called Continuous GRASP (CGRASP) which extends Feo and Resende’s greedy randomized adaptive search procedure (GRASP) from the domain of discrete optimization to that of continuous global optimization. This stochastic local search method is simple to implement, is widely applicable, and does not make use of derivative information, thus making it a wellsuited approach for solving global optimization problems. We illustrate the effectiveness of the procedure on a set of standard test problems as well as two hard global optimization problems. 1.
RealPaver: An Interval Solver using Constraint Satisfaction Techniques
 ACM TRANS. ON MATHEMATICAL SOFTWARE
, 2006
"... RealPaver is an interval software for modeling and solving nonlinear systems. Reliable approximations of continuous or discrete solution sets are computed, using Cartesian products of intervals. Systems are given by sets of equations or inequality constraints over integer and real variables. Moreove ..."
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Cited by 12 (1 self)
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RealPaver is an interval software for modeling and solving nonlinear systems. Reliable approximations of continuous or discrete solution sets are computed, using Cartesian products of intervals. Systems are given by sets of equations or inequality constraints over integer and real variables. Moreover, they may have different natures, being square or non square, sparse or dense, linear, polynomial or involving transcendental functions. The modeling language permits stating constraint models and tuning parameters of solving algorithms, which efficiently combine interval methods and constraint satisfaction techniques. Several consistency techniques (box, hull, 3B) are implemented. The distribution includes C sources, executables for different machine architectures, documentation and benchmarks. The portability is ensured by the GNU C compiler.
Hull consistency under monotonicity
 In Constraint Programming CP’09
"... Abstract. We prove that hull consistency for a system of equations or inequalities can be achieved in polynomial time providing that the underlying functions are monotone with respect to each variable. This result holds including when variables have multiple occurrences in the expressions of the fun ..."
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Cited by 8 (2 self)
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Abstract. We prove that hull consistency for a system of equations or inequalities can be achieved in polynomial time providing that the underlying functions are monotone with respect to each variable. This result holds including when variables have multiple occurrences in the expressions of the functions, which is usually a pitfall for intervalbased contractors. For a given constraint, an optimal contractor can thus be enforced quickly under monotonicity and the practical significance of this theoretical result is illustrated on a simple example. 1
ORIGINAL PAPER Global optimization by continuous grasp
, 2006
"... Abstract We introduce a novel global optimization method called Continuous GRASP (CGRASP) which extends Feo and Resende’s greedy randomized adaptive search procedure (GRASP) from the domain of discrete optimization to that of continuous global optimization. This stochastic local search method is si ..."
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Abstract We introduce a novel global optimization method called Continuous GRASP (CGRASP) which extends Feo and Resende’s greedy randomized adaptive search procedure (GRASP) from the domain of discrete optimization to that of continuous global optimization. This stochastic local search method is simple to implement, is widely applicable, and does not make use of derivative information, thus making it a wellsuited approach for solving global optimization problems. We illustrate the effectiveness of the procedure on a set of standard test problems as well as two hard global optimization problems.
Author manuscript, published in "CP'09 (15th International Conference on Principles and Practice of Constraint Programming), Lisbon: Portugal (2009)" A Constraint on the Number of Distinct Vectors with Application to Localization
, 2009
"... Abstract. This paper introduces a generalization of the nvalue constraint that bounds the number of distinct values taken by a set of variables.The generalized constraint (called nvector) bounds the number of distinct (multidimensional) vectors. The first contribution of this paper is to show that ..."
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Abstract. This paper introduces a generalization of the nvalue constraint that bounds the number of distinct values taken by a set of variables.The generalized constraint (called nvector) bounds the number of distinct (multidimensional) vectors. The first contribution of this paper is to show that this global constraint has a significant role to play with continuous domains, by taking the example of simultaneous localization and map building (SLAM). This type of problem arises in the context of mobile robotics. The second contribution is to prove that enforcing bound consistency on this constraint is NPcomplete. A simple contractor (or propagator) is proposed and applied on a real application. 1
Interval Extensions of Multivalued Inverse Functions The Implementation of Interval Relational Arithmetic in gaol
"... The implementation of inverse functions provided by most interval arithmetic software libraries is restricted to bijective functions and to the principal branch of multivalued functions. On the other hand, some algorithms—most notably, constraint propagation algorithms—require multivalued inverse fu ..."
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The implementation of inverse functions provided by most interval arithmetic software libraries is restricted to bijective functions and to the principal branch of multivalued functions. On the other hand, some algorithms—most notably, constraint propagation algorithms—require multivalued inverse functions as well. We present in details in this paper the algorithms to implement interval arithmetic extensions of the following multivalued inverse functions: the inverse integral power, the inverse cosine, the inverse sine, the inverse tangent, the inverse hyperbolic cosine, and the inverse multiplication. The issues raised by their effective as well as efficient implementation with floatingpoint numbers in the gaol C++ library are carefully addressed.
Interval Extensions of Multivalued Inverse Functions The Implementation of Interval Relational Arithmetic in gaol
"... The implementation of inverse functions provided by most interval arithmetic software libraries is restricted to bijective functions and to the principal branch of multivalued functions. On the other hand, some algorithms—most notably, constraint propagation algorithms—require multivalued inverse fu ..."
Abstract
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The implementation of inverse functions provided by most interval arithmetic software libraries is restricted to bijective functions and to the principal branch of multivalued functions. On the other hand, some algorithms—most notably, constraint propagation algorithms—require multivalued inverse functions as well. We present in details in this paper the algorithms to implement interval arithmetic extensions of the following multivalued inverse functions: the inverse integral power, the inverse cosine, the inverse sine, the inverse tangent, the inverse hyperbolic cosine, and the inverse multiplication. The issues raised by their effective as well as efficient implementation with floatingpoint numbers in the gaol C++ library are carefully addressed.