Results 1  10
of
30
Using Directed Acyclic Graphs to Coordinate Propagation and Search for Numerical Constraint Satisfaction Problems
 In Proceedings of the 16th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2004
, 2004
"... A. NEUMAIER [1] has given the fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation. We show in this paper how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
A. NEUMAIER [1] has given the fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation. We show in this paper how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) enabling the flexible choice of the interval inclusion functions during propagation. We then propose a new simple algorithm which coordinates constraint propagation and exhaustive search for solving numerical constraint satisfaction problems. The experiments carried out on different problems show that the new approach outperforms previously available propagation techniques by an order of magnitude or more in speed, while being roughly the same quality w.r.t. enclosure properties. I.
A Paradigm for the Robust Design of Algorithms for Geometric Modeling
 Computer Graphics Forum
, 1994
"... Geometric modelers are becoming faster and more powerful, but they still suffer from reliability problems because of floating point errors. Previous work in the field of robust geometric modeling tends to be problem specific and has proven hard to generalize. The approach described here is a general ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
Geometric modelers are becoming faster and more powerful, but they still suffer from reliability problems because of floating point errors. Previous work in the field of robust geometric modeling tends to be problem specific and has proven hard to generalize. The approach described here is a general paradigm for handling the accuracy problem for a large set of geometric algorithms. This approach brings together ideas and techniques from interval arithmetic, constraint management, randomization, and algebraic geometry. It acknowledges that input values have tolerances, that objects within tolerance are equivalent, and that certain geometric singularities must be maintained because they reflect design intent or the laws of geometry. Our approach is systematic, and can be applied almost mechanically to the large domain of problems, that can be solved by algorithms using the operations +, , * and /. The required theory and algorithms have been developed, and the viability of the concepts ...
A Heuristic Rejection Criterion in Interval Global Optimization Algorithms
, 1999
"... This paper investigates the properties of the inclusion functions on subintervals while a BranchandBound algorithm is solving global optimization problems. It has been found that the relative place of the global minimum value within the inclusion interval of the inclusion function of the objective ..."
Abstract

Cited by 11 (7 self)
 Add to MetaCart
This paper investigates the properties of the inclusion functions on subintervals while a BranchandBound algorithm is solving global optimization problems. It has been found that the relative place of the global minimum value within the inclusion interval of the inclusion function of the objective function at the actual interval mostly indicates whether the given interval is close to a minimizer point. This information is used in a heuristic interval rejection rule that can save a big amount of computation. Illustrative examples are discussed and a numerical study completes the investigation. AMS subject classication: 65K, 90C. Key words: Global optimization, BranchandBound Algorithm, Inclusion Function. 1
Global, rigorous and realistic bounds for the solution of dissipative differential equations. Part I: Theory
, 1993
"... . It is shown how interval analysis can be used to calculate rigorously valid enclosures of solutions to initial value problems for ordinary differential equations. In contrast to previously known methods, the enclosures obtained are valid over larger time intervals, and for uniformly dissipative sy ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
. It is shown how interval analysis can be used to calculate rigorously valid enclosures of solutions to initial value problems for ordinary differential equations. In contrast to previously known methods, the enclosures obtained are valid over larger time intervals, and for uniformly dissipative systems even globally. This paper discusses the underlying theory; main tools are logarithmic norms and differential inequalities. Numerical results will be given in a subsequent paper. Zusammenfassung. Es wird gezeigt, wie man mit Hilfe von IntervallAnalysis rigorose Einschließungen von Losungen von Anfangswertproblemen bei gewohnlichen Differentialgleichungen berechnen kann. Im Gegensatz zu anderen Methoden sind die Einschließungen uber großere Zeitintervalle, und fur gleichmaßig dissipative Systeme sogar global gultig. Diese Arbeit behandelt die zugrundeliegende Theorie; Hauptwerkzeuge sind logarithmische Normen und Differentialungleichungen. Numerische Ergebnisse werden in einer spateren A...
Taylor Forms  Use and Limits
 Reliable Computing
, 2002
"... This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 19 ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 1984 by Eckmann, Koch and Wittwer, and independently studied and popularized since 1996 by Berz, Makino and Hoefkens. A highlight is their application to the verified integration of asteroid dynamics in the solar system in 2001, although the details given are not sufficient to check the validity of their claims.
Slope Interval, Generalized Gradient, Semigradient, and Slant Derivative
 RELIABLE COMPUTING
, 2001
"... Many practical optimization problems are nonsmooth, and derivativetype methods cannot be applied. To overcome this difficulty, there are dierent approaches to replace the derivative of a function f : R n ! R: interval slopes, semigradients, generalized gradients, and slant derivatives are some ex ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
Many practical optimization problems are nonsmooth, and derivativetype methods cannot be applied. To overcome this difficulty, there are dierent approaches to replace the derivative of a function f : R n ! R: interval slopes, semigradients, generalized gradients, and slant derivatives are some examples. In this paper we study the relationships among these approaches for nonsmooth Lipschitz optimization problems infinite dimensional Euclidean spaces. Inclusion theorems concerning the equivalence of these concepts when there exist one sided derivatives in one dimension and in multidimensional cases are proved separately. Illustrative examples are shown in both cases.
Towards More Efficient Interval Analysis: Corner Forms and a Remainder Interval Newton Method
, 2005
"... c ○ 2005 ..."
Heuristic Rejection in Interval Global Optimization
"... Based on the investigation carried out in Ref. 4, this paper incorporates new studies about the properties of the inclusion functions on subintervals while a BranchandBound algorithm is solving global optimization problems. It had been found that the relative place of the global minimum value w ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Based on the investigation carried out in Ref. 4, this paper incorporates new studies about the properties of the inclusion functions on subintervals while a BranchandBound algorithm is solving global optimization problems. It had been found that the relative place of the global minimum value within the inclusion function value of the objective function at the actual interval indicates mostly whether the given interval is close to a minimizer point. This information is used in a heuristic interval rejection rule that can save a big amount of computation. Illustrative examples are discussed and an extended numerical study shows the advantages of the new approach.