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A Comparison of Complete Global Optimization Solvers
"... Results are reported of testing a number of existing state of the art solvers for global constrained optimization and constraint satisfaction on a set of over 1000 test problems in up to 1000 variables. ..."
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Cited by 23 (4 self)
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Results are reported of testing a number of existing state of the art solvers for global constrained optimization and constraint satisfaction on a set of over 1000 test problems in up to 1000 variables.
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 16 (11 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.
Approximate quantified constraint solving by cylindrical box decomposition
 RESEARCH INSTITUTE FOR SYMBOLIC COMPUTATION (RISC
, 2000
"... This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a firstorder formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols , are in general solvable. However, the problem ..."
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Cited by 14 (7 self)
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This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a firstorder formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols , are in general solvable. However, the problem becomes undecidable, when we add function symbols like sin. Furthermore, all exact algorithms known up to now are too slow for big examples, do not provide partial information before computing the total result, cannot satisfactorily deal with interval constants in the input, and often generate huge output. As a remedy we propose an approximation method based on interval arithmetic. It uses a generalization of the notion of cylindrical decomposition  as introduced by G. Collins. We describe an implementation of the method and demonstrate that, for quantied constraints without equalities, it can efficiently give approximate information on problems that are too hard for current exact methods.
Search Techniques for NonLinear Constraint Satisfaction Problems With Inequalities
 In Canadian Conference on AI
, 2001
"... . In recent years, interval constraintbased solvers have shown their ability to efficiently solve challenging nonlinear real constraint problems. However, most of the working systems limit themselves to delivering pointwise solutions with an arbitrary accuracy. This works well for equalities, or ..."
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Cited by 5 (0 self)
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. In recent years, interval constraintbased solvers have shown their ability to efficiently solve challenging nonlinear real constraint problems. However, most of the working systems limit themselves to delivering pointwise solutions with an arbitrary accuracy. This works well for equalities, or for inequalities stated for specifying tolerances, but less well when the inequalities express a set of equally relevant choices, as for example the possible moving areas for a mobile robot. In that case it is desirable to cover the large number of pointwise alternatives expressed by the constraints using a reduced number of sets, as interval boxes. Several authors [2, 1, 7] have proposed set covering algorithms specific to inequality systems. In this paper we propose a lookahead backtracking algorithm for inequality and mixed equality/inequality constraints. The proposed technique combines a set covering strategy for inequalities with classical interval search techniques for equalities. This allows for a more compact representation of the solution set and improves efficiency. 1